SEAWAY Theory Manual

Theoretical Manual ofStrip Theory Program

“SEAWAY for Windows”

J.M.J. Journée and L.J.M. Adegeest

Report 1370 September 2003

TU DELFT Ship Hydromechanics Laboratory

Delft University of Technology

AMARCON Advanced Maritime Consulting

www.amarcon.com

Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003

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Summary

This report aims to be a guide and help for those people who want to study the theoreticalbackgrounds and the algorithms of a ship motions computer code based on the strip theory.The underlying report describes in detail the theoretical backgrounds and algorithms used bythe first author during the development of his six-degrees-of-freedom ship motions computercode, called SEAWAY.

The six ship motions of and about the centre of gravity G of the vessel have been defined inthe next figure.

Definition of ship motions

According to Newton’s second law, the equations of motion for six degrees of freedom of anoscillating ship in waves in a earth-bounded axes system have to be written as follows:

ixMj

iij direction in momentsor forces all of sum 6

1

=⋅∑=

&& for: 6,...1=i

Because a linear system has been considered here, the forces and moments in the right handside of these equations consist of a superposition of:• so-called hydromechanic forces and moments, caused by a harmonic oscillation of the

rigid body in the undisturbed surface of a fluid being previously at rest, and• so-called exciting wave forces and moments on the restrained body, caused by the

incoming harmonic waves.With this, the system of a with six degrees of freedom moving ship in waves can considered tobe a linear mass-damper-spring system with frequency-dependent coefficients and linearexciting forces and moments:

( ) ij

iijiijiijij FxcxbxaM =⋅+⋅+⋅+∑=

6

1

&&& for: 6,...1=i


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In here, ix with indices 3,2,1=i are the displacements of G (surge, sway and heave) and ixwith indices 6,5,4=i are the rotations about the axes through G (roll, pitch and yaw). Theindices ij present at motion i the coupling with motion j .

The masses in the equations of motion above consist of solid masses or solid mass momentsof inertia of the ship ( ijM ) and “added” masses or “added” mass moments of inertia caused

by the disturbed water, the hydrodynamic masses or mass moments of inertia ( ija ). Anoscillating ship generates waves it self too; energy will be radiated from the ship. Thehydrodynamic damping-terms ( iij xb &⋅ ) account for this. For the heave, roll and pitch motions,

hydrostatic spring-terms ( iij xc ⋅ ) have to be added. The right hand sides of the equations of

motion consist of exciting wave forces and moments ( iF ).

In the so-called strip theory, the ship will be divided in 20 tot 30 cross sections, of which thetwo-dimensional hydromechanic coefficients and exciting wave loads will be calculated. Toobtain the three-dimensional values, these values will be integrated over the ship lengthnumerically. Finally, the differential equations will be solved to obtain the motions. Thesecalculations will be performed in the frequency domain.

It was in 1949 that Ursell published his potential theory for determining the hydrodynamiccoefficients of semicircular cross sections, oscillating in deep water in the frequency domain.Using this, for the first time a rough estimation could be made of the motions of a ship inregular waves at zero forward speed.Shortly after that Tasai, Grim, Gerritsma and many other scientists used various alreadyexisting conformal mapping techniques (to transform ship-like cross sections to a semicircle)together with Ursell’s theory, in such a way that the motions in regular waves of more realistichull forms could be calculated too. Most popular was the 2-parameter Lewis conformalmapping technique. The exciting wave loads were found from the loads in undisturbed waves– the so-called Froude-Krilov forces or moments – completed with diffraction termsaccounting for the presence of the ship in these waves.Borrowed from the broadcasting technology, Denis en Pierson published in 1953 asuperposition method to describe the irregular waves too. The sea was considered to be thesum of many simple harmonic waves; each wave with its own frequency, amplitude, directionand random phase lag. By calculating the responses of the ship on each of these individualharmonic waves and adding up the responses of the ship, the energy distribution of the ship’sbehaviour in irregular waves could be found. These irregular motions are characterised bysignificant amplitude and average period.However, these theories provided the motions at zero forward speed only. In 1957, Korvin-Kroukovski en Jacobs published a method - which was further improved in the sixties - toaccount for the effect of forward ship speed.So at the end of the fifties, all components for an elementary ship motions computer programfor deep water were already available.

Fukuda published in 1962 a calculation technique for the internal sheer forces and bendingmoments in a cross section of a ship.Frank published in 1967 his pulsating source theory to calculate the hydrodynamiccoefficients of a cross section of a ship in deep water directly, without using conformalmapping. The potential coefficients of a fully submerged cross section (bulbous bow) and


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sections with a very low area coefficient (often present in the aft body) could be calculatednow too.Using Lewis conformal mapping, Keil published in 1974 his theory for obtaining the potentialcoefficients in very shallow water.Useful theories to calculate the added resistance of a ship due to waves were given by Boese(integrated pressure method) in 1970 and Gerritsma and Beukelman (radiated energy method)in 1972.So far, all hydrodynamic coefficients had been determined with the potential theory. However,roll requires a viscous correction on that. Ikeda, Himeno and Tanaka published in 1978 a veryuseful semi-empirical method for determining the viscous roll damping components.

The introduction of personal computers in the early eighties increased the accessibility forcarrying out ship motion calculations considerably; even non-specialists could become userstoo. From then on the computer capacity and the computing speed increased very fast, so thatthree-dimensional theories could be developed much easier and cheaper now.Because of the complex problem of forward speed in 3-D theories however, the 2-D approach(strip theory) is still very favourable for calculating the behaviour of a ship at forward speed.The many advantages and just a few disadvantages, when comparing 2-D with 3-D, had beendiscussed very clearly by Faltinsen and Svensen in 1990.

Among others as a consequence of the work of the researchers mentioned above, a DOSpersonal computer strip theory program - called SEAWAY - had been completed by the DelftUniversity of Technology at the end of the eighties. Recently, a Windows version has beencompleted too, see web site www.shipmotions.nl or www.amarcon.com.Based on the linear strip theory, this program calculates for 6 degrees of freedom in thefrequency domain the hydromechanic loads, wave loads, absolute and relative motions, addedresistance and internal loads of displacement ships, barges and yachts in regular and irregularwaves. When ignoring interaction effects between the two individual hulls, the behaviour ofcatamarans and semi-submersibles can be calculated too. The program is suitable for deepwater as well as for very shallow water. Viscous roll damping, bilge keels, free-surface anti-roll tanks, external moments and (linear) mooring springs can be added.The computer code has been verified and validated extensively by the authors, many studentsand a large number of industrial users.

Error messages, advises and all type of comments on this technical report are very welcomeby e-mail to [email protected] or [email protected].


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Table of Contents:

1 Introduction........................................................................................................................111.1 About the Authors..........................................................................................................111.2 About this Manual........................................................................................................ 12

2 Strip Theory Methods ....................................................................................................... 152.1 Definitions .................................................................................................................... 172.2 Incident Wave Potential................................................................................................ 20

2.2.1 Continuity Condition............................................................................................ 212.2.2 Laplace Equation.................................................................................................. 212.2.3 Seabed Boundary Condition................................................................................. 222.2.4 Free Surface Dynamic Boundary Condition......................................................... 222.2.5 Free Surface Kinematic Boundary Condition....................................................... 242.2.6 Dispersion Relationship ........................................................................................ 252.2.7 Relationships in Regular Waves........................................................................... 26

2.3 Floating Rigid Body in Waves...................................................................................... 282.3.1 Fluid Requirements............................................................................................... 282.3.2 Forces and Moments............................................................................................. 302.3.3 Hydrodynamic Loads............................................................................................ 312.3.4 Wave and Diffraction Loads ................................................................................. 362.3.5 Hydrostatic Loads................................................................................................. 38

2.4 Equations of Motion..................................................................................................... 392.5 Strip Theory Approaches.............................................................................................. 43

2.5.1 Zero Forward Ship Speed ..................................................................................... 432.5.2 Forward Ship Speed.............................................................................................. 442.5.3 End-Terms............................................................................................................. 46

2.6 Hydrodynamic Coefficients.......................................................................................... 483 2-D Potential Coefficients ................................................................................................ 51

3.1 Conformal Mapping Methods....................................................................................... 533.1.1 Lewis Conformal Mapping................................................................................... 543.1.2 Extended Lewis Conformal Mapping................................................................... 583.1.3 Close-Fit Conformal Mapping.............................................................................. 593.1.4 Mapping Comparisons .......................................................................................... 63

3.2 Potential Theory of Tasai.............................................................................................. 653.2.1 Heave Motions ...................................................................................................... 663.2.2 Sway Motions ....................................................................................................... 763.2.3 Roll Motions ......................................................................................................... 883.2.4 Low and High Frequencies................................................................................. 100

3.3 Potentia l Theory of Keil.............................................................................................. 1033.3.1 Notations of Keil................................................................................................. 1033.3.2 Basic Assumptions.............................................................................................. 1043.3.3 Vertical Motions.................................................................................................. 1063.3.4 Horizontal Motions ............................................................................................. 1333.3.5 Appendices ......................................................................................................... 144

3.4 Potential Theory of Frank ........................................................................................... 1533.4.1 Notations of Frank .............................................................................................. 1533.4.2 Formulation of the Problem................................................................................ 1553.4.3 Solution of the Problem...................................................................................... 1573.4.4 Low and High Frequencies................................................................................. 1613.4.5 Irregular Frequencies .......................................................................................... 162


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3.4.6 Appendices ......................................................................................................... 1653.5 Comparisons between Calculated Potential Data ....................................................... 1733.6 Estimated Potential Surge Coefficients ...................................................................... 175

4 Viscous Damping............................................................................................................ 1774.1 Surge Damping ........................................................................................................... 178

4.1.1 Total Surge Damping.......................................................................................... 1784.1.2 Viscous Surge Damping...................................................................................... 179

4.2 Roll Damping.............................................................................................................. 1804.2.1 Experimental Determination............................................................................... 1814.2.2 Empirical Formula for Barges ............................................................................ 1834.2.3 Empirical Method of Miller................................................................................ 1834.2.4 Semi-Empirical Method of Ikeda ....................................................................... 184

5 Hydromechanical Loads ................................................................................................. 1975.1 Hydromechanical Forces for Surge ............................................................................ 1985.2 Hydromechanical Forces for Sway............................................................................. 2015.3 Hydromechanical Forces for Heave ........................................................................... 2045.4 Hydromechanical Moments for Roll .......................................................................... 2075.5 Hydromechanical Moments for Pitch......................................................................... 2105.6 Hydromechanical Moments for Yaw.......................................................................... 213

6 Exciting Wave Loads ...................................................................................................... 2176.1 Wave Potential............................................................................................................ 2176.2 Classical Approach..................................................................................................... 219

6.2.1 Exciting Wave Forces for Surge ......................................................................... 2196.2.2 Exciting Wave Forces for Sway.......................................................................... 2216.2.3 Exciting Wave Forces for Heave ........................................................................ 2236.2.4 Exciting Wave Moments for Roll ....................................................................... 2256.2.5 Exciting Wave Moments for Pitch...................................................................... 2276.2.6 Exciting Wave Moments for Yaw....................................................................... 228

6.3 Approximating 2-D Diffraction Approach................................................................. 2296.3.1 Hydromechanical Loads ..................................................................................... 2296.3.2 Energy Considerations ........................................................................................ 2316.3.3 Wave Loads......................................................................................................... 232

6.4 Numerical Comparisons ............................................................................................. 2387 Transfer Functions of Motions ....................................................................................... 239

7.1 Centre of Gravity Motions .......................................................................................... 2407.2 Local Absolute Displacements ................................................................................... 2437.3 Local Absolute Velocities ........................................................................................... 2447.4 Local Absolute Accelerations ..................................................................................... 245

7.4.1 Accelerations in the Earth-Bound Axes System................................................. 2457.4.2 Accelerations in the Ship-Bound Axes System.................................................. 245

7.5 Local Vertical Relative Displacements....................................................................... 2477.6 Local Vertical Relative Velocities............................................................................... 248

8 Anti-Rolling Devices ...................................................................................................... 2498.1 Bilge Keels.................................................................................................................. 2508.2 Passive Free-Surface Tanks........................................................................................ 251

8.2.1 Theoretical Approach......................................................................................... 2518.2.2 Experimental Approach...................................................................................... 2558.2.3 Effect of Free-Surface Tanks .............................................................................. 257

8.3 Active Fin Stabilisers.................................................................................................. 2588.4 Active Rudder Stabilisers ........................................................................................... 261


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9 External Linear Springs .................................................................................................. 2639.1 External Loads ............................................................................................................ 2639.2 Additional Coefficients............................................................................................... 2649.3 Linearised Mooring Coefficients................................................................................ 266

10 Added Resistance due to Waves ..................................................................................... 26710.1 Radiated Energy Method ........................................................................................ 26910.2 Integrated Pressure Method .................................................................................... 27110.3 Comparison of Results............................................................................................ 273

11 Bending and Torsion Moments....................................................................................... 27511.1 Still Water Loads .................................................................................................... 28111.2 Dynamical Lateral Loads........................................................................................ 28211.3 Dynamical Vertical Loads....................................................................................... 28411.4 Dynamical Torsion Loads....................................................................................... 287

12 Statistics in Irregular Waves........................................................................................... 28912.1 Normalised Wave Energy Spectra .......................................................................... 290

12.1.1 Neumann Wave Spectrum................................................................................... 29012.1.2 Bretschneider Wave Spectrum............................................................................ 29012.1.3 Mean JONSWAP Wave Spectrum...................................................................... 29112.1.4 Definition of Parameters..................................................................................... 291

12.2 Response Spectra and Statistics.............................................................................. 29512.3 Shipping Green Water............................................................................................. 30012.4 Bow Slamming........................................................................................................ 302

12.4.1 Criterium of Ochi................................................................................................ 30212.4.2 Criterium of Conolly........................................................................................... 303

13 Twin-Hull Ships.............................................................................................................. 30713.1 Hydromechanical Coefficients ............................................................................... 30713.2 Equations of Motion............................................................................................... 30813.3 Hydromechanical Forces and Moments ................................................................. 30913.4 Exciting Wave Forces and Moments...................................................................... 31013.5 Added Resistance due to Waves ............................................................................. 314

13.5.1 Radiated Energy Method .................................................................................... 31413.5.2 Integrated Pressure Method ................................................................................ 314

13.6 Bending and Torsion Moments............................................................................... 31514 Numerical Recipes.......................................................................................................... 317

14.1 Polynomials ............................................................................................................ 31714.1.1 First Degree Polynomial..................................................................................... 31714.1.2 Second Degree Polynomial................................................................................. 318

14.2 Integrations ............................................................................................................. 31914.2.1 First Degree Integration...................................................................................... 31914.2.2 Second Degree Integration................................................................................. 31914.2.3 Integration of Wave Loads.................................................................................. 320

14.3 Derivatives.............................................................................................................. 32314.3.1 First Degree Derivative....................................................................................... 32314.3.2 Second Degree Derivative .................................................................................. 323

14.4 Curve Lengths......................................................................................................... 32614.4.1 First Degree Curve.............................................................................................. 32614.4.2 Second Degree Curve ......................................................................................... 326

15 References....................................................................................................................... 329


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1 Introduction

SEAWAY is a frequency-domain ship motions PC program, based on the linear strip theoryfor calculating the hydromechanic loads, wave-induced loads, motions, added resistance andinternal loads for six degrees of freedom of displacement ships and yachts, barges, semi-submersibles or catamarans, sailing in regular and irregular waves. The program is suitable fordeep water as well as for shallow water. Viscous roll damping, bilge keels, anti-roll tanks, freesurface effects and (linear springs) can be added.This computer code has been developed in the late eighties and early nineties under DOS bythe first author. His last “SEAWAY for DOS” version was released in 2002.In 2003, the second author took over the software implementation and distribution part of thejob and developed the new “SEAWAY for Windows” release.Information can be found at web site www.shipmotions.nl or www.amarcon.com.

1.1 About the Authors

Johan Journée had obtained his Polytechnical Degree in 1964 at the Avond-HTS Rotterdamand in 1975 his MSc degree at the Delft University of Technology. Both degrees in NavalArchitecture were obtained, alongside a full-time job, by studying in the evening hours.He started his working career in 1958 at the Rotterdam Dockyard Company, first withconstruction work in the shipbuilding factory and two years later with technical ship designwork in the drawing office of this yard. In 1963, he became Technical Officer at the ShipHydromechanics Laboratory of the Delft University of Technology. After obtaining his MScdegree in 1975, he became Scientific Officer there, some years later Assistant Professor and in1992 Associate Professor.During the years 1985 through 1990, Johan had developed - as a more or less derailed hobby -this 2-D ship motions computer code SEAWAY. This development was a very useful exercisefor him to understand in a very detailed way the theory and the practice of the behaviour of aship in waves, see Journée [1992]. Parts of this study were basis for comprehensive lecturenotes on Offshore Hydromechanics, see Journée and Massie [2001].Since 1984, Johan Journée is teaching Ship and Offshore Hydromechanics to students of theMechanical and Civil Engineering Departments and since 1990 also to students of theMaritime Technology Department.

Leon Adegeest was one of the founders of AMARCON in January 2001. AMARCON’s majoractivities are developing software for decision support onboard using seakeeping theory andrelated consultancy and engineering work.Before AMARCON, Leon has worked at Det Norske Veritas (DNV) in Norway and atMARIN in the Netherlands. At DNV (1997 – 2001) he developed methods for the predictionof extreme non-linear ship responses in irregular seas. As Group Leader Hydrodynamics, hewas heavily involved in the development and implementation of DNV's non-linear 3-Dseakeeping software package WASIM. Practical experience in the use of seakeeping codeswas gained during commercial projects, which include non-linear wave load analyses for largecontainer carriers, fatigue analyses for a RoRo-carrier and water on deck evaluations forproduction vessels. From 1994 to 1997, he worked at the Maritime Research InstituteNetherlands (MARIN) as project manager in the Trials and Monitoring group.Leon Adegeest holds a MSc degree in Naval Architecture from the Delft University ofTechnology. In 1994, he finished his PhD study at this university. The title of the thesis was“Non-linear Hull Girder Loads in Ships”, see Adegeest [1994].


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1.2 About this Manual

This manual aim at being a guide and aid for those who want to study the theoreticalbackgrounds and the algorithms of a ship motions computer code, like SEAWAY, based on thestrip theory. The theoretical backgrounds and the algorithms of this program have beendescribed here in detail.

Chapter 1, this introduction, gives a short survey of the contents of all chapters in this report.

Chapter 2 gives a general description of the various strip theory approaches. A generaldescription of the potential flow theory is given. The derivations of the hydromechanic forcesand moments, the wave potential and the wave and diffraction forces and moments have beendescribed.The equations of motion are given with solid mass and inertia terms and hydromechanicforces and moments in the left hand side and the wave exciting forces and moments in theright hand side. The principal assumptions are a linear relation between forces and motionsand the validity of obtaining the total forces by a simple integration over the ship length of thetwo-dimensional cross sectional forces.This includes for all motions a forward speed effect caused by the potential mass, as has beendefined by Korvin-Kroukovsky and Jacobs [1957] for the heave and pitch motions. Thisapproach is called the ''Ordinary Strip Theory Method''. Also an inclusion of the forwardspeed effect caused by the potential damping, as for instance given by Tasai [1969]. Thisapproach is called the ''Modified Strip Theory Method''.The inclusion of so-called ''End-Terms'' has been described too.

Chapter 3 describes the determination of the two-dimensional potential mass and dampingcoefficients for the six modes of motions at infinite and finite water depths.Firstly, it describes several conformal mapping methods. For the determination of the two-dimensional hydrodynamic potential coefficients for sway, heave and roll motions of ship-likecross sections, these sections are conformal mapped to the unit circle. The advantage ofconformal mapping is that the velocity potential of the fluid around an arbitrary shape of across section in a complex plane can be derived from the more convenient circular section inanother complex plane. In this manner hydrodynamic problems can be solved directly withthe coefficients of the mapping function.The close-fit multi-parameter conformal mapping method is given. A very simple and straighton iterative least squares method, used to determine the conformal mapping coefficients, hasbeen described. Two special cases of multi-parameter conformal mapping have beendescribed too: the well known classic transformation of Lewis [1929] with two parametersand an Extended-Lewis transformation with three parameters, as given by Athanassoulis andLoukakis [1985].Then, it describes 3 methods for the determination of the two-dimensional potential mass anddamping coefficients for the six modes of motions at infinite and finite water depths. Atinfinite water depths, the principle of the calculation of these potential coefficients is based onwork of Ursell [1949] for circular cylinders and Frank [1967] for any arbitrary symmetriccross section.Starting from the velocity potentials and the conjugate stream functions of the fluid with aninfinite depth as have been given by Tasai [1959], Tasai [1960], Tasai [1961] and de Jong[1973] and using the multi-parameter conformal mapping technique, the calculation routines


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of the two-dimensional hydrodynamic potential coefficients of ship-like cross sections aregiven for the sway, heave and roll motions.For any arbitrary water depth (deep to very shallow water), the method of Keil [1974] - basedon a variation of the theory of Ursell [1949] with Lewis conformal mapping - has been given.Finally, the pulsating source method of Frank [1967] for deep water has been described.Because of using the strip theory approach here, the pitch and yaw coefficients follow fromthe moments about the ship's centre of gravity of the heave and sway coefficients,respectively.Approximations are given for the surge coefficients.

Chapter 4 gives some corrections on the hydrodynamic damping due to viscous effects. Thesurge-damping coefficient is corrected for viscous effects by an empirical method, based on asimple still water resistance curve as published by Troost [1955].The analysis of free-rolling model experiments and two (semi) empirical methods publishedby Miller [1974] and Ikeda [1978], to determine a viscous correction of the roll-dampingcoefficients, are described in detail.

Chapter 5 describes the determination of the hydromechanic forces and moments in the left-hand side of the six equations of motion of a sailing ship, obtained with the hydromechaniccoefficients as determined in Chapter 3 and 4, for both the ordinary and the modified striptheory method.

Chapter 6 describes the wave exciting forces and moments in the right hand side of the sixequations of motion of a sailing ship in water with an arbitrarily depth, using the relativemotion concept for both the ordinary and the modified strip theory method.First, the classical approach has been described, using equivalent accelerations and velocitiesof the water particles. Then, an alternative approach, based on diffraction of waves, has beendescribed.

Chapter 7 describes the solution of the equations of motion and the determination of thefrequency characteristics of the absolute displacements, rotations, velocities and accelerationsand the vertical relative displacements. The use of a wave potential valid for any arbitrarywater depth makes a calculation method with deep water coefficients, suitable for shipssailing with keel clearances down to about 50 percent of the ship's draft. At lower waterdepths, Keil’s method should be used.

Chapter 8 describes some anti-rolling devices. A description is given of an inclusion ofpassive free-surface tanks as defined by the experiments of van den Bosch and Vugts [1966]and by the theory of Verhagen and van Wijngaarden [1965]. Active fin and rudder stabilisershave been described too.

Chapter 9 describes the inclusion of linear spring terms, to simulate the behavior of anchoredor moored ships.

Chapter 10 describes two methods to determine the transfer functions of the added resistancedue to waves. The first method is a radiated wave energy method, as published by Gerritsmaand Beukelman [1972]. The second method is an integrated pressure method, as published byBoese [1970].


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Chapter 11 describes the determination of the frequency characteristics of the lateral andvertical shear forces and bending moments and the torsion moments in a way as had beenpresented by Fukuda [1962] for the vertical mode. Still water phenomena are described too.

Chapter 12 describes the statistics in irregular waves, by using the superposition principle.Three examples of normalized wave spectra are given: the somewhat wide wave spectrum ofNeumann, an average wave spectrum of Bretschneider and the more narrow Mean JONSWAPwave spectrum.A description is given of the calculation procedure of the energy spectra and the statistics ofthe ship motions for six degrees of freedom, the added resistance, the vertical relative motionsand the mechanic loads on the ship in waves coming from any direction.For the calculation of the probability of exceeding a threshold value by the motions, theRayleigh probability density function has been used.The static and dynamic swell-up of the waves, of importance when calculating the probabilityof shipping green water, are defined according to Tasaki [1963]. A theoretically determineddynamic swell-up had been given too.Bow slamming phenomena are defined by both the relative bow velocity criterion of Ochi[1964] and by the peak bottom-impact-pressure criteria of Conolly [1974].

Chapter 13 describes the additions to all algorithms in case of twin- hull ships, such as semi-submersibles and catamarans. However, for interaction effects between the two individualhulls will not be accounted here.

Chapter 14 shows some typical numerical recipes, as has been used in program SEAWAY.

Finally, Chapter 15 gives a survey of all literature used during the development of thiscomputer code.

Error messages, advises and all type of comments on this technical report are very welcomeby e-mail: [email protected].


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2 Strip Theory Methods

The ship is considered to be a rigid body, floating in the surface of an ideal fluid, which ishomogeneous, incompressible, free of surface tension, irrotational and without viscosity. It isassumed that the problem of the motions of this floating body in waves is linear or can belinearised. Consequently, only the external loads on the underwater part of the ship areconsidered here and the effect of the above water part will be fully neglected.

Faltinsen and Svensen [1990] have discussed the incorporation of seakeeping theories in shipdesign clearly. An overview of seakeeping theories for ships were presented and it wasconcluded that - nevertheless some limitations - strip theories are the most successful andpractical tools for the calculation of the wave induced motions of the ship, at least in an earlydesign stage of a ship.The strip theory solves the three-dimensional problem of the hydromechanical and excitingwave forces and moments on the ship by integrating the two-dimensional potential solutionsover the ship's length. Interactions between the cross sections are ignored for the zero-speedcase. So, each cross section of the ship is considered to be part of an infinitely long cylinder.

The strip theory is a slender body theory, so one should expect less accurate predictions forships with low length to breadth ratios. However, experiments showed that the strip theoryappears to be remarkably effective for predicting the motions of ships with length to breadthratios down to about 3.0, or even sometimes lower.The strip theory is based on the potential flow theory. This holds that viscous effects areneglected, which can deliver serious problems when predicting roll motions at resonancefrequencies. In practice, for viscous roll damping effects can be accounted fairly by empiricalmethods.Because of the way that the forced motion problems are solved, generally in the strip theory,substantial disagreements can be found between the calculated results and the experimentaldata of the wave loads at low frequencies of encounter in following waves. In practicehowever, these ''near zero frequency of encounter problems'' can be solved by forcing thewave loads going to zero, artificially.

For high-speed vessels and for large ship motions, as appear in extreme sea states, the striptheory can deliver less accurate results. Then the so-called ''end-terms'' can become veryimportant.The strip theory accounts for the interaction with the forward speed in a very simple way. Theeffect of the steady wave system around the ship is neglected and the free surface conditionsare simplified, so that the unsteady waves generated by the ship are propagating in directionsperpendicular to the centre plane of the ship. In reality the wave systems around the ship arefar more complex. For high-speed vessels, unsteady divergent wave systems becomeimportant. This effect is neglected in the strip theory.The strip theory is based on linearity. This means that the ship motions are supposed to besmall, relative to the cross sectional dimensions of the ship. Only hydrodynamic effects of thehull below the still water level are accounted for. So, when parts of the ship go out of or in tothe water or when green water is shipped, inaccuracies can be expected. Also, the strip theorydoes not distinguish between alternative above water hull forms.Because of the added resistance of a ship due to the waves is proportional to the relativemotions squared, its inaccuracy will be gained strongly by inaccuracies in the predictedmotions.


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Nevertheless these limitations, seakeeping prediction methods based upon the strip theoryprovide a sufficiently good basis for optimisation studies at an early design stage of the ship.At a more detailed design stage, it can be considered to carry out additional modelexperiments to investigate for instance added resistance or extreme event phenomena, such asshipping green water and slamming.


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2.1 Definitions

Figure 2.1–1 shows a harmonic wave as seen from two different perspectives. Figure 2.1–1-ashows what one would observe in a snapshot photo made looking at the side of a (transparent)wave flume; the wave profile is shown as a function of distance x along the flume at a fixedinstant in time. Figure 2.1–1-b shows a time record of the water level observed at one locationalong the flume; it looks similar in many ways to the other figure, but time t has replaced xon the horizontal axis.

Figure 2.1–1: Harmonic wave definitions

Notice that the origin of the co-ordinate system is at the still water level with the positive z -axis directed upwards; most relevant values of z will be negative.The still water level is the average water level or the level of the water if no waves werepresent. The x -axis is positive in the direction of wave propagation. The water depth, h , (apositive value) is measured between the seabed ( hz −= ) and the still water level ( 0=z ).The highest point of the wave is called its crest and the lowest point on its surface is thetrough. If the wave is described by a harmonic wave, then its amplitude aζ is the distancefrom the still water level to the crest, or to the trough for that matter. The subscript a denotesthe amplitude, here.The horizontal distance (measured in the direction of wave propagation) between any twosuccessive wave crests is the wavelength, λ . The distance along the time axis is the waveperiod, T . The ratio of wave height to wavelength is often referred to as the dimensionlesswave steepness: λζ /2 a⋅ .Since the distance between any two corresponding points on successive harmonic waves is thesame, wave lengths and periods are usually actually measured between two consecutiveupward (or downward) crossings of the still water level. Such points are also called zero-crossings, and are easier to detect in a wave record.Since sine or cosine waves are expressed in terms of angular arguments, the wavelength andwave period are converted to angles using:

πωπλ

⋅=⋅⋅=⋅

2

2

T

k or

T

k

πω

λπ

⋅=

⋅=

2

2

Equation 2.1–1

in which k is the wave number (rad/m) and ω is the circular wave frequency (rad/s).


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Obviously, the wave form moves one wave length during one period, so that its speed orphase velocity, c , is given by:

kTc

ωλ==

Equation 2.1–2

Suppose now a sailing ship in waves, with co-ordinate systems as given in Figure 2.1–2.

Figure 2.1–2: Co-ordinate systems

A right-handed co-ordinate system ( )000 ,, zyxS is fixed in space. The ( )00 , yx -plane lies in the

still water surface, 0x is directed as the wave propagation and 0z is directed upwards.

Another right-handed co-ordinate system ( )zyxO ,, is moving forward with a constant shipspeed V . The directions of the axes are: x in the direction of the forward ship speed V , y inthe lateral port side direction and z vertically upwards. The ship is supposed to carry outoscillations around this moving ( )zyxO ,, co-ordinate system. The origin O lies verticallyabove or under the time-averaged position of the centre of gravity G . The ( )yx, -plane lies inthe still water surface.

A third right-handed co-ordinate system ( )bbb zyxG ,, is connected to the ship with its origin at

G , the ship's centre of gravity. The directions of the axes are: bx in the longitudinal forward

direction, by in the lateral port side direction and bz upwards. In still water, the ( )bb yx , -planeis parallel to the still water surface.

If the wave moves in the positive 0x -direction (defined in a direction with an angle µ relativeto the ship's speed vector, V ), the wave profile - the form of the water surface - can now beexpressed as a function of both 0x and t as follows:

( )txka ⋅−⋅⋅= ωζζ 0cos or ( )0cos xkta ⋅−⋅⋅= ωζζ

Equation 2.1–3


19

The right-handed co-ordinate system ( )zyxO ,, is moving with the ship's speed V , whichyields:

µµµ sincoscos0 ⋅+⋅+⋅⋅= yxtVx

Equation 2.1–4

From the relation between the frequency of encounter eω and the wave frequency ω:

µωω cos⋅⋅−= Vke

Equation 2.1–5

follows:

( )µµωζζ sincoscos ⋅⋅−⋅⋅−⋅⋅= ykxktea

Equation 2.1–6

The resulting six ship motions in the ( )zyxO ,, system are defined by three translations of theship's centre of gravity in the direction of the x -, y - and z -axes and three rotations aboutthem:

( )( )( )( )( )( )ψζ

θζ

φζ

ζ

ζ

ζ

εωψψ

εωθθ

εωφφ

εω

εω

εω

+⋅⋅=

+⋅⋅=

+⋅⋅=

+⋅⋅=

+⋅⋅=

+⋅⋅=

t

t

t

tz

ty

txx

ea

ea

ea

zea

yea

xea

cos :yaw

cos :pitch

cos :roll

cosz :heave

cosy :sway

cos :surge

Equation 2.1–7

The phase shifts of these motions are related to the harmonic wave elevation at the origin ofthe ( )zyxO ,, system, i.e. the average position of the ship's centre of gravity:

( )tea ⋅⋅= ωζζ cos :wave

Equation 2.1–8

The harmonic velocities and accelerations in the ( )zyxO ,, system are found now by takingthe derivatives of the displacements, for instance for surge:

( )( )

( )ζ

ζ

ζ

εωω

εωω

εω

xeae

xeae

xea

txx

txx

txx

+⋅⋅⋅−=

+⋅⋅⋅−=

+⋅⋅=

cos :onaccelerati surge

sin : velocitysurge

cos :ntdisplaceme surge

2&&

&

Equation 2.1–9


20

2.2 Incident Wave Potential

In order to use the linear potential theory for waves, it will be necessary to assume that thewater surface slope is very small. This means that the wave steepness is so small that terms inthe equations of motion of the waves with a magnitude in the order of the steepness-squaredcan be ignored.Suppose a wave moving in the ( )zx, -plane. The profile of that simple wave with a smallsteepness looks like a sine or a cosine and the motion of a water particle in a wave depends onthe distance below the still water level. This is reason why the wave potential can be writtenas:

( ) ( ) ( )txkzPtzxw ⋅−⋅⋅=Φ ωsin,,

Equation 2.2–1

in which ( )zP is an (as yet) unknown function of z .

This velocity potential ( )tzxw ,,Φ of the harmonic waves has to fulfil four requirements:• Continuity condition or Laplace equation• Seabed boundary condition• Free surface dynamic boundary condition• Free surface kinematic boundary conditionThese requirements lead to a more complete expression for the velocity potential as will beexplained in the following subsections.

The relationships presented in these subsections are valid for all water depths, but the fact thatthey contain so many hyperbolic functions makes them cumbersome to use. Engineers - asopposed to (some) scientists - often look for ways to simplify the theory. The simplificationsstem from the following approximations for large and very small arguments, s , as shown inFigure 2.2–1:

For large arguments s : [ ] [ ][ ] 1 tan

coshsinh

≈>>≈

sh

sss

For small arguments s : [ ] [ ][ ] 1cos

tanhsinh

≈≈≈

sh

sss

Equation 2.2–2

Figure 2.2–1: Hyperbolic function limits


21

2.2.1 Continuity Condition

The velocity of the water particles ( )wvu ,, in the three translation directions, or alternatively

( )zyx vvv ,, , follow from the definition of the velocity potential, wΦ :

xvu w

x ∂Φ∂

== y

vv wy ∂

Φ∂==

zvw w

z ∂Φ∂

==

Equation 2.2–3

Since the fluid is homogeneous and incompressible, the continuity condition becomes:

0=∂∂

+∂∂

+∂∂

zw

yv

xu

Equation 2.2–4

2.2.2 Laplace Equation

The continuity condition in Equation 2.2–4 results in the Laplace equation for potential flows:

02

2

2

2

2

22 =

∂Φ∂

+∂Φ∂

+∂Φ∂

=Φ∇zyx

wwww

Equation 2.2–5

Water particles move here in the ( )zx, -plane only, so in the equations above:

0=∂Φ∂

=y

v w and 02

2

=∂Φ∂

=∂∂

yyv w

Equation 2.2–6

Taking this into account, a substitution of Equation 2.2–1 in Equation 2.2–5 yields ahomogeneous solution of this equation:

( ) ( ) 022

2

=⋅− zPkdz

zPd

Equation 2.2–7

with as a homogeneous solution for ( )zP :

( ) zkzk eCeCzP ⋅−⋅+ ⋅+⋅= 21

Equation 2.2–8

Using this result from the continuity condition and the Laplace equation, the wave potentialcan be written now with two unknown coefficients as:

( ) ( ) ( )txkeCeCtzx zkzkw ⋅−⋅⋅⋅+⋅=Φ ⋅−⋅+ ωsin,, 21

Equation 2.2–9

in which:( )tzxw ,,Φ wave potential (m2/s)


22

e base of natural logarithms (-)

21 ,CC as yet undetermined constants (m2/s)k wave number (1/m)t time (s)x horizontal distance (m)z vertical distance, positive upwards (m)ω wave frequency (1/s)

2.2.3 Seabed Boundary Condition

The vertical velocity of water particles at the seabed is zero (no-leak condition):

0=∂Φ∂z

w for: hz −=

Equation 2.2–10

Substituting this boundary condition in Equation 2.2–9 provides:021 =⋅⋅−⋅⋅ ⋅+⋅− hkhk eCkeCk

Equation 2.2–11

By defining:hkhk eCeCC ⋅+⋅− ⋅⋅=⋅⋅= 21 22

or:hke

CC ⋅+⋅=

21 and hkeC

C ⋅−⋅=22

it follows that ( )zP in Equation 2.2–8 can be worked out to:

( ) ( ) ( )( )( )[ ]zhkC

eeC

zP zhkzhk

+⋅⋅=

+⋅= +⋅−+⋅+

cosh2

Equation 2.2–12

and the wave potential Equation 2.2–1 becomes:( ) ( )[ ] ( )txkzhkCtzxw ⋅−⋅⋅+⋅⋅=Φ ωsincosh,,

Equation 2.2–13

in which C is an (as yet) unknown constant.

2.2.4 Free Surface Dynamic Boundary Condition

The pressure, p , at the free surface of the fluid, ζ=z , is equal to the atmospheric pressure,

0p . This requirement for the pressure is called the dynamic boundary condition at the freesurface.The Bernoulli equation for an instationary irrotational flow (with the velocity given in termsof its three components) is in its general form:


23

( ) 021 222 =⋅++++⋅+

∂Φ∂

zgp

wvut

w

ρ

Equation 2.2–14

In two dimensions, 0=v , and since the waves have a small steepness (u and w are small),this equation becomes in a linearised format:

0=⋅++∂Φ∂

zgp

tw

ρ

Equation 2.2–15

At the free surface this condition becomes:

0=⋅++∂Φ∂ ζ

ρg

pt

w for: ζ=z

Equation 2.2–16

The constant value ρ0p can be included in tw ∂Φ∂ ; this will not influence the velocities

being obtained from the potential wΦ .With this the equation becomes:

0=⋅+∂Φ∂

ζgt

w for: ζ=z

Equation 2.2–17

The potential at the free surface can be expanded in a Taylor series, keeping in mind that thevertical displacement of the wave surface ζ is relatively small:

( ) ( ) ( ).........

,,,,,,

00 +

∂Φ∂

⋅+Φ=Φ=

==z

wzwzw z

tzxtzxtzx ζζ

or:( ) ( ) ( )2

0

,,,, εζ

Ot

tzxt

tzx

z

w

z

w +

∂Φ∂

=

∂Φ∂

==

Equation 2.2–18

which yields for the linearised form of the free surface dynamic boundary condition inEquation 2.2–17:

0=⋅+∂Φ∂

ζgt

w for: 0=z

Equation 2.2–19

With this, the wave surface profile becomes:

tgw

∂Φ∂

⋅−=1

ζ for: 0=z

Equation 2.2–20

A substitution of Equation 2.2–13 in Equation 2.2–20 yields the wave surface profile:

[ ] ( )txkhkgC

⋅−⋅⋅⋅⋅⋅

= ωωζ coscosh

or:


24

( )txka ⋅−⋅⋅= ωζζ cos with: [ ]hkgC

a ⋅⋅⋅

= coshωζ

Equation 2.2–21

With this, depending on the water depth h , the wave potential in Equation 2.2–13 willbecome:

( )[ ][ ] ( )txk

hkzhkga

w ⋅−⋅⋅⋅+⋅

⋅⋅

=Φ ωω

ζsin

coshcosh

Equation 2.2–22

or when ω is the first of the sine function arguments, as generally will be used in ship motionequations:

( )[ ][ ] ( )xkt

hkzhkga

w ⋅−⋅⋅⋅+⋅

⋅⋅−

=Φ ωωζ

sincosh

cosh

Equation 2.2–23

In deep water, the expression for the wave potential reduces to:

( )xkteg zka

w ⋅−⋅⋅⋅⋅−

=Φ ⋅ ωω

ζsin (deep water)

Equation 2.2–24

2.2.5 Free Surface Kinematic Boundary Condition

So far the relation between the wave period T and the wavelength, λ , is still unknown. Thisrelation between T and λ (or equivalently ω and k ) follows from the boundary conditionthat the vertical velocity of a water particle in the free surface of the fluid is identical to thevertical velocity of that free surface itself (no-leak condition); this is a kinematic boundarycondition.Using Equation 2.2–21 of the free surface yields:

xu

t

dtdx

xtdtdz

∂∂

⋅+∂∂

=

⋅∂∂

+∂∂

=

ζζ

ζζ

for the wave surface: ζ=z

The second term in this expression is a product of two values, which are both small because ofthe assumed small wave steepness. This product becomes even smaller (second order) and canbe ignored, see Figure 2.2–2.


25

Figure 2.2–2: Kinematic boundary condition

This linearisation provides the vertical velocity of the wave surface:

tdtdz

∂∂

=ζ

for the wave surface: ζ=z

Equation 2.2–25

The vertical velocity of a water particle in the free surface is then:

tzw

∂∂

=∂Φ∂ ζ

for: ζ=z

The vertical velocity of a water particle in the free surface is then:Analogous to Equation 2.2–19 this condition is valid for 0=z too, instead of for ζ=z only:

tzw

∂∂

=∂Φ∂ ζ

for: 0=z

Equation 2.2–26

A differentiation of the free surface dynamic boundary condition (Equation 2.2–19) withrespect to t provides:

02

2

=∂∂

⋅+∂Φ∂

tg

tw ζ

for: 0=z

or after re-arranging terms:

01

2

2

=∂Φ∂

⋅+∂∂

tgtwζ

for: 0=z

Equation 2.2–27

Together with Equation 2.2–25 this delivers the free surface kinematic boundary condition orthe so-called Cauchy-Poisson condition:

01

2

2

=∂Φ∂

⋅+tgdt

dz w for: 0=z

Equation 2.2–28

2.2.6 Dispersion Relationship

The information is now available to establish the relationship between ω and k (orequivalently T and λ ), referred to above. A substitution of the expression for the wave


26

potential (Equation 2.2–22) in Equation 2.2–28 gives the dispersion relation for any arbitrarywater depth h :

[ ]hkgk ⋅⋅⋅= tanh2ω

Equation 2.2–29

In many situations, ω or T will be known; one must determine k or λ . This equation willgenerally has to be solved iteratively, since k appears in a nonlinear way in Equation 2.2–29.

In deep water ( [ ] 1tanh =⋅hk ), Equation 2.2–29 degenerates to a quite simple form which canbe used without difficulty:

gk ⋅=2ω (deep water)

Equation 2.2–30

When calculating the hydromechanical forces and the wave exciting forces on a ship, it isassumed that bxx ≈ , byy ≈ and bzz ≈ . In case of forward ship speed, the wave frequency

ω has to be replaced by the frequency of encounter of the waves eω . This leads to the

following expressions for the wave surface in the ( )bbb zyxG ,, system:

( )µµωζζ sincoscos ⋅⋅−⋅⋅−⋅⋅= bbea ykxkt

Equation 2.2–31

and the expression for the velocity potential of the regular waves, wΦ , becomes:( )[ ][ ] ( )µµω

ωζ

sincossincosh

cosh⋅⋅−⋅⋅−⋅⋅

⋅+⋅

⋅⋅−

=Φ bbeba

w ykxkthk

zhkg

Equation 2.2–32

2.2.7 Relationships in Regular Waves

Figure 2.2–3 shows the relation between λ , T , c and h for a wide variety of conditions.Notice the boundaries 2≈hλ and 20≈hλ in this figure between short (deep water) andlong (shallow water) waves.


27

Figure 2.2–3: Relationships between λ , T , c and h


28

2.3 Floating Rigid Body in Waves

Consider a rigid body, floating in an ideal fluid with harmonic waves. The water depth isassumed to be finite. The time-averaged speed of the body is zero in all directions. For thesake of simple notation, it is assumed here that the ( )zyxO ,, system is identical to the

( )000 ,, zyxS system. The x -axis is coincident with the undisturbed still water free surface

and the z -axis and 0z -axis are positive upwards.The linear fluid velocity potential can be split into three parts:

( ) dwrtzyx Φ+Φ+Φ=Φ ,,,

Equation 2.3–1

in which:rΦ radiation potential for the oscillatory motion of the body in still water

wΦ incident undisturbed wave potential

dΦ diffraction potential of the waves about the restrained body

2.3.1 Fluid Requirements

From the definition of a velocity potential Φ follows the velocity of the water particles in thethree translation directions:

xvx ∂

Φ∂=

yv y ∂

Φ∂=

zvz ∂

Φ∂=

Equation 2.3–2

The velocity potentials, dwr Φ+Φ+Φ=Φ , have to fulfil a number of requirements andboundary conditions in the fluid. Of these, the first three are identical to those in the incidentundisturbed waves. Additional boundary conditions are associated with the oscillating floatingbody.

1. Continuity Condition or Laplace EquationAs the fluid is homogeneous and incompressible, the continuity condition:

0=∂∂

+∂

∂+

∂∂

zv

y

v

x

v zyx

Equation 2.3–3

results into the equation of Laplace:

02

2

2

2

2

22 =

∂Φ∂

+∂

Φ∂+

∂Φ∂

=Φ∇zyx

Equation 2.3–4

2. Seabed Boundary ConditionThe boundary condition on the seabed (no-leak condition), following from the definition ofthe velocity potential, is given by:


29

0=∂Φ∂z

for: hz −=

Equation 2.3–5

3. Dynamic Boundary Condition at the Free SurfaceThe pressure in a point ( )zyxP ,, is given by the linearised Bernoulli equation:

zgt

p ⋅⋅−∂Φ∂

⋅−= ρρ or ρp

zgt

−=⋅+

∂Φ∂

Equation 2.3–6

At the free surface of the fluid, so for ( )tzyxz ,,,ζ= , the pressure p is constant.Because of the linearisation, the vertical velocity of a water particle in the free surfacebecomes:

tzdtdz

∂∂

≈∂Φ∂

=ζ

Equation 2.3–7

Combining these two conditions provides the boundary condition at the free surface:

02

2

=∂Φ∂

⋅+∂

Φ∂z

gt

for: 0=z

Equation 2.3–8

4. Kinematic Boundary Condition on the Oscillating Body SurfaceIt is obvious that the boundary condition at the surface of the rigid body plays a veryimportant role. The velocity of a water particle at a point at the surface of the body is equal tothe velocity of this (watertight) body point itself. The outward normal velocity, nv , at a point

( )zyxP ,, at the surface of the body (positive in the direction of the fluid) is given by:

( )tzyxvn n ,,,=

∂Φ∂

Equation 2.3–9

Because the solution is linearised, this can be written as:

( ) ∑=

⋅==∂Φ∂ 6

1

,,,j

jjn fvtzyxvn

Equation 2.3–10

in terms of oscillatory velocities, jv , and generalised direction-cosines, jf , on the surface of

the body, S , given by:


30

( )( )( )

( ) ( )( ) ( )( ) ( ) 126

315

234

3

2

1

,cos,cos

,cos,cos

,cos,cos

,cos

,cos

,cos

fyfxxnyynxf

fxfzznxxnzf

fzfyynzznyf

znf

ynf

xnf

⋅−⋅=⋅−⋅=⋅−⋅=⋅−⋅=

⋅−⋅=⋅−⋅===

=

Equation 2.3–11

The direction cosines are called generalised, because 1f , 2f and 3f have been normalised

(the sum of their squares is equal to 1) and used to obtain 4f , 5f and 6f .Note: The subscripts 1,2,...6 are used here to indicate the mode of the motion. Alsodisplacements are often indicated in literature in the same way: 1x , 2x ,... 6x , as used here inthe summary.

5. Radiation ConditionThe radiation condition states that when the distance R of a water particle to the oscillatingbody tends to infinity, the potential value tends to zero:

0lim =Φ∞→R

Equation 2.3–12

6. Symmetric or Anti-symmetric ConditionSince ships and many floating bodies are symmetric with respect to its middle line plane, onecan make use of this to simplify the potential equations:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )yxyx

yxyx

yxyx

,, :rollfor

,, :heavefor

,, :swayfor

44

33

22

+Φ−=−Φ

+Φ+=−Φ

+Φ−=−Φ

Equation 2.3–13

in which ( )iΦ is the velocity potential for the given direction i .This indicates that for sway and roll oscillations, the horizontal velocities of the waterparticles, thus the derivative x∂Φ∂ , at any time on both sides of the body must have the samedirection; these motions are anti-symmetric. For heave oscillations these velocities must be ofopposite sign; this is a symmetric motion. However, for all three modes of oscillations thevertical velocities, thus the derivative y∂Φ∂ , on both sides must have the same directions atany time.

2.3.2 Forces and Moments

The forces F and moments M follow from an integration of the pressure, p , over thesubmerged surface, S , of the body:

( )∫∫ ⋅⋅−=S

dSnpF and ( )∫∫ ⋅×⋅−=S

dSnrpM


31

Equation 2.3–14

in which n is the outward normal vector on surface dS and r is the position vector ofsurface dS in the ( )zyxO ,, co-ordinate system.The pressure p - via the linearised Bernoulli equation - is determined from the velocitypotentials by:

zgttt

zgt

p

dwr ⋅⋅−

∂Φ∂

+∂Φ∂

+∂Φ∂

⋅−=

⋅⋅−∂Φ∂

⋅−=

ρρ

ρρ

Equation 2.3–15

which can obviously be split into four separate parts, so that the hydromechanical forces Fand moments M can be split into four parts too:

( )∫∫

∫∫

⋅×⋅

⋅+

∂Φ∂

+∂Φ∂

+∂Φ∂

⋅=

⋅⋅

⋅+

∂Φ∂

+∂Φ∂

+∂Φ∂

⋅=

S

dwr

S

dwr

dSnrzgttt

M

dSnzgttt

F

ρ

ρ

Equation 2.3–16

or:

sdwr FFFFF +++= and sdwr MMMMM +++=

Equation 2.3–17

2.3.3 Hydrodynamic Loads

The hydrodynamic loads are the dynamic forces and moments caused by the fluid on anoscillating body in still water; waves are radiated from the body. The radiation potential,

rΦ , which is associated with this oscillation in still water, can be written in terms, jΦ , for 6degrees of freedom as:

( ) ( )

( ) ( )∑

∑

=

=

⋅=

Φ=Φ

6

1

6

1

,,

,,,,,,

jjj

jjr

tvzyx

tzyxtzyx

φ

Equation 2.3–18

in which the space and time dependent potential term, ( )tzyxj ,,.Φ in direction j , is now

written in terms of a separate space dependent potential, ( )zyxj ,,φ in direction j , multiplied

by an oscillatory velocity, ( )tv j in direction j .

This allows the normal velocity on the surface of the body to be written as:


32

∑

∑

=

=

⋅∂

∂=

Φ∂∂

=∂Φ∂

6

1

6

1

jj

j

jj

r

vn

nn

φ

Equation 2.3–19

and the generalised direction cosines are given by:

nf j

j ∂

∂=

φ

Equation 2.3–20

With this the radiation terms in the hydrodynamic force and moment becomes:

( )

( )∫∫ ∑

∫∫

∫∫ ∑

∫∫

⋅×⋅

⋅

∂∂

=

⋅×⋅

∂Φ∂

=

⋅⋅

⋅

∂∂

=

⋅⋅

∂Φ∂

=

=

=

S jjj

S

r

S jjj

S

r

dSnrvt

dSnrt

M

dSnvt

dSnt

F

6

1

6

1

φρ

ρ

φρ

ρ

Equation 2.3–21

The components of these radiation forces and moments are defined by:

( )321 ,, rrrr XXXF = and ( )654 ,, rrrr XXXM =with:

∫∫ ∑

∫∫ ∑

⋅∂

⋅

⋅

∂∂

=

⋅⋅

⋅

∂∂

=

=

=

S

k

jjj

S

kj

jjrk

dSdn

vt

dSfvt

X

φφρ

φρ

6

1

6

1 for: 6,...1=k

Equation 2.3–22

Since jφ and kφ are not time-dependent in this expression, it reduces to:

∑=

=6

1jrkjrk XX for: 6,...1=k

with:

∫∫ ⋅∂∂

⋅⋅=S

kj

jrkj dS

ndt

dvX

φφρ

Equation 2.3–23


33

This radiation force or moment rkjX in the direction k is caused by a forced harmonicoscillation of the body in the direction j . This is generally true for all j and k in the rangefrom 1 to 6. When kj = , the force or moment is caused by a motion in that same direction.When kj ≠ , the force in one direction results from the motion in another direction. Thisintroduces what is called coupling between the forces and moments (or motions).Equation 2.3–23 expresses the force and moment components, rkjX , in terms of still unknown

potentials, jφ . But not everything is solved yet, a solution for this will be found later in this

Chapter.

2.3.3.1 Oscillatory Motion

Now an oscillatory motion is defined; suppose a motion (in a complex notation) given by:ti

ajj ess ω−⋅=

Equation 2.3–24

Then the velocity and acceleration of this oscillation are:

tiaj

jj

tiajjj

esdt

dvs

esivs

⋅⋅−

⋅⋅−

⋅⋅−==

⋅⋅⋅−==

ω

ω

ω

ω

2&&

&

Equation 2.3–25

The hydrodynamic forces and moments can be split into a load in-phase with the accelerationand a load in-phase with the velocity:

( )ti

S

kjaj

tikjajkjaj

jkjjkjrkj

edSn

s

eNsiMs

sNsMX

⋅⋅−

⋅⋅−

⋅

⋅

∂∂

⋅⋅⋅−=

⋅⋅⋅⋅+⋅⋅=

⋅−⋅−=

∫∫ ω

ω

φφρω

ωω

2

2

&&&

Equation 2.3–26

So in case of an oscillation of the body in the direction j with a velocity potential jφ , thehydrodynamic mass and damping (coupling) coefficients are defined by:

⋅∂∂

⋅−= ∫∫S

kjkj dS

nM

φφρRe and

⋅∂

∂⋅⋅−= ∫∫

S

kjkj dS

nN

φφωρIm

Equation 2.3–27

In case of an oscillation of the body in the direction k with a velocity potential kφ , thehydrodynamic mass and damping (coupling) coefficients are defined by:

⋅∂

∂⋅−= ∫∫

S

jkjk dS

nM

φφρRe and

⋅∂

∂⋅⋅−= ∫∫

S

jkjk dS

nN

φφωρIm

Equation 2.3–28


34

2.3.3.2 Green's Second Theorem

Green's second theorem transforms a large volume-integral into a much easier to handlesurface-integral. Its mathematical background is beyond the scope of this text. It is valid forany potential function, regardless the fact if it fulfils the Laplace condition or not.

Consider two separate velocity potentials jφ and kφ . Green's second theorem, applied tothese potentials, is then:

( ) ∫∫∫∫∫ ⋅

∂

∂⋅−

∂∂

⋅=⋅∇⋅−∇⋅**

**22

S

jk

kj

V

jkkj dSnn

dVφ

φφ

φφφφφ

Equation 2.3–29

As said before, this theorem is generally valid for all kinds of potentials; it is not necessarythat they fulfil the Laplace equation. In Green's theorem, *S is a closed surface with a volume

*V . This volume is bounded by the wall of an imaginary vertical circular cylinder with a verylarge radius R , the seabed at hz −= , the water surface at ζ=z and the wetted surface of thefloating body, S ; see Figure 2.3–1.

Figure 2.3–1: Boundary conditions

Both of the above radiation potentials jφ and kφ must fulfil 022 =∇=∇ kj φφ , the Laplace

equation. So the left-hand side of Equation 2.3–29 becomes zero which yields for the right-hand side of this equation:

**

**

dSn

dSn

S

jk

S

kj ⋅

∂

∂⋅=⋅

∂∂

⋅ ∫∫∫∫φ

φφ

φ

Equation 2.3–30

The boundary condition at the free surface becomes for tie ⋅⋅−⋅=Φ ωφ :

02 =∂∂

⋅+⋅−z

gφφω for: 0=z

Equation 2.3–31

or with the dispersion relation, [ ]hkgk ⋅⋅⋅= tanh2ω :

[ ]z

hkk∂∂

=⋅⋅⋅φφtanh for: 0=z

Equation 2.3–32


35

This implies that at the free surface of the fluid one can write:

[ ] [ ][ ] [ ]

∂∂

⋅⋅⋅

=→∂∂

=∂∂

=⋅⋅⋅

∂∂

⋅⋅⋅

=→∂∂

=∂∂

=⋅⋅⋅

nhkknzhkk

nhkknzhkk

jj

jjj

kk

kkk

φφ

φφφ

φφφφφ

tanh1tanh

tanh1

tanh

at the free surface

Equation 2.3–33

When taking also the boundary condition at the seabed and the radiation condition on the wallof the cylinder in Figure 2.3–1:

0=∂∂

nφ

for: hz −= and 0lim =∞→φ

R

Equation 2.3–34

into account, the integral equation over the surface *S reduces to:

∫∫∫∫ ⋅∂

∂⋅=⋅

∂∂

⋅S

jk

S

kj dS

ndS

n

φφ

φφ

Equation 2.3–35

in which S is the wetted surface of the oscillating body only.Notice that jφ and kφ still have to be evaluated.

2.3.3.3 Potential Coefficients

The previous subsection provides - for the zero forward ship speed case - symmetry in thecoefficients matrices with respect to their diagonals so that:

kjjk MM = and kjjk NN =

Equation 2.3–36

Because of the symmetry of a ship, some coefficients are zero and the two matrices withhydrodynamic coefficients for a ship become:

666462

555351

464442

353331

262422

151311

666462

555351

464442

353331

262422

151311

000

000

000

000

000

000

:matrix damping icHydrodynam

000

000

000

000

000

000

:matrix mass icHydrodynam

NNN

NNN

NNN

NNN

NNN

NNN

MMM

MMM

MMM

MMM

MMM

MMM


36

Equation 2.3–37

For clarity, the symmetry of terms about the diagonal in these matrices (for example that

3113 MM = for zero forward speed) has not been included here. The terms on the diagonals

( nnM ) are the primary coefficients relating properties such as hydrodynamic mass in one

direction to the inertia forces in that same direction. Off-diagonal terms (such as 13M )represent hydrodynamic mass only, which is associated with an inertia dependent force in onedirection caused by a motion component in another.

Forward speed has an effect on the velocity potentials itself, but is not discussed in thisSection. This effect is quite completely explained by Timman and Newman [1962].

2.3.4 Wave and Diffraction Loads

The wave and diffraction terms in the hydrodynamic force and moment are:

( )∫∫

∫∫

⋅×⋅

∂Φ∂

+∂Φ∂

=+

⋅⋅

∂Φ∂

+∂Φ∂

=+

S

dwdw

S

dwdw

dSnrtt

MM

dSntt

FF

ρ

ρ

Equation 2.3–38

The principle of linear superposition allows the determination of these forces on a restrainedbody with zero forward speed: 0=∂Φ∂ n . This simplifies the boundary condition on thesurface of the body to:

0=∂Φ∂

+∂Φ∂

=∂Φ∂

nnndw

Equation 2.3–39

The space and time dependent potentials, ( )tzyxw ,,,Φ and ( )tzyxd ,,,Φ , are written now in

terms of isolated space dependent potentials, ( )zyxw ,,φ and ( )zyxd ,,φ , multiplied by a

normalised oscillatory velocity, ( ) tietv ⋅⋅−⋅= ω1 :

( ) ( )( ) ( ) ti

dd

tiww

ezyxtzyx

ezyxtzyx⋅⋅−

⋅⋅−

⋅=Φ

⋅=Φω

ω

φ

φ

,,,,,

,,,,,

Equation 2.3–40

This results into:

nndw

∂∂

−=∂

∂ φφ

Equation 2.3–41

With this and the expressions for the generalised direction-cosines it is found for the waveforces and moments on the restrained body in waves:


37

( )

( )∫∫

∫∫

⋅∂∂

⋅+⋅⋅−=

⋅⋅+⋅⋅−=

⋅⋅−

⋅⋅−

S

kdw

ti

S

kdwti

wk

dSn

ei

dSfeiX

φφφρ

φφρ

ω

ω

for: 6,...1=k

Equation 2.3–42

in which kφ is the radiation potential.

The potential of the incident waves, wφ , is known, but the diffraction potential, dφ , has to be

determined. Green's second theorem provides a relation between this diffraction potential, dφ ,

and a radiation potential, kφ :

∫∫∫∫ ⋅∂∂

⋅=⋅∂∂

⋅S

dk

S

kd dS

ndS

nφ

φφ

φ

Equation 2.3–43

and with nn dw ∂∂−=∂∂ φφ from Equation 2.3–41 one finds:

∫∫∫∫ ⋅∂∂

⋅−=⋅∂∂

⋅S

wk

S

kd dS

ndS

nφ

φφ

φ

Equation 2.3–44

This elimination of the diffraction potential results into the so-called Haskind relations:

∫∫ ⋅

∂∂

⋅+∂

∂⋅⋅⋅−= ⋅⋅−

S

wk

kw

tiwk dS

nneiX

φφ

φφρ ω for: 6,...1=k

Equation 2.3–45

This limits the problem of the diffraction potential because the expression for wkX depends

only on the undisturbed wave potential wφ and the radiation potential kφ .These relations, found by Haskind [1957], are very important; they underlie the relativemotion (displacement - velocity - acceleration) hypothesis, as used in strip theory. Theserelations are valid only for a floating body with a zero time-averaged speed in all directions.Newman [1962] however, has generalised the Haskind relations for a body with a constantforward speed. He derived equations, which differ only slightly from those found by Haskind.According to Newman's approach the wave potential has to be defined in the moving

( )zyxO ,, system. The radiation potential has to be determined for the constant forward speedcase, taking an opposite sign into account.

The corresponding wave potential for deep water - as given in a previous section - nowbecomes:

( )

( ) tiyxkizka

zkaw

eeegi

ykxkteg

⋅⋅−⋅+⋅⋅⋅

⋅

⋅⋅⋅⋅⋅−

=

⋅⋅−⋅⋅−⋅⋅⋅⋅−

=Φ

ωµµ

ωζ

µµωωζ

sincos

sincossin

Equation 2.3–46

so that the isolated space dependent term is given by:


38

( )µµ

ωζφ sincos ⋅+⋅⋅⋅ ⋅⋅

⋅⋅−= yxkizka

w eegi

Equation 2.3–47

In these equations is µ the wave direction, defined as given in Figure 2.1–2.The velocity of the water particles in the direction of the outward normal n on the surface ofthe body is:

( ) µµφ

µµφφ

sincos

sincos

213 ⋅+⋅⋅+⋅⋅=

⋅

∂∂

+⋅∂∂

⋅+∂∂

⋅⋅=∂∂

ffifk

ny

nx

inz

kn

w

ww

Equation 2.3–48

With this, the wave loads are given by:

( ) ∫∫

∫∫⋅⋅+⋅⋅+⋅⋅⋅⋅⋅⋅+

⋅⋅⋅⋅⋅−=

⋅⋅−

⋅⋅−

dSffifkei

dSfeiX

kwti

Skw

tiwk

µµφφρ

φρ

ω

ω

sincos 213

for: 6,...1=k

Equation 2.3–49

The first term in this expression for the wave forces and moments is the so-called Froude-Krilov force or moment, which is the wave load caused by the undisturbed incident wave. Thesecond term is related to the disturbance caused by the presence of the (restrained) body.

2.3.5 Hydrostatic Loads

In the notations used here, the buoyancy forces and moments are:

∫∫ ⋅⋅⋅=S

s dSnzgF ρ and ( )∫∫ ⋅×⋅⋅=S

s dSnrzgM ρ

Equation 2.3–50

or more generally:

∫∫ ⋅⋅⋅=S

ksk dSfzgX ρ for: 6,...1=k

Equation 2.3–51

in which the skX are the components of these hydrostatic forces and moments.


39

2.4 Equations of Motion

The equations of motion are given here in a - with the ship speed V steadily moving - right-handed co-ordinate system ( )zyxG ,, , with the origin in the average position of the ship’scentre of gravity G .The total mass as well as its distribution over the body is considered to be constant with time.For ships and other floating structures, this assumption is normally valid during a time that islarge relative to the period of the motions. This holds that small effects - such as for instance adecreasing mass due to fuel consumption - can be ignored.The solid mass matrix of a floating structure is given below.

−

−∇⋅

∇⋅

∇⋅

=

zzzx

yy

xzxx

II

I

IIm

0000

00000

0000

00000

00000

00000

:matrix mass Solidρ

ρρ

Equation 2.4–1

The moments of inertia here are often expressed in terms of the radii of inertia and the solidmass of the structure. Since Archimedes’ law ( ∇⋅= ρm ) is valid for a free floating structure:

∇⋅⋅=

∇⋅⋅=

∇⋅⋅=

ρ

ρ

ρ

2

2

2

zzzz

yyyy

xxxx

kI

kI

kI

Equation 2.4–2

When the actual distribution of the solid mass of a ship is unknown, the radii of inertia can beapproximated by:

⋅⋅≈

⋅⋅≈

⋅⋅≈

LL

LL

BB

28.0 to22.0k

28.0 to22.0k

40.0 to30.0k

:shipsfor

zz

yy

xx

Equation 2.4–3

in which L is the length and B is the breadth of the ship.Often, the (generally small) coupling terms, zxxz II = , are simply neglected.Bureau Veritas proposes for the radius of inertia for roll of the ship's solid mass:

⋅+⋅⋅≈

22

0.1289.0BKG

Bk xx

Equation 2.4–4

in which KG is the height of the centre of gravity, G , above the keel.For many ships without cargo on board (ballast condition), the mass is concentrated at theends (engine room aft and ballast water forward to avoid a large trim), while for ships withcargo on board (full load condition) the - more or less amidships laden - cargo plays an


40

important role. Thus for normal ships, the radii of inertia, yyk and zzk , are usually smaller inthe full load condition than in the ballast condition.Notice here that the longitudinal radius of gyration of a long homogeneous rectangular beamwith a length L is equal to about L⋅121 or L⋅289.0 .

The equations of motions of a rigid body in a space fixed co-ordinate system follow fromNewton's second law. The vector equations for the translations of and the rotations about thecentre of gravity are given respectively by:

( )Umdtd

F ⋅= and ( )Hdtd

M =

Equation 2.4–5

in which:F resulting external force acting in the centre of gravitym mass of the rigid body

U instantaneous velocity of the centre of gravity

M resulting external moment acting about the centre of gravity

H instantaneous angular momentum about the centre of gravityt time

Two important assumptions are made for the loads in the right-hand side of these equations:

a) The so-called hydromechanic forces and moments are induced by the harmonicoscillations of the rigid body, moving in the undisturbed surface of the fluid.

b) The so-called wave exciting forces and moments are produced by waves coming in on therestrained body.

Since the system is linear, these loads are added up for obtaining the total loads. Thus, afterassuming small motions, symmetry of the body and that the x -, y - and z -axes are principalaxes, one can write the following six equations of motion for the ship:

( )

( )

( )

( )( )( ) 66

55

44

33

22

11

Yaw

Pitch

:Roll

:Heave

:Sway

:Surge

whzxzzzxzz

whxxyy

whxzxxxzxx

wh

wh

wh

XXIIIIdtd

XXIIdtd

XXIIIIdtd

XXzzdtd

XXyydtd

XXxxdtd

+=⋅−⋅=⋅−⋅

+=⋅=⋅

+=⋅−⋅=⋅−⋅

+=⋅∇⋅=⋅∇⋅

+=⋅∇⋅=⋅∇⋅

+=⋅∇⋅=⋅∇⋅

φψφψ

θθ

ψφψφ

ρρ

ρρ

ρρ

&&&&&&

&&&

&&&&&&

&&&

&&&

&&&

Equation 2.4–6

in which:ρ density of water


41

∇ volume of displacement of the ship

ijI solid mass moment of inertia of the ship

321 ,, hhh XXX hydromechanic forces in the x -, y - and z -directions

654 ,, hhh XXX hydromechanic moments about the x -, y - and z -axes

321 ,, www XXX exciting wave forces in the x -, y - and z -directions

654 ,, www XXX exciting wave moments about the x -, y - and z -axes

Generally, a ship has a vertical-longitudinal plane of symmetry, so that its motions can be splitinto symmetric and anti-symmetric components. Surge, heave and pitch motions aresymmetric motions, that is to say that a point to starboard has the same motion as the mirroredpoint to port side. It is obvious that the remaining motions sway, roll and yaw are anti-symmetric motions. Symmetric and anti-symmetric motions of a free-floating structure are notcoupled; they don't have any effect on each other. For instance, a vertical force acting at thecentre of gravity can cause surge, heave and pitch motions, but will not result in sway, roll oryaw motions.

Because of this symmetry and anti-symmetry, two sets of three coupled equations of motioncan be distinguished for ships:

motions symmetric-anti

:Yaw

:Roll

:Sway

motions symmetric

:Pitch

:Heave

:Surge

66

44

22

55

33

11

=−⋅−⋅

=−⋅−⋅

=−⋅∇⋅

=−⋅

=−⋅∇⋅

=−⋅∇⋅

whzxzz

whxzxx

wh

whxx

wh

wh

XXII

XXII

XXy

XXI

XXz

XXx

φψ

ψφ

ρ

θ

ρρ

&&&&

&&&&

&&

&&

&&

&&

Equation 2.4–7

Note that this distinction between symmetric and anti-symmetric motions disappears when theship is anchored. Then, for instance, the pitch motions can generate roll motions via theanchor lines.

The coupled surge, heave and pitch equations of symmetric motion are:

( )

( )

( ) (pitch)

(heave)

(surge)

5555555

535353

515151

3353535

333333

313131

1151515

131313

111111

wyy

w

w

XcbaI

zczbza

xcxbxa

Xcba

zczbza

xcxbxa

Xcba

zczbza

xcxbxa

=⋅+⋅+⋅++⋅+⋅+⋅+⋅+⋅+⋅

=⋅+⋅+⋅+⋅+⋅+⋅+∇⋅+⋅+⋅+⋅

=⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+∇⋅

θθθ

θθθρ

θθθ

ρ

&&&&&&

&&&

&&&&&&

&&&

&&&&&&

&&&


42

Equation 2.4–8

The coupled sway, roll and yaw equations of anti-symmetric motion are:

( )

( )( )

( )( ) (yaw)

(roll)

(sway)

6666666

646464

626262

4464646

444444

424242

2262626

242424

222222

wzz

zx

wxz

xx

w

XcbaI

cbaI

ycybya

XcbaI

cbaI

ycybya

Xcba

cba

ycybya

=⋅+⋅+⋅++⋅+⋅+⋅+−+⋅+⋅+⋅

=⋅+⋅+⋅+−+⋅+⋅+⋅++⋅+⋅+⋅

=⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+∇⋅

ψψψφφφ

ψψψφφφ

ψψψφφφ

ρ

&&&

&&&

&&&

&&&

&&&

&&&

&&&

&&&

&&&

Equation 2.4–9

In many applications, zxxz II = is not known or small; hence their terms are often omitted. Inprogram SEAWAY they have been introduced in the equations of motion if they can becalculated from an input of the mass distribution along the ship’s length, only.

After the determination of the in and out of phase terms of the hydromechanic and the waveloads, these equations can be solved with a numerical method.


43

2.5 Strip Theory Approaches

Strip theory is a computational method by which the forces on and motions of a three-dimensional floating body can be determined using results from two-dimensional potentialtheory. Strip theory considers a ship to be made up of a finite number of transverse two-dimensional slices, which are rigidly connected to each other. Each of these slices will have aform that closely resembles the segment of the ship that it represents. Each slice is treatedhydrodynamically as if it is a segment of an infinitely long floating cylinder; see Figure 2.5–1.

Figure 2.5–1: Strip theory representation by cross sections

This means that all waves which are produced by the oscillating ship (hydromechanic loads)and the diffracted waves (wave loads) are assumed to travel perpendicular to the middle lineplane - thus parallel to the ( )zy, -plane - of the ship. This holds too that the strip theorysupposes that the fore and aft side of the body (such as a pontoon) does not produce waves inthe x -direction. For the zero forward speed case, interactions between the cross sections areignored as well.Fundamentally, strip theory is valid for long and slender bodies only. In spite of thisrestriction, experiments have shown that strip theory can be applied successfully for floatingbodies with a length to breadth ratio larger than three, 3≥BL , at least from a practical pointof view.

2.5.1 Zero Forward Ship Speed

When applying the strip theory, the loads on the body are found by an integration of the 2-Dloads:


44

∫∫∫∫

∫∫∫∫∫∫∫∫

⋅⋅+=⋅⋅+=

⋅⋅−=⋅⋅−=

⋅=⋅=

⋅=⋅=

⋅=⋅=

⋅=⋅=

Lbbww

Lbbhh

Lbbww

Lbbhh

Lbww

Lbhh

Lbww

Lbhh

Lbww

Lbhh

Lbww

Lbhh

dxxXXdxxXX

dxxXXdxxXX

dxXXdxXX

dxXXdxXX

dxXXdxXX

dxXXdxXX

'26

'26

'35

'35

'44

'44

'33

'33

'22

'22

'11

'11

:Yaw

:Pitch

:Roll

:Heave

:Sway

:Surge

Equation 2.5–1

in which:'

hjX sectional hydromechanic force or moment in direction j per unit ship length'

wjX sectional exciting wave force or moment in direction j per unit ship length

The appearance of two-dimensional surge forces seems strange here. It is strange! A more orless empirical method has been used in SEAWAY for the surge motion, by defining anequivalent longitudinal cross section that is swaying. Then, the 2-D hydrodynamic swaycoefficients of this equivalent cross section are translated to 2-D hydrodynamic surgecoefficients by an empirical method based on theoretical results from three-dimensionalcalculations and these coefficients are used to determine 2-D loads. In this way, all sets of sixsurge loads can be treated in the same numerical way in SEAWAY for the determination of the3-D loads. Inaccuracies of the hydromechanic coefficients for surge of (slender) ships are ofminor importance, because these coefficients are relatively small.

Notice how in the strip theory the pitch and yaw moments are derived from the 2-D heave andsway forces, respectively, while the roll moments are obtained directly.

The equations of motions are defined in the moving axis system with the origin at the time-averaged position of the centre of gravity, G . All two-dimensional potential coefficients havebeen defined here in an axis system with the origin, O , in the water plane; the hydromechanicand exciting wave moments have to be corrected for the distance OG .

2.5.2 Forward Ship Speed

Relative to an oscillating ship moving forward in the undisturbed surface of the fluid, thedisplacements, *

hjζ , velocities, *hjζ& , and accelerations, *

hjζ&& , at forward ship speed V in oneof the 6 directions j of a water particle in a cross section are defined by:

*hjζ **

hjhj DtD ζζ =& **

hjhj DtD ζζ &&& =

Equation 2.5–2

in which:


45

∂∂

⋅−∂∂

=x

VtDt

D

Equation 2.5–3

is a mathematical operator which transforms the potentials ( )tzyx ,,, 000Φ , defined in the

earth bounded (fixed) co-ordinate system, to the potentials ( )tzyx ,,,Φ , defined in the ship'ssteadily translating co-ordinate system with speed V .

In waves the motions of the water particles are depending on its local vertical distance to themean or still water surface. At each cross section of the ship an average (or equivalent)constant value has to be found.Relative to a restrained ship, moving forward with speed V in waves, the equivalent j

constant components of water particle displacements ( *wjζ ), velocities ( *

wjζ& ) and

accelerations ( *wjζ&& ) in a cross section are defined in a similar way by:

*wjζ **

wjwj DtD ζζ =& **

wjwj DtD ζζ &&& =

Equation 2.5–4

The effect of the operator in Equation 2.5–3 can be understood easily when one realises that inthat earth-bound co-ordinate system the sailing ship penetrates through a ''virtual verticaldisk''. For instance, when a ship sails with speed V and constant trim angle θ through stillwater, the relative vertical velocity of a water particle with respect to the bottom of the sailingship becomes θθ ⋅≈⋅ VV sin .

Two different types of strip theory methods (as has been used in SEAWAY) are discussedhere:

1. Ordinary Strip Theory MethodAccording to this classic method, the uncoupled two-dimensional potentialhydromechanic loads and wave loads in an arbitrary direction j are defined by:

'*'*'*

'*'*'*

fkjwjjjwjjjwj

rsjhjjjhjjjhj

XNMDtD

X

XNMDtD

X

+⋅+⋅=

+⋅+⋅=

ζζ

ζζ

&&

&&

Equation 2.5–5

This is the first formulation of the strip theory that can be found in the literature. Itcontains a more or less intuitive approach to the forward speed problem, as published indetail by Korvin-Kroukovski and Jacobs [1957].

2. Modified Strip Theory MethodAccording to this modified method, these loads become:


46

'*''*

'*''*

fkjwjjje

jjwj

rsjhjjje

jjhj

XNi

MDtD

X

XNi

MDtD

X

+

⋅

⋅−=

+

⋅

⋅−=

ζω

ζω

&

&

Equation 2.5–6

This formulation is a more fundamental approach of the forward speed problem, aspublished in detail by Tasai [1969] and many others.

In the equations above, 'jjM and '

jjN are the 2-D potential mass and damping coefficients.'

rsjX is the two-dimensional quasi-static restoring spring term, as generally present for heave,

roll and pitch only. 'fkjX is the two-dimensional Froude-Krilov force or moment which is

calculated by integration of the directional pressure gradient in the undisturbed wave over thecross sectional area of the hull.Equivalent directional components of the orbital acceleration and velocity - derived fromthese Froude-Krilov loads - are used here to calculate the diffraction parts of the total waveforces and moments.From a theoretical point of view, one should prefer the use of the modified strip theorymethod. However, it appeared from the authors’ experiences that for ships with moderateforward speed ( 30.0≤Fn ) the ordinary method could provide in some cases a better fit withexperimental data.

2.5.3 End-Terms

From the previous, it is obvious that in the equations of motion longitudinal derivatives of thetwo-dimensional potential mass '

jjM and damping 'jjN will appear. These derivatives have to

be determined numerically over the whole ship length in such a manner that the followingrelation is fulfilled:

( )( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

( )

( ) ( )( )

( )

( )

0

00

0

0

00

=

−⋅+=

⋅+⋅+⋅=⋅

∫

∫∫∫∫+

−

+

−

Lfdxdx

xdff

dxdx

xdfdx

dxxdf

dxdx

xdfdx

dxxdf

b

Lx

x b

b

b

Lx

Lx b

bb

Lx

x b

bb

x

x b

bb

Lx

x b

b

b

b

b

b

b

b

b

b

b

b

ε

ε

ε

ε

Equation 2.5–7

with L<<ε , while ( )bxf is equal to the local values of ( )bjj xN ' or ( )bjj xM ' ; see Figure 2.5–2.


47

Figure 2.5–2: Integration of derivatives

The numerical integration of the derivatives will be carried out in the region( ) ( )Lxxx bbb ≤≤0 only. So, the additional so-called ''end terms'' are defined by ( )0f and

( )Lf .Because the integration of the derivatives should be carried out in the region just behind untiljust before the ship, so ( ) ( ) εε +≤≤− Lxxx bbb 0 , some can algebra provide the integral andthe first and second order moments (with respect to G ) over the whole ship length (slenderbody assumption):

( )( )

( )

( )( )

( )

( )( )

( )

( )( )

( )

( )( )

( )

bb

Lx

xbbb

Lx

x b

b

b

Lx

xbbb

Lx

x b

b

b

Lx

x b

b

dxxxfdxxdx

xdf

dxxfdxxdx

xdf

dxdx

xdf

b

b

b

b

b

b

b

b

b

b

⋅⋅⋅−=⋅⋅

⋅−=⋅⋅

=⋅

∫∫

∫∫

∫

+

−

+

−

+

−

0

2

0

00

0

2

0

ε

ε

ε

ε

ε

ε

Equation 2.5–8

Notice that these expressions are valid for the integration of the potential coefficients over thefull ship length only. They can not be used for calculating local hydromechanic loads. Also forthe wave loads, these expressions can not be used, because there these derivatives aremultiplied with equivalent bx -depending orbital motion amplitudes.


48

2.6 Hydrodynamic Coefficients

In strip theory, the two-dimensional hydrodynamic sway, heave and roll coefficients can becalculated by several methods:

1. Methods based on Ursell's Theory and Conformal MappingUrsell [1949] derived an analytical solution for solving the problem of calculating thehydrodynamic coefficients of an oscillating circular cylinder in the surface of a fluid:

a) Deep Water Coefficients with Lewis Conformal MappingTasai [1959], Tasai [1961] and many others added the so-called Lewis transformation -which is a very simple and in a lot of cases also more or less realistic method totransform ship-like cross sections to this unit circle - to Ursell's solution. Thistransformation will be carried out using a scale factor and two mapping coefficients.Only the breadth, the draft and the area of the mapped cross section will be similar tothat of the actual cross section.

b) Deep Water Coefficients with Close-Fit Conformal MappingA more accurate way of mapping has been added by Tasai [1960] and others too, byusing more than only two mapping coefficients. The accuracy obtained depends on thenumber of mapping coefficients. Generally, a maximum number of 10 coefficients areused for defining the cross section. These coefficients are determined in such a waythat the Root Mean Square of the differences between the offsets of the mapped andthe actual cross section is minimal.

c) Shallow Water Coefficients with Lewis Conformal MappingFor shallow water, the theory of Keil [1974] - based on an expansion of Ursell'spotential theory for circular cylinders at deep water to shallow water - and Lewisconformal mapping can be used.

2. Frank's Pulsating Source Theory for Deep WaterMapping methods require an intersection of the cross section with the water plane and, asa consequence of this, they are not suitable for submerged cross sections, like at a bulbousbow. Also, conformal mapping can fail for cross sections with very low sectional areacoefficients, such as are sometimes present in the aft body of a ship.Frank [1967] considered a cylinder of constant cross sections with an arbitrarilysymmetrical shape, of which the cross sections are simply a region of connected lineelements. This vertical cross section can be fully or partly immersed in a previouslyundisturbed fluid of infinite depth. He developed an integral equation method utilising theGreen's function, which represents a complex potential of a pulsating point source of unitstrength at the midpoint of each line element. Wave systems were defined in such a waythat all required boundary conditions were fulfilled. The linearised Bernoulli equationprovides the pressures after which the potential coefficients were obtained from the in-phase and out-of-phase components of the resultant hydrodynamic loads.

The 2-D potential pitch and yaw (moment) coefficients follow from the previous heave andsway coefficients and the lever, i.e., the distance of the cross section to the centre of gravityG .


49

A more or less empirical procedure has followed by the author for the surge motion. Anequivalent longitudinal cross section has been defined. For each frequency, the two-dimensional potential hydrodynamic sway coefficient of this equivalent cross section istranslated to two-dimensional potential hydrodynamic surge coefficients, by an empiricalmethod based on theoretical results of three-dimensional calculations.

The 3-D coefficients follow from an integration of these 2-D coefficients over the ship'slength. Viscous terms have been be added for surge and roll.


50


51

3 2-D Potential Coefficients

This Chapter described the various methods, used in the SEAWAY computer code, to obtainthe 2-D potential coefficients:• the theory of Tasai for deep water, based on Ursell's potential theory for circular cylinders

and Lewis and N-parameter conformal mapping• the theory of Keil for very shallow to deep water, based on a variation of Ursell's potential

theory for circular cylinders and Lewis conformal mapping• the theory of Frank for deep water, using pulsating sources on the cross sectional contour.

During the ship motions calculations different co-ordinate systems, as shown before, will beused. The two-dimensional hydrodynamic potential coefficients have been defined here withrespect to the ( )zyxO ,, co-ordinate system for the moving ship in still water.However, in this section deviating axes systems are used for the determination of the two-dimensional hydrodynamic potential coefficients for sway, heave and roll motions. This holdsfor the sway and roll coupling coefficients a change of sign. The signs of the uncoupled sway,heave and roll coefficients do not change.

For each cross section, the following two-dimensional hydrodynamic coefficients have to beobtained:• '

22M and '22N 2-D potential mass and damping coefficients of sway

• '24M and '

24N 2-D potential mass and damping coupling coefficients of roll into sway

• '33M and '

33N 2-D potential mass and damping coefficients of heave

• '44M and '

44N 2-D potential mass and damping coefficients of roll

• '

42M and '42N 2-D potential mass and damping coupling coefficients of sway into roll

The 2-D potential pitch and yaw (moment) coefficients, '55M , '

55N , '66M and '

66N , followfrom the previous heave and sway coefficients and the lever of the loads, i.e., the distance ofthe cross section to the centre of gravity G .

Finally, an approximation is given for the determination of the 2-D potential surge coefficients'

11M and '11N .


52


53

3.1 Conformal Mapping Methods

Ursell's derivation of potential coefficients is valid for semicircular cross sections only. Forthe determination of the two-dimensional added mass and damping in the sway, heave and rollmode of the motions of ship-like cross sections by Ursell's method, the cross sections have tobe mapped conformally to the unit semicircle. The advantage of conformal mapping is that thevelocity potential of the fluid around an arbitrarily shape of a cross section in a complex planecan be derived from the more convenient semicircular section in another complex plane. Inthis manner hydrodynamic problems can be solved directly with the coefficients of themapping function.The general transformation formula – see also Figure 3.1–1 - is given by:

( ) ∑ −−− ⋅⋅= 12

12n

ns aMz ζ

Equation 3.1–1

with:iyxz += plane of the ship's cross section

θαζ ieie −⋅= plane of the unit circle

sM scale factor

1−a 1+=

12 −na conformal mapping coefficients ( Nn ,...1= )N Maximum parameter index number

Figure 3.1–1: Mapping relation between two planes

From this follows the relation between the co-ordinates in the z -plane (the ship's crosssection) and the variables in the ζ -plane (the circular cross section):

( ) ( ) ( )( )

( ) ( ) ( )( ) ∑

∑

=

⋅−−−

=

⋅−−−

⋅−⋅⋅⋅−⋅+=

⋅−⋅⋅⋅−⋅−=

N

n

ann

ns

N

n

ann

ns

neaMy

neaMx

0

1212

0

1212

12cos1

12sin1

θ

θ

Equation 3.1–2


54

The contour of the - by conformal mapping approximated - ship's cross section follows fromputting 0=α in the previous relations in Equation 3.1–2:

( ) ( )( )

( ) ( )( ) ∑

∑

=−

=−

⋅−⋅⋅−⋅+=

⋅−⋅⋅−⋅−=

N

nn

ns

N

nn

ns

naMy

naMx

0120

0120

12cos1

12sin1

θ

θ

Equation 3.1–3

The breadth on the waterline of the approximated ship's cross section is defined by:

asMb λ⋅⋅= 20 with: ∑=

−=N

nna a

012λ and

as

bM

λ⋅=

20

Equation 3.1–4

The draught is defined by:

bsMd λ⋅=0 with: ( ) ∑=

−⋅−=N

nn

nb a

0121λ

Equation 3.1–5

3.1.1 Lewis Conformal Mapping

A very simple and in a lot of cases also a more or less realistic transformation of the crosssectional hull form will be obtained with 2=N in the transformation formula, the wellknown Lewis transformation, see reference Lewis [1929]. An extended and clear descriptionof the representation of ship hull forms by this Lewis two-parameter conformal mapping isgiven by von Kerczek and Tuck [1969].The two-parameter Lewis transformation of a cross section is defined by:

( )33

111

−−− ⋅+⋅+⋅⋅= ζζζ aaaMz s

Equation 3.1–6

In here 11 +=−a and the conformal mapping coefficients 1a and 3a are called Lewis

coefficients, while sM is the scale factor.Then:

( )( )θθθ

θθθααα

ααα

3coscoscos

3sinsinsin3

31

331

⋅⋅+⋅⋅−⋅⋅=

⋅⋅−⋅⋅+⋅⋅=−−

−−

eaeaeMy

eaeaeMx

s

s

Equation 3.1–7

By putting 0=α is the contour of this so-called Lewis form expressed as:( )( )( )( )θθ

θθ3coscos1

3sinsin1

310

310

⋅+⋅−⋅=⋅−⋅+⋅=

aaMy

aaMx

s

s

with scale factor:

3131 112

aaD

aaB

M sss +−

=++

=

Equation 3.1–8


55

in which:

sB sectional breadth on the water line

sD sectional draught

Now the coefficients 1a and 3a and the scale factor sM will be determined in such a mannerthat the sectional breadth, draught and area of the approximated cross section and of the actualcross section are identical.The half breadth to draught ratio 0H is given by:

31

310 1

12aaaa

DB

Hs

s

+−++

==

Equation 3.1–9

An integration of the Lewis form delivers the sectional area coefficient sσ :

( ) 21

23

23

21

1

314 aa

aaDB

A

ss

ss

−+

⋅−−⋅=

⋅=

πσ

Equation 3.1–10

in which sA is the area of the cross section.

Putting 1a , derived from the expression for 0H in Equation 3.1–9, into the expression for sσin Equation 3.1–10 yields a quadratic equation in 3a :

03322

31 =+⋅+⋅ cacacin which:

4

62

114

14

3

13

12

2

0

01

−=−⋅=

+−

⋅

⋅−+

⋅+=

cc

cc

HH

c ss

πσ

πσ

Equation 3.1–11

The (valid) solutions for 3a and 1a become:

( )111

293

30

01

1

113

+⋅+−

=

⋅−++−=

aHH

a

c

cca

Equation 3.1–12

Lewis forms with the other solution of 3a in the quadratic equation, with a minus sign beforethe square root expression:

1

113

293

c

cca

⋅−−+−=


56

are looped; they intersect themselves at a point within the fourth quadrant. Since ships are''better behaved'', these solutions are not considered.

It is obvious that a transformation of a half-immersed circle with radius R will result inRM s = , 01 =a and 03 =a .

Some typical and realistic Lewis forms are presented in Figure 3.1–2.

Figure 3.1–2: Typical Lewis forms

3.1.1.1 Boundaries of Lewis Forms

In some cases the Lewis transformation can give more or less ridiculous results. Thefollowing typical Lewis hull forms, with the regions of the half breadth to draught ratio 0H

and the area coefficient sσ to match as presented before, can be distinguished:

• re-entrant forms, bounded by:

for 0.10 ≤H : ( )0232

3Hs −⋅

⋅<

πσ

for 0.10 ≥H :

−⋅

⋅<

0

12

323

Hs

πσ

Equation 3.1–13

• conventional forms, bounded by:

for 0.10 ≤H : ( )

+⋅

⋅<<−⋅

⋅4

332

32

323 0

0

HH s

πσπ

for 0.10 ≥H :

⋅

+⋅⋅

<<

−⋅

⋅

00 41

332

312

323

HH s

πσ

π

Equation 3.1–14

• bulbous and not-tunneled forms, bounded by:

0.10 ≤H and

⋅

+⋅⋅

<<

+⋅⋅

0

0

41

332

34

332

3H

Hs

πσ

π

Equation 3.1–15

• tunneled and not-bulbous forms, bounded by:

for: 0.10 ≥H and

+⋅

⋅<<

⋅

+⋅⋅

43

323

41

332

3 0

0

HH s

πσπ


57

Equation 3.1–16

• combined bulbous and tunneled forms, bounded by:

for: 0.10 ≤H and

++⋅<<

⋅

+⋅⋅

00

0

110

3241

332

3H

HH s

πσ

π

for: 0.10 ≥H and

++⋅<<

+⋅⋅

00

0 110

3243

323

HH

Hs

πσ

π

Equation 3.1–17

• non-symmetric forms, bounded by:

∞<< 00 H and

++⋅>

00

110

32 HHs

πσ

Equation 3.1–18

These ranges of the half breadth to draught ratio 0H and the area coefficient sσ for thedifferent typical Lewis forms are shown in Figure 3.1–3.

Figure 3.1–3: Ranges of 0H and sσ of Lewis Forms

3.1.1.2 Acceptable Lewis Forms

Not-acceptable forms of ships are supposed to be the re-entrant forms and the asymmetricforms. So conventional forms, bulbous forms and tunneled forms are considered to be validforms here, see Figure 3.1–3. To obtain ship-like Lewis forms, the area coefficient sσ isbounded by a lower limit to omit re-entrant Lewis forms and by an upper limit to omit non-symmetric Lewis forms:

for 0.10 ≤H : ( )

++⋅<<−⋅

⋅

000

110

322

323

HHH s

πσπ

for 0.10 ≥H :

++⋅<<

−⋅

⋅

00

0

110

321

232

3H

HH s

πσ

π


58

Equation 3.1–19

If a value of sσ is outside of this range it has to be set (from a practical point of view) to thevalue of the nearest border of this range, to calculate the Lewis coefficients.Numerical problems, for instance with bulbous or aft cross sections of a ship, are avoidedwhen the following requirements are fulfilled:

ss D

B⋅> γ

2 and

2s

s

BD ⋅> γ with for instance 01.0=γ .

3.1.2 Extended Lewis Conformal Mapping

Somewhat better approximations will be obtained by taking into account also the first ordermoments of half the cross section about the 0x - and 0y -axes. These two additions to theLewis formulation were proposed by Reed and Nowacki [1974] and have been simplified byAthanassoulis and Loukakis [1985] by taking into account the vertical position of the centroidof the cross section. Extending the Lewis transformation from 2=N to 3=N in the generaltransformation formula has done this.

The three-parameter Extended-Lewis transformation is defined by:( )5

53

31

11−−−

− ⋅+⋅+⋅+⋅⋅= ζζζζ aaaaMz s

with 11 +=−a .

Equation 3.1–20

So:( )( )θθθθ

θθθθαααα

αααα

5cos3coscoscos

5sin3sinsinsin5

53

31

55

331

⋅⋅−⋅⋅+⋅⋅−⋅⋅=

⋅⋅+⋅⋅−⋅⋅+⋅⋅=−−−

−−−

eaeaeaeMy

eaeaeaeMx

s

s

Equation 3.1–21

By putting 0=α , the contour of this approximated form is expressed as:( )( )( )( )θθθ

θθθ5cos3coscos1

5sin3sinsin1

5310

5310

⋅−⋅+⋅−⋅=⋅+⋅−⋅+⋅=

aaaMy

aaaMx

s

s

with scale factor:

531531 112

aaaD

aaaB

M sss −+−

=+++

=

Equation 3.1–22

in which:

sB sectional breadth on the water line

sD sectional draught

Now the coefficients 1a , 3a and 5a and the scale factor sM will be determined such that,except the sectional breadth, draught and area, also the centroids of the approximated crosssection and of the actual cross section have a similar position.The half breadth to draught ratio 0H is given by:

531

5310 1

12aaaaaa

DB

Hs

s

−+−+++

==


59

Equation 3.1–23

An integration of the approximated form results into the sectional area coefficient sσ :

( ) ( )251

23

25

23

21

1

5314 aaa

aaaDB

A

ss

ss +−+

⋅−⋅−−⋅=

⋅=

πσ

Equation 3.1–24

A more complex expression has been obtained by Athanassoulis and Loukakis [1985] for therelative distance of the centroid to the keel point:

∑

∑∑∑

=−

=−−−

==

⋅⋅

⋅⋅⋅−== 3

0

3120

3

0121212

3

0

3

01

iis

kkjiijk

ji

s aH

aaaA

DKB

σκ

in which:

( ) ( ) ( ) ( )

++−⋅−⋅−

+−+⋅−

⋅−+

+−⋅−⋅−

−++⋅−

⋅−⋅=

kjik

kjik

kjik

kjik

Aijk 2121

2121

2121

2321

41

Equation 3.1–25

The following requirements should be fulfilled when also bulbous cross sections are allowed:• re-entrant forms are avoided when:

0531

0531

531

531

>⋅+⋅−+>⋅−⋅−−

aaa

aaa

Equation 3.1–26

• existence of a point of self-intersection is avoided when:

020101459

020101459

50532

52

3

50532

52

3

>⋅⋅−⋅⋅−⋅+⋅

>⋅⋅+⋅⋅+⋅+⋅

aHaaaa

aHaaaa

Equation 3.1–27

Taking these restrictions into account, the equations above can be solved in an iterativemanner.

3.1.3 Close-Fit Conformal Mapping

A more accurate transformation of the cross sectional hull form can be obtained by using agreater number of parameters N . A very simple and straight on iterative least squares methodof the first author to determine the Close-Fit conformal mapping coefficients will be describedhere shortly.

The scale factor sM and the conformal mapping coefficients 12 −na , with a maximum value ofn varying from 2=N until 10=N , have been determined successfully from the offsets ofvarious cross sections in such that the sum of the squares of the deviations of the actual crosssection from the approximate described cross section is minimised.


60

The general transformation formula is again given by:

( ) ∑=

−−− ⋅⋅=

N

n

nns aMz

0

1212 ζ

with: 11 +=−a .

Equation 3.1–28

Then the contour of the approximated cross section is given by:

( ) ( )( )

( ) ( )( ) ∑

∑

=−

=−

⋅−⋅⋅−⋅+=

⋅−⋅⋅−⋅−=

N

nn

ns

N

nn

ns

naMy

naMx

0120

0120

12cos1

12sin1

θ

θ

with scale factor:

( ) ∑∑=

−=

− ⋅−== N

nn

n

sN

nn

ss

a

D

a

BM

012

012 1

2

Equation 3.1–29

The procedure starts with initial values for [ ]12 −⋅ ns aM . The initial values of sM , 1a and 3aare obtained with the Lewis method as has been described before, while the initial values of

5a through 12 −Na are set to zero. With these [ ]12 −⋅ ns aM values, a iθ -value is determined for

each offset in such a manner that the actual offset ( )ii yx , lies on the normal of the

approximated contour of the cross section in ( )ii yx 00 , .

Now iθ has to be determined. Therefore a function ( )iF θ , will be defined by the distance

from the offset ( )ii yx , to the normal of the contour to the actual cross section through

( )ii yx 00 , , see Figure 3.1–4.

Figure 3.1–4: Cose-Fit conformal mapping


61

These i offsets ( Ii ,...0= ) have to be selected at approximately equal mutual circumferential

lengths, eventually with somewhat more dense offsets near sharp corners. Then iα is definedby:

( ) ( )

( ) ( )211

211

11

211

211

11

sin

cos

−+−+

−+

−+−+

−+

−+−

+−=

−+−

−+=

iiii

iii

iiii

iii

yyxx

yy

yyxx

xx

α

α

Equation 3.1–30

With this iθ -value, the numerical value of the square of the deviation of ( )ii yx , from ( )ii yx 00 ,is calculated:

( ) ( )20

20 iiiii yyxxe −+−=

Equation 3.1–31

After doing this for all 1+I offsets, the numerical value of the sum of the squares ofdeviations is known:

∑=

=I

iieE

0

Equation 3.1–32

The sum of the squares of these deviations can also be expressed as:

( ) [ ] ( )( )

( ) [ ] ( )( ) ∑

∑

∑=

=−

=−

⋅−⋅⋅⋅−−+

⋅−⋅⋅⋅−++

=I

i N

nins

ni

N

nins

ni

naMy

naMx

E0

2

012

2

012

12cos1

12sin1

θ

θ

Equation 3.1–33

Then, new values of [ ]12 −⋅ ns aM have to be determined such that E is minimised. This means

that the derivative of this equation to each coefficient [ ]12 −⋅ ns aM is zero, so:

012

=⋅∂

∂

−js aME

for: Nj ,...0=

Equation 3.1–34

This provides 1+N equations:

( )( ) ( ) [ ] ( )( )

( )( ) ( ) [ ] ( )( )

( )( ) ( )( ) ∑

∑∑

∑

=

=

=−

=−

⋅−⋅−⋅−⋅=

=

⋅−⋅⋅⋅−⋅⋅−−

⋅−⋅⋅⋅−⋅⋅−−

I

iiiii

I

iN

nins

ni

N

nins

ni

jyjx

naMj

naMj

0

0

012

012

12cos12sin

12cos112cos

12sin112sin

θθ

θθ

θθ

for: Nj ,...0=

which are rewritten as:


62

( ) [ ] ( )( )

( )( ) ( )( ) ∑

∑ ∑

=

= =−

⋅−⋅+⋅−⋅−

=

⋅−⋅⋅⋅−

I

iiiii

N

n

I

iins

n

jyjx

njaM

0

0 012

12cos12sin

22cos1

θθ

θ for: Nj ,...0=

Equation 3.1–35

To obtain the exact actual breadth and draught, the last two equations ( 1−= Nj and Nj = )in Equation 3.1–35 are replaced by the equations for the breadth at the water line and thedraught:

( ) [ ] ( )( )

( )( ) ( )( )

[ ]

( ) [ ] s

N

nns

n

s

N

nns

I

iiiii

N

n

I

iins

n

DaM

BaM

Njjyjx

njaM

=⋅⋅−

=⋅

−=⋅−⋅+⋅−⋅−

=

⋅−⋅⋅⋅−

∑

∑

∑

∑ ∑

=−

=−

=

= =−

012

012

0

0 012

1

2

2,...0 :for 12cos12sin

22cos1

θθ

θ

Equation 3.1–36

These 1+N equations can be solved numerically, so that new values for [ ]12 −⋅ ns aM will be

obtained. These new values are used instead of the initial values to obtain new iθ -values ofthe 1+I offsets again, etc. This procedure will be repeated several times and stops when thedifference between the numerical E -values of two subsequent calculations becomes less thana certain threshold value E∆ , depending on the dimensions of the cross section; for instance:

( )2

2max

2max00005.01

+⋅⋅+=∆ dbIE

Equation 3.1–37

in which:

maxb maximum half breadth of the cross section

maxd maximum draught of the cross section

Because 11 +=−a , the scale factor sM is equal to the final solution of the first coefficient

( 0=n ). The N other coefficients 12 −na can be found by dividing the final solutions of

[ ]12 −⋅ ns aM by this sM -value.

Reference is also given here to a report of de Jong [1973]. In that report several other, suitablebut more complex, methods are described to determine the scale factor sM and the conformal

mapping coefficients 12 −na from the offsets of a cross section.

Attention has been paid in SEAWAY to divergence in the calculation routines and re-entrantforms. In these cases the number N will be decreased until the divergence or re-entrancevanish. In the worse case the ''minimum'' value of N will be attained without success. One


63

can then switch to Lewis coefficients with an area coefficient of the cross section, eventuallyset to the nearest border of the valid Lewis form area.

3.1.4 Mapping Comparisons

A first example has been given here for the amidships cross section of a container vessel, witha breadth of 25.40 meter and a draught of 9.00 meter, with offsets as tabled below.

Table 3.1-1: Offsets of a cross section

For the least square method in the conformal mapping method, 33 new offsets at equidistantlength intervals on the contour of this cross section can be determined by a second degreeinterpolation routine. The calculated data of the two-parameter Lewis and the N -parameterClose-Fit conformal mapping of this amidships cross section are tabled below. The last linelists the RMS -values for the deviations of the 33 equidistant points on the approximatecontour of this cross section.

Table 3.1-2: Conformal mapping coefficients

Another example is given in Figure 3.1–5, which shows the differences between a Lewistransformation and a 10-parameter close-fit conformal mapping of a rectangular cross section.


64

Figure 3.1–5: Lewis and Close-Fit conformal mapping of a rectangle


65

3.2 Potential Theory of Tasai

In this section, the determination of the hydrodynamic coefficients of a heaving, swaying androlling cross section of a ship in deep water at zero forward speed is based on work publishedby Ursell [1949], Tasai [1959], Tasai [1960], Tasai [1961] and de Jong [1973]. Tasai'snotations have been maintained here as far as possible.

The axes system of Tasai (and used here) is given in Figure 3.2–1.

Figure 3.2–1: Tasai’s axes system for heave, sway and roll oscillations

The figure shows a cross section of an infinite long cylinder in the surface of a fluid. Thiscylinder will carry out forced harmonic heave, sway and roll motions, respectively. Using theapproach of Tasai (and de Jong), the determination of the hydrodynamic loads will be showedin the following Sections.


66

3.2.1 Heave Motions

The determination of the hydrodynamic coefficients of a heaving cross section of a ship indeep and still water at zero forward speed, as described here, is based on work published byUrsell [1949], Tasai [1959] and Tasai [1960]. Starting points for the derivation thesecoefficients here are the velocity potentials and the conjugate stream functions of the fluid, asthey have been derived by Tasai and also by de Jong [1973].

Suppose an infinite long cylinder in the surface of a fluid, of which a cross section is given inFigure 3.2–1. The cylinder is forced to carry out a simple harmonic vertical motion about itsinitial position with a frequency of oscillation ω and small amplitude of displacement ay :

( )δω +⋅⋅= tyy a cos

Equation 3.2–1

in which δ is a phase angle.Respectively, the vertical velocity and acceleration of the cylinder are:

( )δωω +⋅⋅⋅−= tyy a sin& and ( )δωω +⋅⋅⋅−= tyy a cos2&&

Equation 3.2–2

This forced vertical oscillation of the cylinder causes a surface disturbance of the fluid.Because the cylinder is supposed to be infinitely long, the generated waves will be two-dimensional. These waves travel away from the cylinder and a stationary state is rapidlyattained.

Two kinds of waves will be produced:• A standing wave system, denoted here by subscript A .

The amplitudes of these waves decrease strongly with the distance to the cylinder.• A regular progressive wave system, denoted here by subscript B .

These waves dissipate energy. At a distance of a few wavelengths from the cylinder, thewaves on each side can be described by a single regular wave train. The wave amplitude atinfinity aη is proportional to the amplitude of oscillation of the cylinder ay , provided thatthis amplitude is sufficiently small compared with the radius of the cylinder and thewavelength is not much smaller than the diameter of the cylinder.

The two-dimensional velocity potential of the fluid has to fulfil the following requirements:

1. The velocity potential must satisfy to the equation of Laplace:

02

2

2

22 =

∂Φ∂

+∂

Φ∂=Φ∇

yx

Equation 3.2–3

2. Because the heave motion of the fluid is symmetrical about the (vertical) y -axis, thisvelocity potential has the following relation:

( ) ( )yxyx ,, +Φ=−Φ

Equation 3.2–4

from which follows:


67

0=∂Φ∂θ

for: 0=θ

Equation 3.2–5

3. The linearised free surface condition in deep water is expressed as follows:

02

=∂Φ∂

+Φ⋅yg

ω for:

2sB

x ≥ and 0=y

Equation 3.2–6

In consequence of the conformal mapping, the free surface condition in Equation 3.2–6 can bewritten as:

( ) ( ) 0120

1212 =

∂Φ∂

±⋅⋅−⋅Φ⋅ ∑=

⋅−−− θσ

ξ αN

n

nn

a

b ean for: 0≥α and 2πθ ±=

in which:

sa

b Mg

⋅=2ω

σξ

or gb

b ⋅⋅

=2

02ωξ (non-dimensional frequency squared)

Equation 3.2–7

From the definition of the velocity potential follows the boundary condition on the surface ofthe cylinder for 0=α :

( )ny

yn ∂

∂⋅=

∂Φ∂ 00 &

θ

Equation 3.2–8

in which n is the outward normal of the cylinder surface.Using the stream function Ψ , this boundary condition on the surface of the cylinder ( 0=α )reduces to:

( )

( ) ( ) ( )( ) ∑=

− ⋅−⋅⋅−⋅−⋅⋅−=

∂∂

⋅=∂Ψ∂−

N

nn

ns nanMy

yy

012

00

12cos121 θ

αθθ

&

&

Equation 3.2–9

Integration results into the following requirement for the stream function on the surface of thecylinder:

( ) ( ) ( )( ) ( )tCnaMyN

nn

ns +⋅−⋅⋅−⋅⋅=Ψ ∑

=−

0120 12sin1 θθ &

Equation 3.2–10

in which ( )tC is a function of time only.

When defining:


68

( )

( ) ( )( ) ∑=

− ⋅−⋅⋅−⋅−=

⋅=

N

nn

n

a

na

bx

h

012

0

0

12sin11

2

θσ

θ

Equation 3.2–11

the stream function on the surface of the cylinder is given by:

( ) ( ) ( )tChb

y +⋅⋅−=Ψ θθ20

0 &

Equation 3.2–12

Because of the symmetry of the fluid about the y -axis, it is clear that ( ) 0=tC , so that:

( ) ( )θθ hb

y ⋅⋅−=Ψ20

0 &

Equation 3.2–13

For the standing wave system a velocity potential and a stream function satisfying theequation of Laplace, the symmetrical motion of the fluid and the free surface condition has tobe found.The following set of velocity potentials, as they are given by Tasai [1959], Tasai [1960] andde Jong [1973], fulfil these requirements:

( ) ( ) ( ) ( )

⋅⋅⋅+⋅⋅⋅⋅

⋅⋅

=Φ ∑∑∞

=

∞

= 122

122 sin,cos,

mmAm

mmAm

aA tQtP

gωθαφωθαφ

ωπη

in which:( ) ( )

( ) ( ) ( )( )∑=

⋅−+−−

⋅−

⋅−+⋅⋅⋅

−+−

⋅−⋅−

⋅⋅=N

n

nmn

n

a

b

mmA

nmeanm

n

me

0

12212

22

122cos122

121

2cos,

θσξ

θθαφ

α

α

Equation 3.2–14

The set of conjugate stream functions is expressed as:

( ) ( ) ( ) ( )

⋅⋅⋅+⋅⋅⋅⋅

⋅⋅

=Ψ ∑∑∞

=

∞

= 122

122 sin,cos,

mmAm

mmAm

aA tQtP

gωθαψωθαψ

ωπη

in which:( ) ( )

( ) ( ) ( )( )∑=

⋅−+−−

⋅−

⋅−+⋅⋅⋅

−+−

⋅−⋅−

⋅⋅=N

n

nmn

n

a

b

mmA

nmeanm

n

me

0

12212

22

122sin122

121

2sin,

θσξ

θθαψ

α

α

Equation 3.2–15

These sets of functions tend to zero as α tends to infinity.In these expressions the magnitudes of the mP2 and mQ2 series follow from the boundaryconditions as will be explained further on.

Another requirement is a diverging wave train for α goes to infinity. It is therefore necessaryto add a stream function, satisfying the free surface condition and the symmetry about the y -


69

axis, representing such a train of waves at infinity. For this, a function describing a source atthe origin O is chosen.Tasai [1959], Tasai [1960] and de Jong [1973] gave the velocity potential of the progressivewave system as:

( ) ( ) ( ) ( ) tyxtyxg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅

=Φ ωφωφωπη

sin,cos,

in which:( )

( ) ( ) ( )∫∞

⋅−⋅−

⋅−

⋅⋅+

⋅⋅−⋅⋅+⋅⋅⋅+=

⋅⋅⋅+=

022

cossinsin

cos

dkek

ykkykxe

xe

xkyBs

yBc

νννπφ

νπφ

ν

ν

while:

g

2ων = (wave number for deep water)

Equation 3.2–16

Changing the parameters provides:

( ) ( ) ( ) ( ) ttg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅

=Φ ωθαφωθαφωπη

sin,cos,

Equation 3.2–17

The conjugate stream function is given by:

( ) ( ) ( ) ( ) tyxtyxg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅

=Ψ ωψωψωπη

sin,cos,

in which:( )

( ) ( ) ( )∫∞

⋅−⋅−

⋅−

⋅⋅+

⋅⋅+⋅⋅+⋅⋅⋅−=

⋅⋅⋅+=

022

sincoscos

sin

dkek

ykkykxe

xe

xkyBs

yBc

νννπψ

νπψ

ν

ν

Equation 3.2–18


( ) ( ) ( ) ( ) ttg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅

=Ψ ωθαψωθαψωπη

sin,cos,

Equation 3.2–19

When calculating the integrals in the expressions for Bsψ and Bcψ numerically, theconvergence is very slowly.Power series expansions, as given by Porter [1960], can be used instead of these last integralsover k . Summations in these expansions converge much faster than the numeric integrationprocedure. This will be shown in the Section 3.2.2 for the sway case.

The total velocity potential and stream function to describe the waves generated by a heavingcylinder are:

BA

BA

Ψ+Ψ=ΨΦ+Φ=Φ

Equation 3.2–20


70

So the velocity potential and the conjugate stream function are expressed by:

( )( ) ( ) ( )

( ) ( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Ψ

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Φ

∑

∑

∑

∑

∞

=

∞

=

∞

=

∞

=

tQ

tPg

tQ

tPg

mmAmBs

mmAmBc

a

mmAmBs

mmAmBc

a

ωθαψθαψ

ωθαψθαψ

ωπηθα

ωθαφθαφ

ωθαφθαφ

ωπηθα

sin,,

cos,,

,

sin,,

cos,,

,

122

122

122

122

Equation 3.2–21

When putting 0=α , the stream function is equal to the expression in Equation 3.2–13, foundfrom the boundary condition on the surface of the cylinder:

( )( ) ( ) ( )

( ) ( ) ( )

( )θ

ωθψθψ

ωθψθψ

ωπηθ

hb

y

tQ

tPg

mmAmsB

mmAmcB

a

⋅⋅−=

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Ψ

∑

∑∞

=

∞

=

2

sin

cos

0

10220

10220

0

&

in which:( ) ( )

( ) ( )( )∑=

−

⋅−+⋅⋅

−+−

⋅−⋅−

⋅=N

nn

n

a

b

mA

nmanm

n

m

012

02

122sin122

121

2sin

θσξ

θθψ

Equation 3.2–22

In this expression, ( )θψ cB 0 and ( )θψ sB 0 are the values of ( )θαψ ,Bc and ( )θαψ ,Bs at thesurface of the cylinder, so for 0=α .

So for each θ , the following equation has been obtained from Equation 3.2–22:

( ) ( ) ( )

( ) ( ) ( )( )θ

ηωπ

ωθψθψ

ωθψθψh

gb

y

tQ

tP

a

mmAmsB

mmAmcB

⋅⋅⋅⋅⋅

⋅−=

⋅⋅

⋅++

⋅⋅

⋅++

∑

∑∞

=

∞

=

2sin

cos0

10220

10220

&

Equation 3.2–23

The right hand side of this equation can be written as:

( ) ( ) ( )

( ) ( ) ( ) tBtAh

ty

hhg

by b

a

a

a

⋅⋅+⋅⋅⋅=

+⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅

⋅−

ωωθ

δωξπη

θθη

ωπ

sincos

sin2

00

0&

in which:


71

δξπη

sin0 ⋅⋅⋅= ba

ayA and δξπ

ηcos0 ⋅⋅⋅= b

a

ayB

Equation 3.2–24

This results for each θ into a set of two equations:

( ) ( ) ( )

( ) ( ) ( ) 01

0220

01

0220

BhQ

AhP

mmAmsB

mmAmcB

⋅=⋅+

⋅=⋅+

∑

∑∞

=

∞

=

θθψθψ

θθψθψ

Equation 3.2–25

When putting 2πθ = , so at the intersection of the surface of the cylinder with the freesurface of the fluid where ( ) 1=θh , we obtain the coefficients 0A and 0B :

( ) ( )

( ) ( ) ∑

∑∞

=

∞

=

⋅+=

⋅+=

102200

102200

22

22

mmAmsB

mmAmcB

QB

PA

πψπψ

πψπψ

in which:

( ) ( ) ∑=

−

⋅

−+−

⋅−⋅=N

nn

m

a

bmA a

nmn

01202 122

1212

σξ

πψ

Equation 3.2–26

A substitution of 0A and 0B into the set of two equations for each θ , results for each θ-value

less than 2π in a set of two equations with the yet unknown parameters mP2 and mQ2 , so:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ∑

∑∞

=

∞

=

⋅=⋅−

⋅=⋅−

12200

12200

2

2

mmmsBsB

mmmcBcB

Qfh

Pfh

θπψθθψ

θπψθθψ

in which:( ) ( ) ( ) ( )202022 πψθθψθ mAmAm hf ⋅+−=

Equation 3.2–27

The series in these two sets of equations converges uniformly with an increasing value of m .For practical reasons the maximum value of m is limited to M , for instance 10=M .

Each θ-value less than 2π will provide an equation for the mP2 and mQ2 series. For a lot of

θ-values, the best fit values of mP2 and mQ2 are supposed to be those found by means of a

least square method. Notice that at least M values of θ , less than 2π , are required to solvethese equations.

Another favourable method is to multiply both sides of the equations with θ∆ . Then thesummation over θ can be replaced by integration.Herewith, two sets of M equations have been obtained, one set for mP2 and one set for mQ2 :


72

( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )∫∑ ∫

∫∑ ∫

⋅⋅⋅−=

⋅⋅⋅

⋅⋅⋅−=

⋅⋅⋅

=

=

2

0200

1

2

0222

2

0200

1

2

0222

2

2

ππ

ππ

θθπψθθψθθθ

θθπψθθψθθθ

dfhdffQ

dfhdffP

nsBsB

M

mnmm

ncBcB

M

mnmm

for: Mn ,...1=

Equation 3.2–28

Now the mP2 and mQ2 series can be solved by a numerical method and with these values, the

coefficients 0A and 0B are known too.

From the definition of these coefficients in Equation 3.2–24 follows the amplitude ratio of theradiated waves and the forced heave oscillation:

20

20 BAy

b

a

a

+

⋅=

ξπη

Equation 3.2–29

With the solved mP2 and mQ2 values, the velocity potential on the surface of the cylinder( 0=α ) is known too:

( )( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Φ

∑

∑∞

=

∞

=

tQ

tPg

mmAmsB

mmAmcB

a

ωθφθφ

ωθφθφ

ωπηθ

sin

cos

10220

10220

0

in which:( ) ( )

( ) ( )( )∑=

−

⋅−+⋅⋅

−+−

⋅−⋅−

⋅=N

nn

n

a

b

mA

nmanm

n

m

012

02

122cos122

121

2cos

θσξ

θθφ

Equation 3.2–30

In this expression, ( )θφ cB0 and ( )θφ sB0 are the values of ( )θαφ ,Bc and ( )θαφ ,Bs at thesurface of the cylinder, so for 0=α .

3.2.1.1 Pressure Distribution During Heave Motions

Now the hydrodynamic pressure on the surface of the cylinder can be obtained from thelinearised equation of Bernoulli:


73

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅+−

⋅⋅

⋅++

⋅⋅⋅−

=

∂Φ∂

⋅−=

∑

∑

=

=

tP

tQg

tp

M

mmAmcB

M

mmAmsB

a

ωθφθφ

ωθφθφ

πηρ

θρθ

sin

cos

10220

10220

0

Equation 3.2–31

It is obvious that this pressure is symmetric in θ .

3.2.1.2 Heave Coefficients

The two-dimensional hydrodynamic vertical force, acting on the cylinder in the direction ofthe y -axis, can be found by integrating the vertical component of the hydrodynamic pressureon the surface of the cylinder:

( )

( ) θθ

θ

θ

π

π

π

dddx

p

dsdsdx

pFy

⋅⋅⋅−=

⋅⋅−=

∫

∫+

−

02

0

02

2

'

2

Equation 3.2–32

With this the two-dimensional hydrodynamic vertical force due to heave oscillations can bewritten as follows:

( ) ( )( )tNtMbg

F ay ⋅⋅−⋅⋅⋅

⋅⋅⋅= ωω

πηρ

sincos 000'

in which:

( ) ( ) ( ) ( )( )

( ) ( )( ) ( )

( ) ( )

⋅⋅−⋅⋅−+⋅⋅

+

⋅−−

−⋅⋅−⋅−

⋅⋅−⋅⋅−⋅−⋅⋅−=

∑ ∑

∑ ∑

∫ ∑

=

−

=−+−

= =−

=−

N

m

mN

nnmnm

m

a

b

M

m

N

nnm

m

a

N

nn

nsB

a

aanQQ

anm

nQ

dnanM

1 012212222

1 01222

2

2

2

0 01200

1214

12212

11

12cos1211

σξπ

σ

θθθφσ

π

and 0N as obtained from this expression above for 0M , by replacing there ( )θφ sB0 by

( )θφ cB0 and mQ2 by mP2 .

Equation 3.2–33

For the determination of 0M and 0N , it is required that NM ≥ . These expressions coincidewith those as given by Tasai [1960].

With Equation 3.2–33 in some other format:


74

( ) ( )( )

0

0

000'

cos

sin

sincos

By

Ay

tNtMbg

F

ba

a

ba

a

ay

⋅⋅⋅

=

⋅⋅⋅

=

−+⋅⋅−−+⋅⋅⋅⋅⋅⋅

=

ξπη

δ

ξπηδ

δδωδδωπ

ηρ

Equation 3.2–34

the two-dimensional hydrodynamic vertical force can be resolved into components in phaseand out phase with the vertical displacement of the cylinder:

( ) ( ) ( ) ( ) δωδω

ξπηρ

+⋅⋅⋅−⋅++⋅⋅⋅+⋅

⋅⋅⋅⋅⋅⋅

=

tBNAMtANBM

ybg

Fab

ay

sincos 00000000

2

20'

Equation 3.2–35

This hydrodynamic vertical force can also be written as:

( ) ( )δωωδωω +⋅⋅⋅⋅++⋅⋅⋅⋅=

⋅−⋅−=

tyNtyM

yNyMF

aa

y

sincos '33

2'33

'33

'33

' &&&

Equation 3.2–36

in which:'

33M 2-D hydrodynamic mass coefficient of heave'

33N 2-D hydrodynamic damping coefficient of heave

When using also the amplitude ratio of the radiated waves and the forced heave oscillation,found before in Equation 3.2–29, the two-dimensional hydrodynamic mass and dampingcoefficients of heave are given by:

ωρ

ρ

⋅+

⋅−⋅⋅

⋅=

+

⋅+⋅⋅

⋅=

20

20

00002

0'33

20

20

00002

0'33

2

2

BA

BNAMbN

BA

ANBMbM

Equation 3.2–37

The signs of these two coefficients are proper in both, the axes system of Tasai and the shipmotions ( )zyxO ,, co-ordinate system.

The energy delivered by the exciting forces should be equal to the energy radiated by thewaves, so:

( ) ( )2

1 2

0

'33

cgdtyyN

Ta

T

osc

osc ⋅⋅⋅=⋅⋅⋅⋅ ∫

ηρ&&

Equation 3.2–38

in which oscT is the period of oscillation.


75

With the relation for the wave speed ωgc = at deep water, follows the relation between thetwo-dimensional heave damping coefficient and the amplitude ratio of the radiated waves andthe forced heave oscillation:

2

3

2'

33

⋅

⋅=

a

a

yg

Nη

ωρ

Equation 3.2–39

With this amplitude ratio the two-dimensional hydrodynamic damping coefficient of heave isalso given by:

ωπρ

⋅+

⋅⋅⋅

= 20

20

20

2'

33

14 BA

bN

Equation 3.2–40

When comparing this expression for '33N with the expression found before, the following

energy balance relation is found:

2

2

0000

π=⋅−⋅ BNAM

Equation 3.2–41


76

3.2.2 Sway Motions

The determination of the hydrodynamic coefficients of a swaying cross section of a ship indeep and still water at zero forward speed is based here on work published by Tasai [1961] forthe Lewis conformal mapping method. Starting points for the derivation these coefficientshere are the velocity potentials and the conjugate stream functions of the fluid as they havebeen derived by Tasai [1961] and also by de Jong [1973].Suppose an infinite long cylinder in the surface of a fluid, of which a cross section is given inFigure 3.2–1. The cylinder is forced to carry out a simple harmonic lateral motion about itsinitial position with a frequency of oscillation ω and small amplitude of displacement ax :

( )εω +⋅⋅= txx a cos

Equation 3.2–42

in which ε is a phase angle.Respectively, the lateral velocity and acceleration of the cylinder are:

( )εωω +⋅⋅⋅−= txx a sin& and ( )εωω +⋅⋅⋅−= txx a cos2&&

Equation 3.2–43

This forced lateral oscillation of the cylinder causes a surface disturbance of the fluid.Because the cylinder is supposed to be infinitely long, the generated waves will be two-dimensional. These waves travel away from the cylinder and a stationary state is rapidlyattained.



These waves dissipate energy. At a distance of a few wavelengths from the cylinder, thewaves on each side can be described by a single regular wave train. The wave amplitude atinfinity aη is proportional to the amplitude of oscillation of the cylinder ax , provided thatthis amplitude is sufficiently small compared with the radius of the cylinder and thewavelength is not much smaller than the diameter of the cylinder.



02

2

2

22 =

∂Φ∂

+∂

Φ∂=Φ∇

yx

Equation 3.2–44

2. Because the sway motion of the fluid is not symmetrical about the y -axis, this velocitypotential has the following anti-symmetric relation:

( ) ( )yxyx ,, +Φ−=−Φ

Equation 3.2–45



77

02

=∂Φ∂

+Φ⋅yg

ω for:

2sB

x ≥ and 0=y

Equation 3.2–46

In consequence of the conformal mapping, this free surface condition can be written as:

( ) ( ) 0120

1212 =

∂Φ∂

±⋅⋅−⋅Φ⋅ ∑=

⋅−−− θσ

ξ αN

n

nn

a


in which:

sa

b Mg

⋅=2ω

σξ

or gb

b ⋅⋅

=2


Equation 3.2–47

From the definition of the velocity potential follows the boundary condition on the surface ofthe cylinder S for 0=α :

( )nx

xn ∂

∂⋅=

∂Φ∂ 00 &

θ

Equation 3.2–48

in which n is the outward normal of the cylinder surface S .

Using the stream function Ψ , this boundary condition on the surface of the cylinder ( 0=α )reduces to:

( )

( ) ( ) ( )( ) ∑=

− ⋅−⋅⋅−⋅−⋅⋅−=

∂∂

⋅−=∂

Ψ∂

N

nn

ns nanMx

xx

012

00

12sin121 θ

αθθ

&

&

Equation 3.2–49


( ) ( ) ( )( ) ( )tCnaMxN

nn

ns +⋅−⋅⋅−⋅⋅=Ψ ∑

=−

0120 12cos1 θθ &

Equation 3.2–50


When defining:

( )

( ) ( )( ) ∑=

− ⋅−⋅⋅−⋅=

⋅=

N

nn

n

a

na

by

g

012

0

0

12cos11

2

θσ

θ

Equation 3.2–51



78

( ) ( ) ( )tCgb

x +⋅⋅−=Ψ θθ20

0 &

Equation 3.2–52

For the standing wave system a velocity potential and a stream function satisfying to theequation of Laplace, the non-symmetrical motion of the fluid and the free surface conditionhas to be found.

The following set of velocity potentials, as they are given by Tasai [1961] and de Jong [1973],fulfil these requirements:

( ) ( ) ( ) ( )

⋅⋅⋅+⋅⋅⋅⋅

⋅⋅

=Φ ∑∑∞

=

∞

= 122

122 sin,cos,

mmAm

mmAm

aA tQtP

gωθαφωθαφ

ωπη

in which:( ) ( ) ( )( )

( ) ( ) ( )( )∑=

⋅+−−

⋅+−

⋅+⋅⋅⋅

+−

⋅−⋅−

⋅+⋅+=N

n

nmn

n

a

b

mmA

nmeanm

n

me

0

2212

122

22sin2212

1

12sin,

θσξ

θθαφ

α

α

Equation 3.2–53


( ) ( ) ( ) ( )

⋅⋅⋅+⋅⋅⋅⋅

⋅⋅

=Ψ ∑∑∞

=

∞

= 122

122 sin,cos,

mmAm

mmAm

aA tQtP

gωθαψωθαψ

ωπη

in which:( ) ( ) ( )( )

( ) ( ) ( )( )∑=

⋅+−−

⋅+−

⋅+⋅⋅⋅

+−

⋅−⋅+

⋅+⋅−=N

n

nmn

n

a

b

mmA

nmeanm

n

me

0

2212

122

22cos2212

1

12cos,

θσξ

θθαψ

α

α

Equation 3.2–54

These sets of functions tend to zero as α tends to infinity.

In these expressions the magnitudes of the mP2 and mQ2 series follow from the boundaryconditions as will be explained further on.

Another requirement is a diverging wave train for α goes to infinity. It is therefore necessaryto add a stream function, satisfying the equation of Laplace, the non-symmetrical motion andthe free surface condition, representing such a train of waves at infinity. For this, a functiondescribing a two-dimensional horizontal doublet at the origin O is chosen.Tasai [1961] and de Jong [1973] gave the velocity potential of the progressive wave systemas:

( ) ( ) ( ) ( ) tyxtyxg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅

=Φ ωφωφωπη

sin,cos,

in which:


79

( )

( ) ( ) ( )

( )22

022

sincoscos

sin

yxx

dkek

ykkykxej

xej

xkyBs

yBc

+⋅+

⋅⋅+

⋅⋅+⋅⋅−⋅⋅⋅+=⋅

⋅⋅⋅−=⋅

∫∞

⋅−⋅−

⋅−

ν

νννπφ

νπφ

ν

ν

while:1+=j for: 0>x1−=j for: 0<x

g


Equation 3.2–55


( ) ( ) ( ) ( ) ttg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅


sin,cos,

Equation 3.2–56


( ) ( ) ( ) ( ) tyxtyxg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅

=Ψ ωψωψωπη

sin,cos,

in which:( )

( ) ( ) ( )

( )22

022

cossinsin

cos

yxy

dkek

ykkykxe

xe

xkyBs

yBc

+−

⋅⋅+

⋅⋅−⋅⋅+⋅⋅⋅+=

⋅⋅⋅+=

∫∞

⋅−⋅−

⋅−

ν

νννπψ

νπψ

ν

ν

Equation 3.2–57


( ) ( ) ( ) ( ) ttg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅


sin,cos,

Equation 3.2–58

When calculating the integrals in the expressions for Bsφ and Bcφ numerically, theconvergence is very slowly.Power series expansions, as given by Porter [1960], can be used instead of these last integralsover k :

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) yxk

yxk

exSxQdkek

ykkyk

exSxQdkek

ykkyk

⋅−∞

⋅−

⋅−∞

⋅−

⋅⋅⋅−+⋅⋅=⋅⋅+

⋅⋅−⋅⋅

⋅⋅⋅−−⋅⋅=⋅⋅+

⋅⋅+⋅⋅

∫

∫

ν

ν

νπνν

ν

νπνν

ν

sincoscossin

cossinsincos

022

022

in which:


80

( ) ( )

( )

( )constant)(Euler .....57722.0

!

arctan

sin

cosln

22

1

1

22

=⋅+⋅

=

=

⋅⋅+=

⋅⋅++⋅+=

∑

∑∞

=

∞

=

γ

ν

β

ββ

βνγ

nnyx

p

yx

npS

npyxQ

n

n

nn

nn

Equation 3.2–59

The summations in these expansions converge much faster than the numeric integrationprocedure. A suitable maximum value for n should be chosen, Nn ,...1= .

The total velocity potential and stream function to describe the waves generated by a swayingcylinder are:

BA

BA

Ψ+Ψ=ΨΦ+Φ=Φ

Equation 3.2–60


( )( ) ( ) ( )

( ) ( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Ψ

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Φ

∑

∑

∑

∑

∞

=

∞

=

∞

=

∞

=

tQ

tPg

tQ

tPg

mmAmBs

mmAmBc

a

mmAmBs

mmAmBc

a

ωθαψθαψ

ωθαψθαψ

ωπηθα

ωθαφθαφ

ωθαφθαφ

ωπηθα

sin,,

cos,,

,

sin,,

cos,,

,

122

122

122

122

Equation 3.2–61

When putting 0=α , the stream function is equal to the expression found before in Equation3.2–52 from the boundary condition on the surface of the cylinder:

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( )tCgb

x

tQ

tPg

mmAmsB

mmAmcB

a

+⋅⋅=

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Ψ

∑

∑∞

=

∞

=

θ

ωθψθψ

ωθψθψ

ωπηθ

2

sin

cos

0

10220

10220

0

&


81

in which:( ) ( )( )

( ) ( )( )∑=

−

⋅+⋅⋅

+−

⋅−⋅+

⋅+−=N

nn

n

a

b

mA

nmanm

n

m

012

02

22cos2212

1

12cos

θσξ

θθψ

Equation 3.2–62


So for each θ , the following equation has been obtained from Equation 3.2–62:

( ) ( ) ( )

( ) ( ) ( )( ) ( )tCg

gbx

tQ

tP

a

mmAmsB

mmAmcB

*0

10220

10220

2sin

cos

+⋅⋅⋅⋅⋅⋅=

⋅⋅

⋅++

⋅⋅

⋅++

∑

∑∞

=

∞

= θη

ωπ

ωθψθψ

ωθψθψ&

Equation 3.2–63

When putting 2πθ = , so at the intersection of the surface of the cylinder with the free

surface of the fluid where ( ) 0=θg , we obtain the constant ( )tC* :

( )( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅++

⋅⋅

⋅++

=

∑

∑∞

=

∞

=

tQ

tP

tC

mmAmsB

mmAmcB

ωπψπψ

ωπψπψ

sin22

cos22

10220

10220

*

in which:

( ) ( ) ∑=

−

⋅

+−

⋅−⋅=N

nn

m

a

bmA a

nmn

01202 22

1212

σξπψ

Equation 3.2–64

A substitution of ( )tC* in the equation for each θ-value, results for each θ-value less than2π into the following equation:

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

( )θη

ωπ

ωπψθψπψθψ

ωπψθψπψθψ

gg

bx

tQ

tP

a

mmAmAmsBsB

mmAmAmcBcB

⋅⋅⋅⋅⋅

⋅

=

⋅⋅

−⋅+−+

⋅⋅

−⋅+−+

∑

∑∞

=

∞

=

2

sin22

cos22

0

10202200

10202200

&

Equation 3.2–65


( ) ( ) ( )

( ) ( ) ( ) tQtPg

tx

ggg

bx b

a

a

a

⋅⋅+⋅⋅⋅=

+⋅⋅⋅⋅⋅−=⋅⋅⋅⋅⋅

⋅

ωωθ

εωξπη

θθη

ωπ

sincos

sin2

00

0&

in which:


82

εξπη

sin0 ⋅⋅⋅−= ba

axP and εξπ

ηcos0 ⋅⋅⋅−= b

a

axQ

Equation 3.2–66

This provides for each θ-value less than 2π a set of two equations with the unknownparameters mP2 and mQ2 :

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ∑

∑∞

=

∞

=

⋅+⋅=−

⋅+⋅=−

122000

122000

2

2

mmmsBsB

mmmcBcB

QfQg

PfPg

θθπψθψ

θθπψθψ

in which:( ) ( ) ( )202022 πψθψθ mAmAmf +−=

Equation 3.2–67

These equations can also be written as:

( ) ( ) ( )

( ) ( ) ( ) ∑

∑∞

=

∞

=

⋅=−

⋅=−

02200

02200

2

2

mmmsBsB

mmmcBcB

Qf

Pf

θπψθψ

θπψθψ

in which:for :0=m ( ) ( )θθ gf =0

for 0>m : ( ) ( ) ( )202022 πψθψθ mAmAmf +−=

Equation 3.2–68



θ-values, the best fit values of mP2 and mQ2 are supposed to be those found by means of aleast squares method. Notice that at least 1+M values of θ , less than 2π , are required tosolve these equations.

Another favourable method is to multiply both sides of the equations with θ∆ . Then thesummation over θ can be replaced by integration.Herewith, two sets of 1+M equations have been obtained, one set for mP2 and one set for

mQ2 :

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )∫∑ ∫

∫∑ ∫

⋅⋅−=

⋅⋅⋅

⋅⋅−=

⋅⋅⋅

=

=

2

0

2000

2

0

222

2

0200

0

2

0222

2

2

ππ

ππ

θθπψθψθθθ

θθπψθψθθθ

dfdffQ

dfdffP

nsBsB

M

mnmm

ncBcB

M

mnmm

for: Mn ,...0=

Equation 3.2–69


83


coefficients 0P and 0Q are known now and from the definition of these coefficients inEquation 3.2–66 follows the amplitude ratio of the radiated waves and the forced swayoscillation:

20

20 QPx

b

a

a

+

⋅=

ξπη

Equation 3.2–70


( )( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Φ

∑

∑

=

=

tQ

tPg

M

mmAmsB

M

mmAmcB

a

ωθφθφ

ωθφθφ

ωπηθ

sin

cos

10220

10220

0

in which:( ) ( )( )

( ) ( )( )∑=

−

⋅+⋅⋅

+−

⋅−⋅−

⋅++=N

nn

n

a

b

mA

nmanm

n

m

012

02

22sin2212

1

12sin

θσξ

θθφ

Equation 3.2–71

In this expression, ( )θφ cB0 and ( )θφ sB0 are the values of ( )θαφ ,Bc and ( )θαφ ,Bs at thesurface of the cylinder, so for 0=α .

3.2.2.1 Pressure Distribution During Sway Motions


( ) ( )

( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅+−

⋅⋅

⋅++

⋅⋅⋅−

=

∂Φ∂

⋅−=

∑

∑∞

=

∞

=

tP

tQg

tp

mmAmcB

mmAmsB

a

ωθφθφ

ωθφθφ

πηρ

θρθ

sin

cos

10220

10220

0

Equation 3.2–72

It is obvious that this pressure is skew-symmetric in θ .

3.2.2.2 Sway Coefficients


84

The two-dimensional hydrodynamic horizontal force, acting on the cylinder in the direction ofthe x -axis, can be found by integrating the horizontal component of the hydrodynamicpressure on the surface S of the cylinder:

( ) ( )

( ) θθ

θ

θθ

π

π

dddy

p

dsdsdy

ppFx

⋅⋅⋅=

⋅−

⋅−−+−=

∫

∫+

0

2

0

0

2

0

'

2

Equation 3.2–73

With this the two-dimensional hydrodynamic horizontal force due to sway oscillations can bewritten as follows:

( ) ( )( )tNtMbg

F ax ⋅⋅−⋅⋅⋅

⋅⋅⋅−= ωω

πηρ

sincos 000'

in which:

( ) ( ) ( ) ( )( )

( ) ( )

( ) ( ) ( )( ) ( )∑ ∑∑

∑

∫ ∑

= = =−−

−

=+

=−

⋅⋅−−+

−⋅−⋅⋅−+

⋅+⋅⋅−⋅⋅

+

⋅⋅−⋅⋅−⋅−⋅⋅−=

M

m

N

n

N

iinm

m

a

b

N

mmm

m

a

N

nn

nsB

a

aanim

inQ

amQ

dnanM

1 0 012122222

1

1122

2

0 01200

12221212

1

1214

12sin1211

σξ

σπ

θθθφσ

π



Equation 3.2–74

For the determination of 0M and 0N , it is required that NM ≥ .


( ) ( )( )

0

0

000'

cos

sin

sincos

Qx

Px

tNtMbg

F

ba

a

ba

a

ax

⋅⋅⋅

−=

⋅⋅⋅

−=

−+⋅⋅−−+⋅⋅⋅⋅⋅⋅−

=

ξπηε

ξπηε

εεωεεωπ

ηρ

Equation 3.2–75

the two-dimensional hydrodynamic horizontal force can be resolved into components in phaseand out phase with the horizontal displacement of the cylinder:

( ) ( ) ( ) ( ) εωεω

ξπηρ

+⋅⋅⋅−⋅++⋅⋅⋅+⋅

⋅⋅⋅⋅⋅⋅

=

tQNPMtPNQM

xbg

Fab

ax

sincos 00000000

2

20'

Equation 3.2–76


85


( ) ( )εωωεωω +⋅⋅⋅⋅++⋅⋅⋅⋅=

⋅−⋅−=

txNtxM

xNxMF

aa

x

sincos '22

2'22

'22

'22

' &&&

Equation 3.2–77

in which:'

22M 2-D hydrodynamic mass coefficient of sway'

22N 2-D hydrodynamic damping coefficient of swayWhen using also the amplitude ratio of the radiated waves and the forced sway oscillation,found before in Equation 3.2–70, the two-dimensional hydrodynamic mass and dampingcoefficients of sway are given by:

ωρ

ρ

⋅+

⋅−⋅⋅

⋅=

+⋅+⋅

⋅⋅

=

20

20

00002

0'22

20

20

00002

0'22

2

2

QPQNPMb

N

QPPNQMb

M

Equation 3.2–78


The energy delivered by the exciting forces should be equal to the energy radiated by thewaves, so:

( ) ( )2

1 2

0

'22

cgdtxxN

Ta

T

osc

osc ⋅⋅⋅=⋅⋅⋅⋅ ∫

ηρ&&

Equation 3.2–79


With the relation for the wave speed ωgc = at deep water, follows the relation between thetwo-dimensional heave damping coefficient and the amplitude ratio of the radiated waves andthe forced sway oscillation:

2

3

2'

22

⋅

⋅=

a

a

xg

Nη

ωρ

Equation 3.2–80


ωπρ⋅

+⋅

⋅⋅= 2

02

0

20

2'

22

14 QP

bN

Equation 3.2–81




86

2

2

0000

π=⋅−⋅ QNPM

Equation 3.2–82

3.2.2.3 Coupling of Sway into Roll

In the case of a sway oscillation generally a roll moment is produced. The hydrodynamicpressure is skew-symmetric in θ .The two-dimensional hydrodynamic moment acting on the cylinder in the clockwise directioncan be found by integrating the roll component of the hydrodynamic pressure on the surfaceS of the cylinder:

( ) ( )

( ) θθθ

θ

θθ

π

π

dddy

yddx

xp

dsdsdy

ydsdx

xppM R

⋅

⋅+⋅⋅⋅−=

⋅

−

⋅++

⋅−⋅−−+=

∫

∫+

00

00

2

0

00

00

2

0

'

2

Equation 3.2–83

With this the two-dimensional hydrodynamic roll moment due to sway oscillations can bewritten as follows:

( ) ( )( )tXtYbg

M RRa

R ⋅⋅−⋅⋅⋅⋅⋅⋅

= ωωπ

ηρsincos

20'

in which:

( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( )

( )( ) ( )

( ) ( )∑

∑ ∑

∑ ∑

∑ ∑∑

∫ ∑∑

=−

= +=−+−−−−

=

−

=−−+−−−

= = =−−

= =−−

+

⋅⋅⋅

−−⋅−+−−

+

⋅⋅⋅

−−⋅−−+−

+

⋅⋅−

⋅

⋅+

⋅⋅−−+

−⋅−⋅⋅−⋅

⋅+

⋅⋅−⋅⋅⋅−⋅−⋅⋅⋅

=

N

mmN

n

N

nmiinmin

N

mn

mn

iinmin

mm

a

b

M

m

N

n

N

iinm

m

a

N

n

N

iin

insB

a

R

aaain

iinm

aaain

iinm

Q

aainm

iniQ

dinaaiY

1

012221212

012221212

2

3

1 0 012122222

2

0 0 0121202

22121222

22121222

1

8

2212

22121

2

1

22sin1212

1

σξπ

σ

θθθφσ

π

and RX as obtained from this expression above for RY , by replacing there ( )θφ sB0 by


Equation 3.2–84



87

( ) ( )( )

0

0

20'

cos

sin

sincos

Qx

Px

tXtYbg

M

ba

a

ba

a

RRa

R

⋅⋅⋅

−=

⋅⋅⋅

−=

−+⋅⋅−−+⋅⋅⋅⋅⋅⋅

=

ξπηε

ξπηε

εεωεεωπ

ηρ

Equation 3.2–85

the two-dimensional hydrodynamic roll moment can be resolved into components in phaseand out phase with the lateral displacement of the cylinder:

( ) ( ) ( ) ( )( )εωεωξπ

ηρ

+⋅⋅⋅−⋅++⋅⋅⋅+⋅

⋅⋅⋅

⋅⋅⋅−=

tQXPYtPXQY

xbg

M

RRRR

ab

aR

sincos 0000

2

220'

Equation 3.2–86

This hydrodynamic roll moment can also be written as:

( ) ( )εωωεωω +⋅⋅⋅⋅++⋅⋅⋅⋅=

⋅−⋅−=

txNtxM

xNxMM

aa

R

sincos '42

2'42

'42

'42

' &&&

Equation 3.2–87

in which:'

42M 2-D hydrodynamic mass coupling coefficient of sway into roll'

42N 2-D hydrodynamic damping coupling coefficient of sway into roll

When using also the amplitude ratio of the radiated waves and the forced sway oscillation,found before, the two-dimensional hydrodynamic mass and damping coupling coefficients ofsway into roll in Tasai's axes system are given by:

ωρ

ρ

⋅+

⋅−⋅⋅

⋅−=

+⋅+⋅

⋅⋅−

=

20

20

003

0'42

20

20

003

0'42

2

2

QPQXPYb

N

QPPXQYb

M

RR

RR

Equation 3.2–88

In the ship motions ( )zyxO ,, co-ordinate system these two coupling coefficients will changesign.


88

3.2.3 Roll Motions

The determination of the hydrodynamic coefficients of a rolling cross section of a ship in deepand still water at zero forward speed, is based here on work published by Tasai [1961] for theLewis method. Starting points for the derivation these coefficients here are the velocitypotentials and the conjugate stream functions of the fluid as they have been derived by Tasaiand also by de Jong [1973].Suppose an infinite long cylinder in the surface of a fluid, of which a cross section is given inFigure 3.2–1. The cylinder is forced to carry out a simple harmonic roll motion about theorigin O with a frequency of oscillation ω and small amplitude of displacement aβ :

( )γωββ +⋅⋅= ta cos

Equation 3.2–89

in which γ is a phase angle.Respectively, the angular velocity and acceleration of the cylinder are:

( )γωβωβ +⋅⋅⋅−= ta sin& and ( )γωβωβ +⋅⋅⋅−= ta cos2&&

Equation 3.2–90

This forced angular oscillation of the cylinder causes a surface disturbance of the fluid.Because the cylinder is supposed to be infinitely long, the generated waves will be two-dimensional. These waves travel away from the cylinder and a stationary state is rapidlyattained.



These waves dissipate energy. At a distance of a few wavelengths from the cylinder, thewaves on each side can be described by a single regular wave train. The wave amplitude atinfinity aη is proportional to the amplitude of oscillation of the cylinder aβ , provided thatthis amplitude is sufficiently small compared with the radius of the cylinder and the wavelength is not much smaller than the diameter of the cylinder.



02

2

2

22 =

∂Φ∂

+∂

Φ∂=Φ∇

yx

Equation 3.2–91

2. Because the sway motion of the fluid is not symmetrical about the y -axis, this velocitypotential has the following anti-symmetric relation:

( ) ( )yxyx ,, +Φ−=−Φ

Equation 3.2–92



89

02

=∂Φ∂

+Φ⋅yg

ω for:

2sB

x ≥ and 0=y

Equation 3.2–93

In consequence of the conformal mapping, this free surface condition can be written as:

( ) ( ) 0120

1212 =

∂Φ∂

±⋅⋅−⋅Φ⋅ ∑=

⋅−−− θσ

ξ αN

n

nn

a


in which:

sa

b Mg

⋅=2ω

σξ

or gb

b ⋅⋅

=2


Equation 3.2–94

From the definition of the velocity potential follows the boundary condition on the surface ofthe cylinder S for 0=α :

( )sr

rn ∂

∂⋅⋅=

∂Φ∂ 0

00 βθ &

Equation 3.2–95

in which n is the outward normal of the cylinder surface S and 0r is the radius from theorigin to the surface of the cylinder.Using the stream function Ψ , this boundary condition on the surface of the cylinder ( 0=α )reduces to:

( )

+∂∂⋅=

∂Ψ∂−

2

20

200 yx

ssβθ &

Equation 3.2–96


( ) ( ) ( )tCyx ++⋅−=Ψ 20

200 2

βθ&

Equation 3.2–97


The vertical oscillation at the intersection of the surface of the cylinder and the waterline isdefined by:

( )γωχβχ +⋅⋅=⋅= tb

a sin20

Equation 3.2–98

When defining:


90

( )( )

( ) ( )( )

( ) ( )( ) 2

012

2

012

20

20

20

12cos11

12sin11

2

⋅−⋅⋅−⋅++

⋅−⋅⋅−⋅−=

+=

∑

∑

=−

=−

N

nn

n

a

N

nn

n

a

na

na

byx

θσ

θσ

θµ

Equation 3.2–99


( ) ( ) ( )tCb

+⋅⋅−=Ψ θµχθ40

0 &

Equation 3.2–100

For the standing wave system a velocity potential and a stream function satisfying to theequation of Laplace, the non-symmetrical motion of the fluid and the free surface conditionhas to be found.

The following set of velocity potentials, as they are given by Tasai [1961] and de Jong [1973],fulfil these requirements:

( ) ( ) ( ) ( )

⋅⋅⋅+⋅⋅⋅⋅

⋅⋅

=Φ ∑∑∞

=

∞

= 122

122 sin,cos,

mmAm

mmAm

aA tQtP

gωθαφωθαφ

ωπη

in which:( ) ( ) ( )( )

( ) ( ) ( )( )∑=

⋅+−−

⋅+−

⋅+⋅⋅⋅

+−

⋅−⋅−

⋅+⋅+=N

n

nmn

n

a

b

mmA

nmeanm

n

me

0

2212

122

22sin2212

1

12sin,

θσξ

θθαφ

α

α

Equation 3.2–101


( ) ( ) ( ) ( )

⋅⋅⋅+⋅⋅⋅⋅

⋅⋅

=Ψ ∑∑∞

=

∞

= 122

122 sin,cos,

mmAm

mmAm

aA tQtP

gωθαψωθαψ

ωπη

in which:( ) ( ) ( )( )

( ) ( ) ( )( )∑=

⋅+−−

⋅+−

⋅+⋅⋅⋅

+−

⋅−⋅+

⋅+⋅−=N

n

nmn

n

a

b

mmA

nmeanm

n

me

0

2212

122

22cos2212

1

12cos,

θσξ

θθαψ

α

α

Equation 3.2–102

These sets of functions tend to zero as α tends to infinity.In these expressions the magnitudes of the mP2 and the mQ2 series follow from the boundaryconditions as will be explained further on.

Another requirement is a diverging wave train for α goes to infinity. It is therefore necessaryto add a stream function, satisfying the equation of Laplace, the non-symmetrical motion and


91

the free surface condition, representing such a train of waves at infinity. For this, a functiondescribing a two-dimensional horizontal doublet at the origin O is chosen.Tasai [1961] and de Jong [1973] gave the velocity potential of the progressive wave systemas:

( ) ( ) ( ) ( ) tyxtyxg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅

=Φ ωφωφωπη

sin,cos,

in which:( )

( ) ( ) ( )

( )22

022

sincoscos

sin

yxx

dkek

ykkykxej

xej

xkyBs

yBc

+⋅+

⋅⋅+

⋅⋅+⋅⋅−⋅⋅⋅+=⋅

⋅⋅⋅−=⋅

∫∞

⋅−⋅−

⋅−

ν

νννπφ

νπφ

ν

ν

while:1+=j for: 0>x1−=j for: 0<x

g


Equation 3.2–103


( ) ( ) ( ) ( ) ttg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅


sin,cos,

Equation 3.2–104


( ) ( ) ( ) ( ) tyxtyxg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅

=Ψ ωψωψωπη

sin,cos,

in which:( )

( ) ( ) ( )

( )22

022

cossinsin

cos

yxy

dkek

ykkykxe

xe

xkyBs

yBc

+−

⋅⋅+

⋅⋅−⋅⋅+⋅⋅⋅+=

⋅⋅⋅+=

∫∞

⋅−⋅−

⋅−

ν

νννπψ

νπψ

ν

ν

Equation 3.2–105


( ) ( ) ( ) ( ) ttg

BsBca

B ⋅⋅+⋅⋅⋅⋅⋅


sin,cos,

Equation 3.2–106

When calculating the integrals in the expressions for Bsψ and Bcψ numerically, theconvergence is very slowly.


92

Power series expansions, as given by Porter [1960], can be used instead of these last integralsover k . Summations in these expansions converge much faster than the numeric integrationprocedure. This has been showed for the sway case.

The total velocity potential and stream function to describe the waves generated by a swayingcylinder are:

BA

BA

Ψ+Ψ=ΨΦ+Φ=Φ

Equation 3.2–107


( )( ) ( ) ( )

( ) ( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Ψ

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Φ

∑

∑

∑

∑

∞

=

∞

=

∞

=

∞

=

tQ

tPg

tQ

tPg

mmAmBs

mmAmBc

a

mmAmBs

mmAmBc

a

ωθαψθαψ

ωθαψθαψ

ωπηθα

ωθαφθαφ

ωθαφθαφ

ωπηθα

sin,,

cos,,

,

sin,,

cos,,

,

122

122

122

122

Equation 3.2–108

When putting 0=α , the stream function is equal to the expression found before in Equation3.2–100 from the boundary condition on the surface of the cylinder:

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( )tCb

tQ

tPg

mmAmsB

mmAmcB

a

+⋅⋅−=

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Ψ

∑

∑∞

=

∞

=

θµχ

ωθψθψ

ωθψθψ

ωπηθ

4

sin

cos

0

10220

10220

0

&

in which:( ) ( )( )

( ) ( )( )∑=

−

⋅+⋅⋅

+−

⋅−⋅+

⋅+−=N

nn

n

a

b

mA

nmanm

n

m

012

02

22cos2212

1

12cos

θσξ

θθψ

Equation 3.2–109


So for each θ , the following equation has been obtained:


93

( ) ( ) ( )

( ) ( ) ( )( ) ( )tC

gb

tQ

tP

a

mmAmsB

mmAmcB

*0

10220

10220

4sin

cos

+⋅⋅⋅⋅⋅⋅−=

⋅⋅

⋅++

⋅⋅

⋅++

∑

∑∞

=

∞

= θµη

ωπχωθψθψ

ωθψθψ&

Equation 3.2–110

When putting 2πθ = , so at the intersection of the surface of the cylinder with the free

surface of the fluid where ( ) 1=θµ , we obtain the constant ( )tC* :

( ) ( ) ( ) ( )

( ) ( ) ( )

a

mmAmsB

mmAmcB

gb

tQ

tPtC

ηωπχ

ωπψπψ

ωπψπψ

⋅⋅⋅⋅

⋅+

⋅⋅

⋅++

⋅⋅

⋅+=

∑

∑∞

=

∞

=

4

sin22

cos22

0

10220

10220

*

&

in which:

( ) ( ) ∑=

−

⋅

+−

⋅−⋅=N

nn

m

a

bmA a

nmn

01202 22

1212

σξπψ

Equation 3.2–111

A substitution of ( )tC* in the equation for each θ-value, results for any θ-value less than2π into the following equation:

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

( ) 14

sin22

cos22

0

10202200

10202200

−⋅⋅⋅⋅⋅

⋅−

=

⋅⋅

−⋅+−+

⋅⋅

−⋅+−+

∑

∑∞

=

∞

=

θµη

ωπχ

ωπψθψπψθψ

ωπψθψπψθψ

a

mmAmAmSBsB

mmAmAmcBcB

gb

tQ

tP

&

Equation 3.2–112


( ) ( ) ( )

( ) ( ) ( ) tQtP

tg

bb

a

a

a

⋅⋅+⋅⋅⋅−=

+⋅⋅⋅⋅⋅

−⋅−=−⋅⋅⋅⋅⋅

⋅−

ωωθµ

γωξηχπθµθµ

ηωπχ

sincos1

sin2

114

00

0&

in which:

( )γξηχπ

sin20 ⋅⋅

⋅⋅

= ba

aP and ( )γξηχπ

cos20 ⋅⋅

⋅⋅

= ba

aQ

Equation 3.2–113

This provides for each θ-value less than 2π a set of two equations with the unknownparameters mP2 and mQ2 :


94

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ∑

∑∞

=

∞

=

⋅+⋅−=−

⋅+⋅−=−

122000

122000

12

12

mmmsBsB

mmmcBcB

QfQ

PfP

θθµπψθψ

θθµπψθψ

in which:( ) ( ) ( )202022 πψθψθ mAmAmf +−=

Equation 3.2–114

These equations can also be written as:

( ) ( ) ( )

( ) ( ) ( ) ∑

∑∞

=

∞

=

⋅=−

⋅=−

02200

02200

2

2

mmmsBsB

mmmcBcB

Qf

Pf

θπψθψ

θπψθψ

in which:for :0=m ( ) ( ) 10 −= θµθf

for 0>m : ( ) ( ) ( )202022 πψθψθ mAmAmf +−=

Equation 3.2–115



θ-values, the best fit values of mP2 and mQ2 are supposed to be those found by means of aleast squares method. Note that at least 1+M values of θ , less than 2π , are required tosolve these equations.

Another favourable method is to multiply both sides of the equations with θ∆ . Then thesummation over θ can be replaced by integration. Herewith, two sets of 1+M equationshave been obtained, one set for mP2 and one set for mQ2 :

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )∫∑ ∫

∫∑ ∫

⋅⋅−=

⋅⋅⋅

⋅⋅−=

⋅⋅⋅

=

=

2

0

2000

2

0

222

2

0200

0

2

0222

2

2

ππ

ππ

θθπψθψθθθ

θθπψθψθθθ

dfdffQ

dfdffP

nsBsB

M

mnmm

ncBcB

M

mnmm

for: Mn ,...0=

Equation 3.2–116


coefficients 0P and 0Q are known now and from the definition of these coefficients followsthe amplitude ratio of the radiated waves and the forced sway oscillation:

20

202 QP

b

a

a

+⋅

⋅=

ξπχη

Equation 3.2–117


95


( )( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅++

⋅⋅

⋅++

⋅⋅⋅

=Φ

∑

∑

=

=

tQ

tPg

M

mmAmsB

M

mmAmcB

a

ωθφθφ

ωθφθφ

ωπηθ

sin

cos

10220

10220

0

in which:( ) ( )( )

( ) ( )( )∑=

−

⋅+⋅⋅

+−

⋅−⋅−

⋅++=N

nn

n

a

b

mA

nmanm

n

m

012

02

22sin2212

1

12sin

θσξ

θθφ

Equation 3.2–118

In this expression ( )θφ cB0 and ( )θφ sB0 are the values of ( )θαφ ,Bc and ( )θαφ ,Bs at the surfaceof the cylinder, so for 0=α .

3.2.3.1 Pressure Distribution During Roll Motions


( ) ( )

( ) ( ) ( )

( ) ( ) ( )

⋅⋅

⋅+−

⋅⋅

⋅++

⋅⋅⋅−

=

∂Φ∂

⋅−=

∑

∑∞

=

∞

=

tP

tQg

tp

mmAmcB

mmAmsB

a

ωθφθφ

ωθφθφ

πηρ

θρθ

sin

cos

10220

10220

0

Equation 3.2–119

It is obvious that this pressure is skew-symmetric in θ .

3.2.3.2 Roll Coefficients

The two-dimensional hydrodynamic moment acting on the cylinder in the clockwise directioncan be found by integrating the roll component of the hydrodynamic pressure on the surfaceS of the cylinder:

( ) ( )

( ) θθθ

θ

θθ

π

π

dddy

yddx

xp

dsdsdy

ydsdx

xppM R

⋅

⋅+⋅⋅⋅−=

⋅

−

⋅++

⋅−⋅−−+=

∫

∫+

00

00

2

0

00

00

2

0

'

2

Equation 3.2–120

With this the two-dimensional hydrodynamic moment due to roll oscillations can be written asfollows:


96

( ) ( )( )tXtYbg

M RRa

R ⋅⋅−⋅⋅⋅⋅⋅⋅

= ωωπ

ηρsincos

20'

in which:

( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( )

( )( ) ( )

( ) ( )∑

∑ ∑

∑ ∑

∑ ∑∑

∫ ∑∑

=−

= +=−+−−−−

=

−

=−−+−−−

= = =−−

= =−−

+

⋅⋅⋅

−−⋅−+−−

+

⋅⋅⋅

−−⋅−−+−

+

⋅⋅−

⋅

⋅−

⋅⋅−−+

−⋅−⋅⋅−⋅

⋅+

⋅⋅−⋅⋅⋅−⋅−⋅⋅⋅

=

M

mmN

n

N

nmiinmin

N

mn

mn

iinmin

mm

a

b

M

m

N

n

N

iinm

m

a

N

n

N

iin

insB

a

R

aaain

iinm

aaain

iinm

Q

aainm

iniQ

dinaaiY

1

012221212

012221212

2

3

1 0 012122222

2

0 0 0121202

22121222

22121222

1

8

2212

22121

2

1

22sin1212

1

σξπ

σ

θθθφσ

π

and RX as obtained from this expression above for RY , by replacing there ( )θφ sB0 by


Equation 3.2–121

These expressions are similar to those found before for the hydrodynamic roll moment due tosway oscillations.


( ) ( )( )

0

0

20'

2cos

2sin

sincos

Qx

Px

tXtYbg

M

ba

a

ba

a

RRa

R

⋅⋅⋅

⋅=

⋅⋅⋅

⋅=

−+⋅⋅−−+⋅⋅⋅⋅⋅⋅

=

ξπηγ

ξπηγ

γγωγγωπ

ηρ

Equation 3.2–122

the two-dimensional hydrodynamic roll moment can be resolved into components in phaseand out phase with the angular displacement of the cylinder:

( ) ( ) ( ) ( ) γωγω

χξπηρ

+⋅⋅⋅−⋅++⋅⋅⋅+⋅

⋅⋅⋅

⋅⋅⋅⋅=

tQXPYtPXQY

bgM

RRRR

ab

aR

sincos

2

0000

2

220'

Equation 3.2–123

This hydrodynamic roll moment can also be written as:

( ) ( )γωβωγωβω

ββ

+⋅⋅⋅⋅++⋅⋅⋅⋅=

⋅−⋅−=

tNtM

NMM

aa

R

sincos '44

2'44

'44

'44

' &&&

Equation 3.2–124

in which:


97

'44M 2-D hydrodynamic mass moment of inertia coefficient of roll

'44N 2-D hydrodynamic damping coefficient of roll

When using also the amplitude ratio of the radiated waves and the forced roll oscillation,found before, the two-dimensional hydrodynamic mass and damping coefficients of roll inTasai's axes system are given by:

ωρ

ρ

⋅+

⋅−⋅⋅

⋅+=

+⋅+⋅

⋅⋅+

=

20

20

004

0'44

20

20

004

0'44

8

8

QPQXPYb

N

QPPXQYb

M

RR

RR

Equation 3.2–125


The energy delivered by the exciting moments should be equal to the energy radiated by thewaves, so:

( ) ( )2

1 2

0

'44

cgdtN

Ta

T

osc

osc ⋅⋅⋅=⋅⋅⋅⋅ ∫

ηρββ &&

Equation 3.2–126


With the relation for the wave speed ωgc = in deep water, follows the relation between thetwo-dimensional heave damping coefficient and the amplitude ratio of the radiated waves andthe forced sway oscillation:

2

3

2'

44

⋅

⋅=

a

agN

βη

ωρ

Equation 3.2–127


ωπρ⋅

+⋅

⋅⋅= 2

02

0

40

2'

44

164 QP

bN

Equation 3.2–128



8

2

00

π=⋅−⋅ QXPY RR

Equation 3.2–129


98

3.2.3.3 Coupling of Roll into Sway

In the case of a roll oscillation generally a sway force produced too. The hydrodynamicpressure is skew-symmetric in θ .The two-dimensional hydrodynamic lateral force, acting on the cylinder in the direction of thex -axis, can be found by integrating the horizontal component of the hydrodynamic pressureon the surface S of the cylinder:

( ) ( )

( ) θθ

θ

θθ

π

π

dddy

p

dsdsdy

ppFx

⋅⋅⋅=

⋅−

⋅−−+−=

∫

∫+

0

2

0

0

2

0

'

2

Equation 3.2–130

With this the two-dimensional hydrodynamic horizontal force due to sway oscillations can bewritten as follows:

( ) ( )( )tNtMbg

F ax ⋅⋅−⋅⋅⋅

⋅⋅⋅−= ωω

πηρ

sincos 000'

in which:

( ) ( ) ( ) ( )( )

( ) ( )

( ) ( ) ( )( ) ( )∑ ∑∑

∑

∫ ∑

= = =−−

−

=+

=−

⋅⋅−−+

−⋅−⋅⋅−+

⋅+⋅⋅−⋅⋅

+

⋅⋅−⋅⋅−⋅−⋅⋅−=

M

m

N

n

N

iinm

m

a

b

N

mmm

m

a

N

nn

nsB

a

aanim

inQ

amQ

dnanM

1 0 012122222

1

1122

2

0 01200

12221212

1

1214

12sin1211

σξ

σπ

θθθφσ

π



Equation 3.2–131

For the determination of 0M and 0N , it is required that NM ≥ .


( ) ( )( )

0

0

000'

2cos

2sin

sincos

Qx

Px

tNtMbg

F

ba

a

ba

a

ax

⋅⋅⋅

⋅=

⋅⋅⋅

⋅=

−+⋅⋅−−+⋅⋅⋅⋅⋅⋅−

=

ξπηγ

ξπηγ

γγωγγωπ

ηρ

Equation 3.2–132

the two-dimensional hydrodynamic horizontal force can be resolved into components in phaseand out phase with the horizontal displacement of the cylinder:


99

( ) ( ) ( ) ( )( )γωγωχξπ

ηρ

+⋅⋅⋅−⋅++⋅⋅⋅+⋅

⋅⋅⋅

⋅⋅⋅⋅−=

tQNPMtPNQM

bgF

ab

ax

sincos

2

00000000

2

20'

Equation 3.2–133


( ) ( )γωβωγωβω

ββ

+⋅⋅⋅⋅++⋅⋅⋅⋅=

⋅−⋅−=

tNtM

NMF

aa

x

sincos '24

2'24

'24

'24

' &&&

Equation 3.2–134

in which:'

24M 2-D hydrodynamic mass coupling coefficient of roll into sway'

24N 2-D hydrodynamic damping coupling coefficient of roll into sway

When using also the amplitude ratio of the radiated waves and the forced sway oscillation,found before, the two-dimensional hydrodynamic mass and damping coupling coefficients ofroll into sway are given by:

ωρ

ρ

⋅+

⋅−⋅⋅

⋅−=

+⋅+⋅

⋅⋅−

=

20

20

00003

0'24

20

20

00003

0'24

8

8

QPQNPMb

N

QPPNQMb

M

Equation 3.2–135

In the ship motions ( )zyxO ,, co-ordinate system these two coupling coefficients will changesign.


100

3.2.4 Low and High Frequencies

The potential coefficients for very small and very large frequencies in the ship motions( )zyxO ,, co-ordinate system have been given in the following subsections.

3.2.4.1 Near-Zero Frequency Coefficients

The 2-D hydrodynamic mass coefficient for sway of a Lewis cross section is given by Tasai[1961] as:

( ) ( ) 23

21

2

31

'22 31

120 aa

aaD

M s ⋅+−⋅

+−

⋅⋅

=→πρω

Equation 3.2–136

The 2-D hydrodynamic mass coupling coefficient of sway into roll of a Lewis cross section isgiven by Grim [1955] as:

( ) ( )

( ) 23

21

233

2331311

31

'22

'42

31712

54

53

54

1

1316

00

aa

aaaaaaaa

aaD

MM s

⋅+−

⋅−⋅+

⋅+⋅−⋅+−⋅

⋅+−

⋅⋅

⋅→−=→π

ωω

Equation 3.2–137

In Tasai's axes system, '42M will change sign.

The 2-D hydrodynamic mass coefficient of heave of a Lewis cross section goes to infinite, so:( ) ∞=→ 0'

33 ωM

Equation 3.2–138

The 2-D hydrodynamic mass moment of inertia coefficient of roll of a Lewis cross section isgiven by Grim [1955] as:

( ) ( )

( ) ( )

⋅++⋅⋅⋅++⋅

⋅

++⋅

⋅⋅

=→

23331

23

21

4

31

'44

916

198

1

1216

0

aaaaaa

aaB

M s

πρω

Equation 3.2–139

The 2-D hydrodynamic mass coupling coefficient of roll into sway of a Lewis cross section isgiven by Grim [1955] as:


101

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) 23331

23

21

2313331311

31'44

'24

916

198

1

712

154

153

11

16

00

aaaaaa

aaaaaaaaa

Daa

MMs

⋅++⋅⋅⋅++⋅

⋅−−⋅⋅++⋅⋅⋅++⋅−⋅

⋅+−

⋅⋅→−=→πωω

Equation 3.2–140

In Tasai's axes system, '24M will change sign.

All potential damping values for zero frequency will be zero:( )( ) 00

00'

42

'22

=→

=→

ω

ω

N

N ( ) 00'

33 =→ωN ( )( ) 00

00'

24

'44

=→

=→

ω

ω

N

N

Equation 3.2–141

3.2.4.2 Infinite Frequency Coefficients

The 2-D hydrodynamic mass coefficient of sway of a Lewis cross section is given byLandweber and de Macagno [1957, 1959] as:

( ) ( )

⋅++−⋅

+−

⋅⋅

=∞→ 23

231

2

31

'22 3

161

12

aaaaa

DM s

πρω

Equation 3.2–142

The 2-D hydrodynamic mass coefficient of heave of a Lewis cross section is given by Tasai[1959] as:

( ) ( ) ( )( )23

21

2

31

'33 31

122aa

aaB

M s ⋅++⋅

++⋅

⋅⋅

=∞→πρω

Equation 3.2–143

The 2-D hydrodynamic mass moment of inertia coefficient of roll of a Lewis cross section isgiven by Kumai [1959] as:

( ) ( ) ( )( )23

23

21

4

31

'44 21

12aaa

aaB

M s ⋅++⋅⋅

++⋅

⋅⋅=∞→ πρω

Equation 3.2–144

Information about the 2-D hydrodynamic mass coupling coefficients between sway and rollof a Lewis cross section could not be found in literature, so:

( )( ) ?

?'

24

'42

=∞→

=∞→

ω

ω

M

M

Equation 3.2–145


102

All potential damping values for infinite frequency will be zero:

( )( ) 0

0'

42

'22

=∞→

=∞→

ω

ω

N

N ( ) 0'

33 =∞→ωN ( )( ) 0

0'

24

'44

=∞→

=∞→

ω

ω

N

N

Equation 3.2–146


103

3.3 Potential Theory of Keil

In this section, the determination of the hydrodynamic coefficients of a heaving, swaying androlling cross section of a ship in shallow at zero forward speed is based on work published byKeil [1974]. His method is based on Lewis conformal mapping of the ships' cross sections tothe unit circle and the shallow water potential theory.Journée [2001] has given a few comparisons of predicted and measured data on verticalmotions at various water depths. Recently, Vantorre and Journée [2003] have given moreextended comparisons of computed results by Keil’s theory with model test data on verticalmotions of a slender and a full ship, sailing at very shallow water.

For a significant part, the detailed description of the shallow water potential theory in thisSection is simply a translation of Keil’s original 1974 German report into the Englishlanguage. However, it has been supplemented with some numerical improvements andoutcomes of discussions with the author in the early eighties. The theory has been presentedhere in a layout, more or less as used in the computer code SEAWAY.

3.3.1 Notations of Keil

Keil’s notations have been maintained here as far as possible:

a Lewis coefficient

indexA source strength

indexA amplitude ratiob Lewis coefficientB breadth of bodyc wave velocity

indexC non-dimensional force or moment

indexE non-dimensional exciting force or moment

indexF hydrodynamic forceg acceleration of gravityG function (real part)h water depthH function (imaginary part)

indicesH fictive moment armHT water depth - draft ratio

"I hydrodynamic moment of inertia

xk wave number in x -direction

yk wave number in y -direction"m hydrodynamic mass

indicesM hydrodynamic moment

indicesN hydrodynamic damping coefficientp pressure

22 yxr += polar coordinate


104

t time or integer valueT draughtU velocity amplitude of horizontal oscillationV velocity amplitude of vertical oscillation

xA cross sectional areazyx ,, earth-bounded co-ordinates

indicesY transfer functionsγ Euler constant (= 0.57722)ε phase shiftζ wave amplitudeθ polar co-ordinate or pitch amplitudeλ wavelengthµ wave direction

g2ων = wave number at deep waterλπν 20 = wave number

ρ density of waterϕ roll angle

indicesΦ time-dependent potential

indicesφ part of potential

indicesΨ time-dependent stream function

indicesψ part of stream function ω circular frequency of oscillation

In here, the indices – being used by Keil - are:

E related to excitationH horizontal or related to horizontal motionsj imaginary partn numbering of potential partsQ related to transverse motionsr real partR related to roll motionsV related to vertical or vertical motionsW related to waves

3.3.2 Basic Assumptions

Figure 3.3–1 shows the co-ordinate system as used by Keil and maintained here.The potentials of the incoming waves have been described in Appendix I of this Section.


105

Figure 3.3–1: Keil’s axes system

The wave number, λπν 20 = , follows from:

[ ]hh

g⋅⋅=

⋅⋅

⋅⋅

== 00

2

tanh2

tanh2 νν

λπ

λπων

Equation 3.3–1

The fluid is supposed to be incompressible and inviscid. The flow caused by the oscillatingbody in the surface of this fluid can be described by a potential flow. The problem will belinearised, i.e., contributions of second and higher order in the definition of the boundaryconditions will be ignored. Physically, this yields an assumption of small amplitude motions.The earth-bounded axes system of the sectional contour is given in Figure 3.3–2-a.

Figure 3.3–2: Definition of sectional contour

Velocities are positive if they are directed in the positive co-ordinate direction:

yvy

=∂Φ∂

zvz

=∂Φ∂

The value of the stream function increases when - going in the positive direction - the flowgoes in the negative y -direction:

21 Ψ<Ψ →

∂Ψ∂

+=∂Φ∂

zy


106

43 Ψ>Ψ →

∂Ψ∂

+=∂Φ∂

∂Ψ∂

−=∂Φ∂

∂Ψ∂

−=∂Φ∂

ns

sn

yz

Equation 3.3–2

For the imaginary parts, the symbols i and j have been used: i for geometrical variables(potential and stream function) and j for functions of time.

3.3.3 Vertical Motions

3.3.3.1 Boundary Conditions


1. The fluid is incompressible and the velocity potential must satisfy to the ContinuityCondition and the Equation of Laplace:

02

2

2

22 =

∂Φ∂

+∂

Φ∂=Φ∇

zy

Equation 3.3–3

2. The linearised free surface condition follows from the condition that the pressure at thefree surface is not time-depending but constant:

02

2

=

∂

Φ∂−

∂Φ∂

⋅⋅=∂∂

tzg

tp ρ for:

2B

y ≥ and 0=z

from which follows:

02

=∂Φ∂

+Φ⋅zg

ω or 0=

∂Φ∂

+Φ⋅z

ν for: 2B

y ≥ and 0=z

Equation 3.3–4

3. The seabed is impervious, so the vertical fluid velocity at hz = is zero:

0=∂Φ∂z

for: hz =

Equation 3.3–5

4. The harmonic oscillating cylinder produces a regular progressive wave system, travellingaway from the cylinder, which fulfils the Sommerfeld radiation condition:

0ImRelim 0 =

Φ⋅−Φ

∂∂

⋅∞→

νy

yy

Equation 3.3–6

In here, λπν 20 = is the wave number of the radiated wave.


107

5. The oscillating cylinder is impervious too; thus at the surface of the body is the fluidvelocity equal to the body velocity, see Figure 3.3–2-b. This yields that the boundaryconditions on the surface of the body are given by:

body

body

bodynbody

dsd

dsdz

ydsdy

z

vn

Ψ−=

⋅∂Φ∂

−⋅∂Φ∂

=

=∂Φ∂

Equation 3.3–7

Two cases have to be distinguished:

a) The hydromechanical loads, which have to be obtained for the vertically oscillatingcylinder in still water with a vertical velocity equal to:

tjeVV ⋅⋅⋅= ω

The boundary condition on the surface of the body becomes:

tj

bodybody

edsdy

Vdsd ⋅⋅⋅⋅=Ψ

− ω

or:( ) CyeVtzy

bodytj

body+⋅⋅−=Ψ ⋅⋅ω,,

Equation 3.3–8

b) The wave loads, which have to be obtained for the restrained cylinder in regular wavesfrom the incoming undisturbed wave potential WΦ and the diffraction potential SΦ :

body

SW

body

SSWW

body

SW

dsd

dsd

dsdz

ydsdy

zdsdz

ydsdy

z

nn

Ψ+

Ψ−=

⋅∂Φ∂

−⋅∂Φ∂

+⋅∂Φ∂

−⋅∂Φ∂

=

=∂Φ∂

+∂Φ∂

0

or:( ) ( )

body

WW

bodyWbodyS

dzy

dyz

tzydtzyd

⋅∂Φ∂

−⋅∂Φ∂

=

Ψ−=Ψ ,,,,

Equation 3.3–9

The stream function of an incoming wave - which travels in the negative y -direction, so090+=µ - is given in Appendix I of this Section by:


108

( ) [ ] [ ] [ ] zhzej ytjW ⋅⋅⋅−⋅⋅⋅

⋅⋅=Ψ ⋅+⋅⋅

000 coshtanhsinh0 ννννωζ νω

Equation 3.3–10

Because only vertical forces have to be determined, only the in y -symmetric part of thepotential and stream functions have to be considered. From this follows the boundarycondition on the surface of the body for beam waves, so wave direction 090+=µ :

( ) ( )

[ ] [ ] [ ] [ ]body

tj

bodyWbodyS

yzhze

tzytzy

⋅⋅⋅⋅⋅−⋅⋅⋅⋅

=

Ψ−=Ψ

⋅⋅0000 sinhcoshtanhsinh

,,,,

νννννωζ ω

Equation 3.3–11

In case of another wave direction, this problem becomes three-dimensional and a streamfunction can not be written. However, boundary condition (Equation 3.3–11) provides us a''quasi stream function'' sΨ

~, i.e. this is the amount of fluid which has to come out of the body

per unit of length, so that - in total - no fluid of the incoming wave enters into the body.This function can be used as an approximation of the problem:

( )

( )

( ) [ ] [ ] [ ]( )

( )( ) [ ] [ ] [ ]( )

⋅⋅⋅⋅−⋅⋅⋅⋅

⋅−⋅+

⋅⋅⋅−⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅⋅=

⋅∂Ψ∂

−⋅∂Ψ∂

=Ψ

∫

∫∫

⋅⋅

1

1

0

1

00000

20

1001010

0

0111

coshtanhsinhsincos

sin1

coshtanhsinhsinsinsin

coscos

,,,~

y

tj

body

z

z

W

y

W

bodys

dyzhzy

zhzy

xe

dzy

dyz

tzyx

νννµν

µν

νννµνµ

µννωζ ω

Equation 3.3–12

3.3.3.2 Potentials

3.3.3.2.1 3-D Radiation Potential

Suppose a three-dimensional oscillating cylindrical body in previously still water. To find thepotential of the resulting fluid motions, an oscillating pressure p at the free surface will

replace this body. The unknown amplitude p of this pressure has to follow from the boundaryconditions.This pressure is not supposed to act over the full breadth of the body; it is supposed to act -over the full length L of the body - only over a small distance 2y∆ on both sides of 0=y ,so:


109

( ) ( )

( ) 2 and 2 :for 0,,

:or

,,,,,

0

00

Lxyyzzyxp

ezzyxpjtzzyxp tj

≥∆≥==

⋅=⋅−== ⋅⋅ω

Equation 3.3–13

in which 0z is the z -co-ordinate of the fluid surface.

The resulting force P in the z -direction becomes:

( )

( )∫

∫ ∫+

−

+

−

∆+

∆−

∞<⋅=

⋅⋅==

2

2

'

2

2

2

20,,

L

L

L

L

y

y

dxxP

dxdyzzyxpP

Equation 3.3–14

The boundary condition in Equation 3.3–13 can be fulfilled by pressure amplitude( )0,, zzyxp = , which is found by a superposition of an infinite number of harmonic pressures.

From Equation 3.3–14 follows that the pressure amplitude ( )xp can be integrated, so aFourier series expansion follows from:

( ) ( ) ( )( )∫ ∫∞ +∞

∞−

⋅⋅−⋅⋅⋅=0

cos1

xx dkdxkpxp ξξξπ

Because the pressure amplitude p depends on two variables, the Fourier series expansion hasto be two-dimensional too:

( ) ( )

( )( ) ( )( ) xyxy dkdkddxkyk

pzzyxp

⋅⋅⋅⋅−⋅⋅−⋅

⋅⋅== ∫ ∫ ∫ ∫∞ ∞ +∞

∞−

+∞

∞−

ξηξη

ηξπ

coscos

,1

,,0 0

20

in which xk is the wave number in the x -direction and yk is the wave number in the y -direction.According to Equation 3.3–13, the pressure amplitude p disappears for 2yy ∆≥ and

2Lx ≥ , so for this pressure expression remains:

( ) ( )

( )( ) ( )( ) xyxy

L

L

y

y

dkdkddxkyk

pzzyxp

⋅⋅⋅⋅−⋅⋅−⋅

⋅⋅== ∫ ∫ ∫ ∫∞ ∞ +

−

∆+

∆−

ξηξη

ηξπ

coscos

,1,,0 0

2

2

2

220

It is assumed that the value of y∆ is small. This means that η remains small too. Thus, onecan safely suppose that:

( )( ) ( )ykyk yy ⋅≈−⋅ coscos ηwhich results in:

( ) ( ) ( )( ) ( )

( ) ( ) ( )( )∫∫ ∫

∫ ∫ ∫ ∫+

−

∞ ∞

∞ ∞ +

−

∆+

∆−

⋅⋅⋅−⋅⋅⋅⋅⋅=

⋅⋅⋅⋅⋅−⋅⋅⋅⋅==

2

2

'

0 02

0 0

2

2

2

220

coscos1

coscos,1

,,

L

L

xyxy

xyyx

L

L

y

y

dkdkdxkPyk

dkdkykdxkdpzzyxp

ξξξπ

ξξηηξπ


110

Equation 3.3–15

This pressure definition leads - as a start - to the following initial definition of the radiationpotential:

( ) ( ) ( )( ) ( )

( )ξ

ξω

ddkdkhkk

hzkk

ykxkkkCetzyx

yx

yx

yx

yxyxtj

r

⋅⋅⋅

⋅+

−⋅+

⋅⋅⋅−⋅⋅⋅=Φ ∫ ∫ ∫∞ ∞ ∞

⋅⋅

22

22

0 0 00

sinh

cosh

coscos,,,,

Equation 3.3–16

in which the function ( )yx kkC , is still unknown.

This expression in Equation 3.3–16 for the radiation potential fulfils the Equation of Laplace:

02

2

2

2

2

2

=∂

Φ∂+

∂Φ∂

+∂

Φ∂zyx

Now, the harmonic pressure at the free surface 1p can be obtained from an integration of the -with the Bernoulli Equation obtained - derivative to the time of the pressure:

( ) ( )( ) ( )

ξω

ξρω

ddkdkhkk

hkkkkg

ykxkkkCetp

yx

yx

yxyx

yxyxtj

⋅⋅⋅

⋅+

⋅+⋅+⋅−

⋅⋅⋅−⋅⋅⋅⋅=∂∂

∫ ∫ ∫∞ ∞ ∞

⋅⋅

22

22222

0 0 0

1

tanh

tanh

coscos,

Equation 3.3–17

The harmonic oscillating pressure is given by:( ) ( ) tjezzyxpjtzzyxp ⋅⋅⋅=⋅−== ω

0101 ,,,,,

Equation 3.3–18

and its amplitude becomes:

( ) ( ) ( )( ) ( )

ξω

ξωρ

ddkdkhkk

hkkkkg

ykxkkkCzzyxp

yx

yx

yxyx

yxyx

⋅⋅⋅

⋅+

⋅+⋅+⋅−

⋅⋅⋅−⋅⋅⋅== ∫ ∫ ∫∞ ∞ ∞

22

22222

0 0 001

tanh

tanh

coscos,,,

Equation 3.3–19

If this pressure amplitude 1p is supposed to be equal to the amplitude p , then combining

Equation 3.3–15 and Equation 3.3–19 provides the unknown function ( )yx kkC , :


111

( ) ( ) ( ) ( )( )

( )

( ) ( ) ( )( )

ξν

ξω

ρ

ξξξπ

ddkdkhkk

hkkkk

xkkkCykg

zzyxp

dkdkdxkPykzzyxp

yx

yx

yxyx

xyxy

xy

L

L

xy

⋅⋅⋅

⋅+

⋅+⋅+−

⋅−⋅⋅⋅⋅⋅

=

==

⋅⋅⋅−⋅⋅⋅⋅==

∫ ∫∫

∫ ∫∫

∞ ∞∞

∞ +

−

∞

22

2222

0 00

01

0

2

2

'

020

tanh

tanh

cos,cos

,,

coscos1

,,

Equation 3.3–20

Comparing the two integrands provides:

( ) ( )( )

( ) ( )( ) ξξξπ

ξν

ωρξ

dxkP

dhkk

hkkkkgxkkkC

L

Lx

yx

yxyx

xyx

⋅−⋅⋅⋅

=⋅

⋅+

⋅+⋅+−

⋅⋅

⋅−⋅⋅

∫

∫

+

−

∞

2

2

'

2

22

2222

0

cos1

tanh

tanhcos,

or:

( ) ( )( )

( ) ( )( ) ξξξπρ

ω

νξξ

dxkPg

hkkkk

hkkdxkkkC

L

Lx

yxyx

yx

xyx

⋅−⋅⋅⋅⋅⋅

⋅

⋅+⋅+−

⋅+

=⋅−⋅⋅

∫

∫

+

−

∞

2

2

'

2

2222

22

0

cos

tanh

tanhcos,

Equation 3.3–21

When defining:

( ) ( )2

'

0 πρξωξ

⋅⋅⋅

=gP

A

and substituting Equation 3.3–21 in Equation 3.3–16 provides the radiation potential:

( ) ( ) ( )( )

( ) ( )ξ

ν

ξξω

ddkdkhkkkkhkk

ykhzkk

xkAetzyx

yx

yxyxyx

yyx

x

L

L

tjr

⋅⋅⋅

⋅+⋅+−

⋅+⋅

⋅⋅

−⋅+

⋅−⋅⋅⋅=Φ

∫

∫∫

∞

∞+

−

⋅⋅

0222222

22

0

2

2

00

sinhcosh

coscosh

cos,,,

Equation 3.3–22

This potential fulfils both, the radiation condition at infinity and the boundary condition at thefree surface.



112

In case of an oscillating two-dimensional body, no waves are travelling in the x -direction, so0=xk and kk y = . The distribution of ( )ξA is constant over the full length of the body from

−∞=ξ until +∞=ξ and the radiation potential - given in Equation 3.3–22 - reduces to:

( ) ( )[ ][ ] [ ] ( ) dkyk

hkkhkhzk

Aetzy tjr ∫

∞⋅⋅ ⋅⋅⋅

⋅⋅−⋅⋅−⋅

⋅⋅=Φ0

00 cossinhcosh

cosh,,

νω

Equation 3.3–23

To fulfil also the Sommerfeld radiation condition in Equation 3.3–6, a term has to be added.For this, use will be made here of the value of the potential given in Equation 3.3–23 at a largedistance from the body:

( ) ( )[ ][ ] [ ] ( )

( ) ( )

⋅+⋅⋅⋅⋅=

⋅⋅⋅⋅⋅−⋅⋅

−⋅⋅⋅=Φ

∫∫

∫∞

⋅⋅−∞

⋅⋅+⋅⋅

∞⋅⋅

00

0

0

00

21

cossinhcosh

cosh,,

dkekFdkekFAe

dkykhkkhk

hzkAetzy

ykiykitj

tjr

ω

ω

ν

with:

( ) ( )[ ][ ] [ ]hkkhk

hzkkF

⋅⋅−⋅⋅−⋅

=sinhcosh

coshν

The treatment of the singularities is visualised in Figure 3.3–3.

Figure 3.3–3: Treatment of singularities

When substituting for k the term liku ⋅+= , the first integral integrates for 0>y over theclosed line I-II-III-IV in the first quadrant of the complex domain and the second integralintegrates for 0>y over the closed line I-V-VI-VII in the fourth quadrant, so:

1. For line I-II-III-IV :


113

( )

0

............0

=+++=

+++=⋅⋅ ∫∫∫ ∫∫ ⋅⋅+

IVIIIIII

IVIII

R

II

yui

JJJJ

dudududkdueuF

with:

( ) ( )IVIIIIIR

yki JJJdkekF ++−=⋅⋅∫∞

∞→

⋅⋅+

0

lim

The location of the singular point follows from the denominator in the expression for( )kF :

[ ] [ ] 0sinhcosh 000 =⋅⋅−⋅⋅ hh ννννBecause ( ) 0lim =

∞→ IIIRJ and IVJ disappears too for a large y , the singular point itself

delivers a contribution only:( )

( )[ ][ ] [ ] [ ]

[ ] ( )[ ][ ] [ ]

yi

yi

II

ehhh

hzhi

ehhhhh

hzi

iJ

⋅⋅+

⋅⋅+

⋅⋅⋅⋅+⋅

−⋅⋅⋅⋅⋅+=

⋅⋅⋅⋅−⋅−⋅⋅⋅

−⋅⋅⋅−=

⋅⋅−=

0

0

000

00

0000

0

0

coshsinhcoshcosh

coshsinhsinhcosh

Residue

ν

ν

νννννπ

ννννννπ

νπ

and the searched integral becomes for ∞→y :

( ) [ ] ( )[ ][ ] [ ]

yiyki ehhh

hzhidkekF ⋅⋅+

∞⋅⋅+ ⋅

⋅⋅⋅+⋅−⋅⋅⋅

⋅⋅−=⋅⋅∫ 0

000

00

0 coshsinhcoshcosh ν

νννννπ

2. For line I-V-VI-VII:

( )

0

............0

=+++=

+++=⋅⋅ ∫∫∫ ∫∫ ⋅⋅−

VIIVIVI

VIIVI

R

V

yui

JJJJ

dudududkdueuF

with:

( ) ( )VIIVIVR

yki JJJdkekF ++−=⋅⋅∫∞

∞→

⋅⋅−

0

lim

Because ( ) 0lim =∞→ VIR

J and VIIJ disappears too for a large y , the singular point itself

delivers a contribution only:( )

[ ] ( )[ ][ ] [ ]

yi

V

ehhh

hzhi

iJ

⋅⋅−⋅⋅⋅⋅+⋅

−⋅⋅⋅⋅⋅−=

⋅⋅+=

0

000

00

0

coshsinhcoshcosh

Residue

ν

ννννν

π

νπ

and the searched integral becomes for ∞→y :

( ) [ ] ( )[ ][ ] [ ]

yiyki ehhh

hzhidkekF ⋅⋅−

∞⋅⋅− ⋅

⋅⋅⋅+⋅−⋅⋅⋅

⋅⋅+=⋅⋅∫ 0

000

00

0 coshsinhcoshcosh ν

νννννπ

This provides for the potential in Equation 3.3–23 for ∞→y :


114

( ) [ ] ( )[ ][ ] [ ] ( )y

hhhhzh

Aetzy tjr ⋅⋅

⋅⋅⋅+⋅−⋅⋅⋅

⋅⋅⋅=Φ ⋅⋅0

000

0000 sin

coshsinhcoshcosh

,, νννν

ννπω

Equation 3.3–24

The Sommerfeld radiation condition in Equation 3.3–6 will be fulfilled when:[ ] ( )[ ]

[ ] [ ] ( ) ( ) 0,,Imcoscoshsinh

coshcosh000

000

0000 =Φ⋅−⋅⋅

⋅⋅⋅+⋅−⋅⋅⋅

⋅⋅⋅⋅⋅⋅ tzyyhhh

hzhAe tj νν

ννννννπω

or:( ) ( )

[ ] ( )[ ][ ] [ ] ( )y

hhhhzh

Ae

tzytzy

tj

j

⋅⋅⋅⋅⋅+⋅

−⋅⋅⋅⋅⋅⋅=

Φ=Φ

⋅⋅0

000

000

00

coscoshsinh

coshcosh

,,,,Im

νννν

ννπω

Equation 3.3–25

With Equation 3.3–24 and Equation 3.3–25, the radiation potential becomes:( ) ( )

( )[ ][ ] [ ] ( )

[ ] ( )[ ][ ] [ ] ( )

⋅⋅⋅⋅⋅+⋅

−⋅⋅⋅⋅⋅

+⋅⋅⋅⋅⋅−⋅⋅

−⋅

⋅⋅=Φ⋅+Φ

∫∞

⋅⋅

yhhh

hzhj

dkykhkkhk

hzk

Aetzyjtzy tjjr

0000

00

0

000

coscoshsinh

coshcosh

cossinhcosh

cosh

,,,,

νννν

ννπ

ν

ω

Equation 3.3–26

From this follows for ∞→y :

( ) [ ] ( )[ ][ ] [ ]

yjtj ehhh

hzhAejtzy ⋅⋅−⋅⋅ ⋅

⋅⋅⋅+⋅−⋅⋅⋅

⋅⋅⋅⋅=∞→Φ 0

000

0000 coshsinh

coshcosh,, νω

νννννπ

This means that Equation 3.3–26 describes a flow, consisting of waves with amplitude:[ ]

[ ] [ ]hhhh

gA

⋅⋅⋅+⋅⋅

⋅⋅

⋅=000

02

0 coshsinhcosh

ννννπωζ

Equation 3.3–27

travelling away from both sides of the cylinder.

From the orthogonality condition:

zy ∂Ψ∂

+=∂Φ∂

follows the stream function:( ) ( )

( )[ ][ ] [ ] ( )

[ ] ( )[ ][ ] [ ] ( )

⋅⋅⋅⋅⋅+⋅

−⋅⋅⋅⋅⋅

+⋅⋅⋅⋅⋅−⋅⋅

−⋅

⋅⋅−=Ψ⋅+Ψ

∫∞

⋅⋅

yhhh

hzhj

dkykhkkhk

hzk

Aetzyjtzy tjjr

0000

00

0

000

sincoshsinh

sinhcosh

sinsinhcosh

sinh

,,,,

νννν

ννπ

ν

ω

Equation 3.3–28

For an infinite water depth, Equation 3.3–26 and Equation 3.3–28 reduce to:


115

( ) ( )

( ) ( )

( ) ( )

( ) ( )

⋅⋅⋅⋅+⋅⋅⋅−

⋅⋅=Ψ⋅+Ψ

⋅⋅⋅⋅+⋅⋅⋅−

⋅⋅=Φ⋅+Φ

⋅−∞ ⋅−

⋅⋅∞∞

⋅−∞ ⋅−

⋅⋅∞∞

∫

∫

yejdkykk

e

Aetzyjtzy

yejdkykk

e

Aetzyjtzy

zzk

tjjr

zzk

tjjr

νπν

νπν

ν

ω

ν

ω

sinsin

,,,,

coscos

,,,,

0

000

0

000

Equation 3.3–29

Now, the potential and stream functions can be written as:

( )

( ) [ ] [ ][ ] [ ]

[ ] ( )[ ][ ] [ ] ( )

( )

( ) [ ] [ ][ ] [ ]

[ ] ( )[ ][ ] [ ] ( )

⋅⋅⋅⋅⋅+⋅

−⋅⋅⋅⋅⋅

−⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅⋅⋅−

−⋅⋅⋅−

⋅⋅=

Ψ⋅+Ψ+Ψ=Ψ

⋅⋅⋅⋅⋅+⋅

−⋅⋅⋅⋅⋅

+⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅⋅⋅−

+⋅⋅⋅−

⋅⋅=

Φ⋅+Φ+Φ=Φ

∫

∫

∫

∫

∞ ⋅−

∞ ⋅−

⋅⋅

∞

∞ ⋅−

∞ ⋅−

⋅⋅

∞

yhhh

hzhj

dkhkkhkzkkzk

ykk

e

dkykk

e

Ae

j

yhhh

hzhj

dkhkkhkzkkzk

ykk

e

dkykk

e

Ae

j

hk

zk

tj

jradr

hk

zk

tj

jradr

0000

00

0

0

0

0000

0000

00

0

0

0

0000

sincoshsinh

sinhcosh

sinhcoshsinhcosh

sin

sin

coscoshsinh

coshcosh

sinhcoshcoshsinh

cos

cos

νννν

ννπ

νν

ν

ν

νννν

ννπ

νν

ν

ν

ω

ω

Equation 3.3–30

In here, ∞Φ r0 is the potential at deep water and rad0Φ is the additional potential due to the

finite water depth. jΦ can be written in the same way.

3.3.3.2.3 Alternative Derivation

Assuming that the real part of the potential at an infinite water depth, ∞Φr , is known, anotherderivation of the 2-D potential is given by Porter [1960]. The additional potential for arestricted water depth, radΦ , will be determined in such a way that it fulfils the free surfacecondition and - together with ∞Φr - also the boundary condition at the seabed.As a start for the real additional potential will be chosen:

( ) ( ) [ ] ( ) ( )[ ] ( ) dkykhzkkCzkkCAetzy tjrad ⋅⋅⋅−⋅⋅+⋅⋅⋅⋅=Φ ∫

∞⋅⋅ coscoshsinh,,

02100

ω

Equation 3.3–31


116

From the free surface condition in Equation 3.3–4 follows for 2By ≥ :

( ) [ ] ( ) ( ) [ ] ( ) 0cossinhcosh0

2120 =⋅⋅⋅⋅⋅⋅−⋅+⋅⋅⋅⋅⋅ ∫∞

⋅⋅ dkykhkkkCkkChkkCAe tj νω

The solution of this Fourier integral equation:

( ) ( ) ( )ξξ gdkkkf =⋅⋅⋅∫∞

cos0

is known:

( ) ( ) ( ) ξξξπ

dkgkf ⋅⋅⋅⋅= ∫∞

cos1

0

Also will be obtained:( ) ( ) ( ) [ ] [ ] 0sinhcosh21 =⋅⋅−⋅⋅⋅+⋅= hkkhkkCkkCkf ν

from which follows:

( ) [ ] [ ] ( )kChkkhk

kkC 12 sinhcosh

⋅⋅⋅−⋅⋅

−=

ν

With this will be obtained:( )

( ) [ ] ( )[ ][ ] [ ] ( ) dkyk

hkkhkhzkk

zkkC

Aetzy tjrad

⋅⋅⋅

⋅⋅−⋅⋅−⋅⋅

−⋅⋅

⋅⋅=Φ

∫∞

⋅⋅

cossinhcosh

coshsinh

,,

01

00

ν

ω

The still unknown function ( )kC1 follows from the boundary condition at the seabed:

( )

( ) [ ] ( )

⋅⋅⋅⋅⋅⋅

−⋅⋅⋅−

⋅

⋅⋅−=

=∂

Φ∂+

∂Φ∂

∫

∫∞

∞ ⋅−

⋅⋅

=

∞

0

1

00

00

coscosh

cos

0

dkykhkkkC

dkykk

ek

Ae

zzhk

tj

hz

radr

νω

So:

( ) ( ) [ ]

( ) ( ) [ ] [ ] [ ]hkhkkhkkek

kC

hkke

kC

hk

hk

⋅⋅⋅⋅−⋅⋅⋅−⋅−

=

⋅⋅−=

⋅−

⋅−

coshsinhcosh

cosh

2

1

νν

ν

With this, the real additional potential, as given in Equation 3.3–30, becomes:( )

[ ] [ ][ ] [ ] ( ) dkyk

hkkhkzkkzk

ke

Aetzyhk

tjrad

⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅−

⋅⋅=Φ

∫∞ ⋅−

⋅⋅

cossinhcoshcoshsinh

,,

0

00

νν

ν

ω

Equation 3.3–32

The imaginary part can be obtained as described before.


117

3.3.3.2.4 2-D Multi-Potential

The free surface conditions can not be fulfilled with the potential and the stream function inEquation 3.3–30 only.Additional potentials nΦ are required which fulfil the boundary conditions in Equation 3.3–3

through Equation 3.3–6 and together with 0Φ also fulfil the boundary conditions in Equation3.3–7 through Equation 3.3–9:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ∑

∑

∞

=∞

∞

∞

=

Φ⋅+Φ+Φ⋅

+Φ⋅+Φ+Φ⋅=

Φ⋅+Φ⋅=Φ

1

'''

'0

'0

'00

1

''00

,,,,,,

,,,,,,

,,,,,,

nnjnradnrn

jradr

nnn

tzyjtzytzyA

tzyjtzytzyA

tzyAtzyAtzy

Equation 3.3–33

Use will be made here of multi-potentials given by Grim [1956, 1957] of which - using theSommerfeld radiation condition - the real additional potential nradΦ and the imaginary

potential part njΦ will be determined. This results in:

( )

( ) ( ) ( )

( )

( ) ( ) [ ] [ ][ ] [ ] ( )

( )

[ ]( )[ ]

[ ] [ ] ( )yhhh

hzh

Aetzy

dkykhkkhkzkkzk

ekk

Aetzy

dkykekk

Aetzy

n

ntj

nj

khn

ntj

nrad

kzn

ntj

nr

⋅⋅⋅⋅⋅+⋅

−⋅⋅

⋅⋅

⋅⋅−=Φ

⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅⋅⋅+

⋅⋅+=Φ

⋅⋅⋅⋅⋅+

⋅⋅+=Φ

⋅⋅

∞−−⋅

⋅⋅

−−⋅∞

⋅⋅∞

∫

∫

0000

0

0

20

0

12

12

0

coscoshsinh

coshcosh

,,

cossinhcoshcoshsinh

,,

cos

,,

νννν

νννπ

ννν

ν

ω

ω

ω

Equation 3.3–34

The orthogonality condition provides the stream functions:( )

( ) ( ) ( )

( )

( ) ( ) [ ] [ ][ ] [ ] ( )

( )

[ ]( )[ ]

[ ] [ ] ( )yhhh

hzh

Aetzy

dkykhkkhkzkkzk

ekk

Aetzy

dkykekk

Aetzy

n

ntj

nj

khn

ntj

nrad

kzn

ntj

nr

⋅⋅⋅⋅⋅+⋅

−⋅⋅

⋅⋅

⋅⋅+=Ψ

⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅⋅⋅+

⋅⋅−=Ψ

⋅⋅⋅⋅⋅+

⋅⋅+=Ψ

⋅⋅

∞−−⋅

⋅⋅

−−⋅∞

⋅⋅∞

∫

∫

0000

0

0

20

0

12

12

0

sincoshsinh

sinhcosh

,,

sinsinhcoshsinhcosh

,,

sin

,,

νννν

νννπ

ννν

ν

ω

ω

ω


118

Equation 3.3–35

The potentials njΦ and nradΦ disappear in deep water.

3.3.3.2.5 Total Potentials

Only the complex constant nA with ∞≤≤ n0 in the potential has to be determined:

( ) ( ) ( ) ( )

∑

∑

∑

∞

= ∞

∞⋅⋅

∞

= ∞

∞

∞

=∞

⋅++⋅⋅+

⋅−+⋅⋅=

Φ⋅+Φ+Φ⋅⋅+

Φ⋅−Φ+Φ⋅=

Φ⋅+Φ+Φ⋅⋅+=Φ

0

0'''

'''

0

''' ,,,,,,,,

n njnrnradnrnj

njnjnradnrnrtj

n njnrnradnrnj

njnjnradnrnr

nnjnradnrnjnr

AAj

AAe

AAj

AA

tzyjtzytzyAjAtzy

φφφ

φφφω

( ) ( ) ( ) ( )

∑

∑

∑

∞

= ∞

∞⋅⋅

∞

= ∞

∞

∞

=∞

⋅++⋅⋅+

⋅−+⋅⋅=

Ψ⋅+Ψ+Ψ⋅⋅+

Ψ⋅−Ψ+Ψ⋅=

Ψ⋅+Ψ+Ψ⋅⋅+=Ψ

0

0'''

'''

0

''' ,,,,,,,,

n njnrnradnrnj

njnjnradnrnrtj

n njnrnradnrnj

njnjnradnrnr

nnjnradnrnjnr

AAj

AAe

AAj

AA

tzyjtzytzyAjAtzy

ψψψ

ψψψω

Equation 3.3–36

Summarised, the complex total potential can now be written as:( ) ( )

( )( )( )[ ] [ ]

[ ] ( )( )( )[ ] [ ]

( ) ( ) ( )( )( )[ ] [ ]

[ ]( )( )( )

[ ] [ ]

∑ ∫

∫

∞

=

∞−⋅

⋅⋅

∞

⋅⋅

⋅⋅⋅+⋅−⋅+⋅

⋅⋅

⋅⋅−

⋅⋅⋅−⋅⋅

−⋅+⋅⋅⋅−

⋅⋅+

⋅

+

⋅⋅⋅+⋅−⋅+⋅⋅⋅

⋅⋅+

⋅⋅⋅−⋅⋅

−⋅+⋅

⋅⋅+⋅=Ψ⋅+Φ

1

000

0

0

20

0

1222

000

00

0

00

coshsinhcos

cosh

sinhcoshcos

coshsinhcoscosh

sinhcoshcos

,,,,

nn

n

njnr

tj

jrtj

hhhhziy

hj

dkhkkhk

hziykkk

AjA

e

hhhhziyh

j

dkhkkhk

hziyk

AjAetzyitzy

νννν

ννπ

νν

νννννπ

ν

ω

ω

Equation 3.3–37

The coefficients nrA and njA with ∞≤≤ n0 have to be determined in such a way that the

instantaneous boundary conditions on the body surface have been fulfilled. These coefficientsare dimensional and it is very practical to determine them for the amplitude of the flowvelocity V ; also if they then have the dimension [ ]12 +nL :


119

( )''jr

jrjr

AjAV

V

AjAVAjA

⋅+⋅=

⋅+⋅=⋅+

Then, nnA φ⋅' and nnA ψ⋅' have the dimensions of a length [ ]L .

3.3.3.3 Expansion of Potential Parts

The expansion of the potential parts at an infinite water depth is given by Grim, see Kirsch[1969].

For 022 →+⋅=⋅ zyr νν :

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

⋅⋅

⋅+⋅

⋅+

−⋅⋅

⋅+⋅

⋅+⋅+

⋅=

⋅⋅

⋅+⋅

⋅+

+⋅⋅

⋅+⋅

⋅+⋅+

⋅=

∑

∑

∑

∑

∞

=

∞

=⋅−∞

∞

=

∞

=⋅−∞

yyizmmz

y

yyizmm

r

e

yyizmmz

y

yyizmm

r

e

m

mm

m

m

m

zr

m

mm

m

m

m

zr

νν

νν

νγψ

νν

νννγ

φ

ν

ν

cosIm!

arctan

sinRe!

ln

sinIm!

arctan

cosRe!

ln

1

1

0

1

1

0

Equation 3.3–38

with the Euler constant: 57722.0=γ .

For ∞→+⋅=⋅ 22 zyr νν :

( ) ( ) ( )

( ) ( ) ( )yyy

eyizr

m

yyy

eyizr

m

zM

m

mmmr

zM

m

m

mmr

⋅⋅⋅⋅−

⋅+⋅

⋅−

=

⋅⋅⋅⋅+

⋅+⋅

⋅−=

⋅−

=∞

⋅−

=∞

∑

∑

νπν

ψ

νπν

φ

ν

ν

cosIm!1

sinRe!1

120

120

Equation 3.3–39

Mind you that ( ) ( )m

mm yizr

m⋅+⋅

⋅−

2

!1ν

is semi-convergent.

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

⋅+⋅

−−⋅+⋅−⋅−=

⋅+⋅

−+⋅+⋅−⋅−=

−−−∞

−−−∞

122

122

Re12

Im!121

Im12

Re!121

nnnnr

nnnnr

ziyn

ziyn

ziyn

ziyn

νψ

νφ

Equation 3.3–40

For the expansion of the remaining potential parts use has been made of the followingrelations as derived in Appendix II:


120

[ ] ( ) ( ) ( )

[ ] ( ) ( ) ( )

[ ] ( ) ( ) ( )

[ ] ( ) ( ) ( ) ∑

∑

∑

∑

∞

=

++

∞

=

++

∞

=

∞

=

⋅+⋅+

=⋅⋅⋅

⋅+⋅+

=⋅⋅⋅

⋅+⋅=⋅⋅⋅

⋅+⋅=⋅⋅⋅

0

1212

0

1212

0

22

0

22

Im!12

sincosh

Re!12

cossinh

Im!2

sinsinh

Re!2

coscosh

t

tt

t

tt

t

tt

t

tt

yiztk

ykzk

yiztk

ykzk

yizt

kykzk

yizt

kykzk

With these relations follows from Equation 3.3–32:

( ) [ ] [ ]( )

( ) ( ) ( ) ( )

( ) ( ) [ ] [ ]( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )∑

∑ ∫

∑∑

∫

∞

=

+

∞

= +

∞ ⋅−+

∞

=

∞

=

++

∞ ⋅−

⋅+⋅+−⋅+⋅⋅++−=

⋅+⋅+−⋅+⋅

⋅

⋅

⋅⋅−⋅⋅⋅−⋅

⋅+=

⋅+⋅⋅−

⋅+⋅+

⋅

⋅

⋅

⋅⋅−⋅⋅⋅−=

0

212

0 212

0

12

0

22

0

1212

0

0

Re12Re!1212

Re12Re

sinhcosh!121

Re!2

Re!12

sinhcosh

t

tt

t tt

hkt

t

tt

t

tt

hk

rad

yiztyizttG

yiztyiz

dkhkkhkk

ekt

yizt

kkyiz

tk

dkhkkhkk

e

ν

ν

νν

ν

ννφ

Equation 3.3–41

It is obvious that:( )

( ) ( ) ( ) ( ) ( )∑∞

=

+

⋅+⋅−⋅+⋅+⋅++

−=0

1220 ImIm12

!1212

t

ttrad yizyizt

ttG νψ

Equation 3.3–42

The function:

( ) ( ) [ ] [ ]( ) dkhkkhkk

ektG

hkt

⋅⋅⋅−⋅⋅⋅−

⋅= ∫

∞ ⋅−

0 sinhcoshννwill be treated in the next Section.

Further follows from Equation 3.3–34:


121

( ) ( )

( ) [ ] [ ]( )

( ) ( ) ( ) ( )

( )( ) [ ] [ ]( )

( ) [ ] [ ]( )( ) ( ) ( ) ( )

( ) ( )( )

( ) ( ) ( ) ( )∑

∑ ∫

∫

∑∑

∫

∞

= +

∞

=

+

∞ ⋅−−+

∞ ⋅−++

∞

=

∞

=

++

∞ ⋅−−

⋅+⋅+−⋅+⋅

⋅+

−+⋅−++=

⋅+⋅+−⋅+⋅

⋅

⋅⋅⋅−⋅⋅⋅−

⋅⋅

−⋅⋅⋅−⋅⋅⋅−

⋅

⋅+=

⋅+⋅⋅−

⋅+⋅+

⋅

⋅

⋅

⋅⋅−⋅⋅⋅−⋅⋅−

=

0 212

2

0

212

0

1222

0

122

0

22

0

1212

0

1222

Re12Re

!12122122

Re12Re

sinhcosh

sinhcosh

!121

Re!2

Re!12

sinhcosh

t tt

t

tt

hknt

hknt

t

tt

t

tt

hkn

nrad

yiztyiz

tntGntG

yiztyiz

dkhkkhkk

ek

dkhkkhkk

ek

t

yizt

kkyiz

tk

dkhkkhkk

ekk

ν

ν

ν

ννν

νν

ν

νννφ

Equation 3.3–43

It is clear that:

( ) ( )( )

( ) ( ) ( ) ( )∑

∞

= +

⋅+⋅+−⋅+⋅

⋅+

−+⋅−++−=

0 212

2

Im12Im

!12122122

t tt

nrad

yiztyiz

tntGntG

ν

νψ

Equation 3.3–44

For the imaginary parts can be written:[ ]

[ ] [ ]

( ) ( ) [ ] ( ) ( ) [ ]

[ ] [ ]

( ) ( ) [ ] ( ) ( )

⋅+⋅+

⋅⋅−

⋅+⋅

⋅⋅⋅⋅+⋅

⋅⋅−=

⋅+⋅+

⋅⋅−

⋅+⋅

⋅⋅⋅⋅+⋅

⋅⋅+=

∑∑

∑∑

∞

=

++∞

=

∞

=

++∞

=

0

1212

00

0

22

0

000

02

0

0

1212

00

0

22

0

000

02

0

Im!12

tanhIm!2

coshsinhcosh

Re!12

tanhRe!2

coshsinhcosh

t

tt

t

tt

j

t

tt

t

tt

j

yizt

hyizt

hhhh

yizt

hyizt

hhhh

νν

ν

ννννπψ

νν

ν

ννννπ

φ

Equation 3.3–45


122

[ ] [ ]

( ) ( ) [ ] ( ) ( ) ( ) ( )

[ ] [ ]

( ) ( ) [ ] ( ) ( ) ( ) ( ) jn

t

tt

t

tt

n

nj

jn

t

tt

t

tt

n

nj

yizt

hyizt

hhh

yizt

hyizt

hhh

022

012

0

0

1212

00

0

22

0

000

20

022

012

0

0

1212

00

0

22

0

000

20

Im!12

tanhIm!2

coshsinh

Re!12

tanhRe!2

coshsinh

ψννν

νν

ν

ννννπ

ψ

φννν

ννν

ννννπ

φ

⋅−⋅−=

⋅+⋅+

⋅⋅−

⋅+⋅

⋅⋅⋅⋅+⋅

⋅+=

⋅−⋅−=

⋅+⋅+

⋅⋅−

⋅+⋅

⋅⋅⋅⋅+⋅

⋅−=

−⋅

∞

=

++∞

=

−⋅

∞

=

++∞

=

∑∑

∑∑

Equation 3.3–46

3.3.3.4 Function G(t)

The function:

( ) ( ) [ ] [ ]( ) dkhkkhkk

ektG

hkt

⋅⋅⋅−⋅⋅⋅−

⋅= ∫

∞ ⋅−

0 sinhcoshνν

with unit [ ]tL −1 has two singular points: ν=k and 0ν=k , see Figure 3.3–4.

Figure 3.3–4: Singularities in the G-function

Thus, it is not possible to solve this integral directly.

First, this integral will be normalised:( ) ( )

( ) [ ] [ ]( ) duuuuhhu

eu

htGtGut

t

⋅⋅−⋅⋅⋅⋅−

⋅=

⋅=

∫∞ −

−

0

1'

sinhcoshνν

Equation 3.3–47

A substitution of:σς ⋅⋅⋅=⋅+= ieviuw 2

2


123

provides:

( ) [ ] [ ]

0

...

...............

sinhcosh

0

0

=

++++=

++++=

⋅⋅−⋅⋅⋅⋅−

⋅=

∫

∫∫∫∫∫

∫

∞

∞

−

IVIIIIII

IVIIIIII

wt

JJJJdu

dwdwdwdwdu

dwwwwhhw

ewJ

νν

From this follows:

( ) [ ] [ ] IVIIIIII

ut

JJJJduuuuhhu

eu+++−=⋅

⋅−⋅⋅⋅⋅−⋅

∫∞ −

Resinhcosh0 νν

IJ and IIJ are imaginary because they are residues and 0=IIIJ for ∞→R .

So, it remains:

( ) [ ] [ ]

( ) [ ] [ ]

⋅⋅−⋅⋅⋅⋅−

⋅−=

−=

⋅⋅−⋅⋅⋅⋅−

⋅

∫

∫

−

∞ −

IV

wt

IV

ut

dwwwwhhw

ew

J

duuuuhhu

eu

sinhcoshRe

Re

sinhcosh0

νν

νν

With the complex function:

( ) [ ] [ ] wwwhhw ~sinh~~cosh~ ⋅−⋅⋅⋅⋅− νν with real: σς ⋅⋅⋅= iew 22

~

the nominator of this integral will be made real by removing.

So:


124

( ) ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ]

( ) ( )

( )( )

[ ] [ ]( )( )

∫

∫

∞−

−

−

⋅

+−+

−+⋅

⋅−⋅−

−−−

+⋅+

+⋅−

+++

−⋅

⋅

⋅

⋅

−=

⋅

⋅−⋅⋅⋅⋅−

⋅⋅−⋅⋅⋅⋅−

⋅−⋅⋅⋅⋅−⋅⋅

−=

0

222

222

2

2

2

'

sin2cos2

tanh22cosh

2

sin4

tan1cos4

tan1

sin4

tancos4

sin4

tan1cos4

tan12

2

4cos

~sinh~~cosh~sinhcosh

~sinh~~cosh~Re

ς

ςςνςςν

ςςνςνς

νςνς

ςπςπς

ςπςςν

ςπ

ςπ

ν

ς

π

νννννν

ς

ς

d

hh

hh

hh

te

t

th

te

th

t

dw

wwwhhw

wwwhhw

wwwhhw

ew

tG

t

IV

wt

Equation 3.3–48

Because t is always odd:

( ) ( ) ( ) ( )( )

[ ] [ ]( )( )

ςς

ςςνςςν

ςςνςνς

νςνςςςςνςςςνπ ς

d

hh

hh

hh

hehttG

t

⋅

⋅

+−+

−+⋅

⋅−−+−−+

⋅−= ∫∞ −

2

sin2cos2

tanh22cosh

2

sincos4cos2sin44

cos0

222

222

2

22'

for ,......9,5,1=t

( ) ( ) ( ) ( )( )

[ ] [ ]( )( )

ςς

ςςνςςν

ςςνςνς

νςνςςςςνςςςνπ ς

d

hh

hh

hh

hehttG

t

⋅

⋅

+−+

−+⋅

⋅−−−−−+

⋅−= ∫∞ −

2

sin2cos2

tanh22cosh

2

sincos4sin2cos44

cos0

222

222

2

22'

for ,......11,7,3=t

For 1>t the function ( )tG ' becomes finite.

However, ( )1'G does not converge for 0→hν ; the integral increases monotone withdecreasing h⋅ν . This will be investigated first.

( ) ( ) ( ) [ ] [ ]( ) duuuuhhu

euGG

u

⋅⋅−⋅⋅⋅⋅−

⋅== ∫

∞ −

0

'

sinhcosh11

νν

Equation 3.3–49


125

This integral converges fast for small h⋅ν -values. This will be approximated by:

( ) ( ) ( ) duuhhu

euG

u

h⋅

−⋅⋅⋅−⋅

= ∫∞ −

→0

2101lim

ννν

Equation 3.3–50

This can be written as:

( ) ( )

( )

⋅⋅+

⋅+⋅⋅⋅

+⋅⋅−

⋅−⋅⋅⋅

+⋅⋅−

⋅⋅−

=

∫

∫

∫

∞ −

∞ −

∞ −

→→

duhu

ehh

duhu

ehh

duhu

eh

G

u

u

u

hh

0

0

0

010

121

121

11

lim1lim

ννν

ννν

νν

νν

From:

⋅++⋅−=⋅

− ∑∫∞

=

−∞ −

10 !ln

m

ma

u

mma

aeduau

e γ

follows:

( )

( ) ( )

( )( ) ( )

( )( ) ( ) ( )

⋅⋅

⋅−+⋅

+⋅⋅+⋅⋅

−

+

⋅⋅

+⋅

+⋅⋅−⋅⋅

+

+

⋅⋅

+⋅+⋅⋅−

−

=

∑

∑

∑

∞

=

⋅+

∞

=

⋅−

∞

=

⋅−

→→

1

2

1

2

1

010

!1

2ln

12

!2ln

12

!ln

1

lim1lim

m

m

mh

m

mh

m

mh

hh

mmhh

hh

e

mmhh

hh

e

mmh

hh

e

G

ννγνν

ννγνν

ννγν

ν

ν

ν

νν

or:

( )

( ) ( )

( )( )

( ) ( )

( )( )

( ) ( ) ( )

⋅⋅

⋅−+⋅−⋅

+⋅⋅−⋅⋅

⋅⋅−

−

⋅⋅

+⋅+⋅

+⋅⋅−⋅⋅

⋅⋅+

+

⋅⋅

+⋅+⋅⋅−

−

=

∑

∑

∑

∞

=

⋅+

∞

=

⋅−

∞

=

⋅−

→→

2

2

2

2

1

010

!1

2ln

12

1

!2ln

12

1

!ln

1

lim1lim

m

m

mh

m

mh

m

mh

hh

mmh

hh

hh

eh

mmh

hh

hh

eh

mmh

hh

e

G

νννγνν

ν

νννγνν

ν

ννγν

ν

ν

ν

νν

or:


126

( )

( ) ( )

[ ] [ ] ( )

[ ] [ ]( )

( ) ( )

( ) ( ) ( ) ( )

⋅⋅

⋅−⋅+⋅

⋅⋅−⋅−

⋅⋅

⋅+⋅

⋅⋅+

⋅⋅−⋅

+⋅

⋅−

⋅−⋅

+

+

⋅

+⋅

⋅⋅

−⋅⋅⋅−

+

⋅⋅

+⋅+⋅⋅−

−

=

∑∑

∑∑

∑

∞

=

⋅+∞

=

−

⋅+

∞

=

⋅−∞

=

−

⋅−

∞

=

⋅−

→→

2

2

2

21

2

2

2

21

1

01

0

!1

!1

!!

121sinh

1cosh

2lnsinh

cosh1

1

!ln

1

lim1lim

m

m

mh

m

m

mh

m

m

h

m

m

h

m

mh

hh

mmh

emm

he

mmh

emm

he

hhh

hh

hh

hh

h

mmh

hh

e

G

νν

νν

ννν

νν

νγν

ννν

ννγν

νν

νν

ν

νν

or:( ) ( )hG

h⋅−−=

→νγ

νln11lim 10

Equation 3.3–51

The imaginary part of integral in Equation 3.3–48 has been treated in Appendix III.

3.3.3.5 Hydrodynamic Loads

The hydrodynamic loads can be found from an integration of the pressures on the hull of theoscillating body in (previously) still water. With a known potential, these pressures can befound from the linear part of the instationary pressures as follows from the Bernoulli equation:

Φ⋅⋅⋅−=∂Φ∂

⋅−=

−=

ρω

ρ

jt

ppp statdyn

The potential is in-phase with the oscillation velocity. To obtain the phase of the pressureswith respect to the oscillatory motion a phase shift of 090− is required, which means amultiplication with j− . Then the pressure is:

Φ⋅⋅−= ωρdynpThe hydrodynamic force on the body is equal to the integrated pressure on the body. In thetwo-dimensional case, this is a force per unit length.The vertical force becomes:

( )( ) ∑∫

∫

∞

=

⋅⋅ ⋅⋅−⋅⋅+⋅−⋅⋅⋅⋅⋅−=

⋅+⋅⋅⋅=

⋅=

⋅+=

0

''''

''

n Snrnjnjnrnjnjnrnr

tj

VjVr

S

VjVrV

dyAAjAAeV

FjFV

dyp

FjFF

φφφφωρ

ωρ

ω

Equation 3.3–52


127

The real part of this force is equal to the hydrodynamic mass coefficient times the oscillatoryacceleration, from which the hydrodynamic mass coefficient follows:

VF

bF

m VrVr

⋅==

ω"

or non-dimensional:

VB

F

B

mC Vr

V

⋅⋅⋅⋅=

⋅⋅=

ωπρπρ 22

"

88

Equation 3.3–53

The imaginary part of the force must be equal to the hydrodynamic damping coefficient timesthe oscillatory velocity, from which the damping coefficient follows:

V

FN

Vj

V =

Instead of this coefficient, generally the ratio between amplitude of the radiated wave ζ and

the oscillatory motion z will be used. The energy balance provides:

[ ][ ]

[ ][ ] [ ] Vj

V

group

V

V

Fhhh

hV

Nhh

hg

cN

g

zA

⋅⋅⋅⋅+⋅

⋅⋅

⋅⋅=

⋅⋅⋅+⋅⋅

⋅⋅⋅

⋅⋅

=

⋅⋅

⋅=

=

000

022

00

00

2

2

2

coshsinhcosh

2sinh22sinh

2

νννν

ωρν

ννν

ρνω

ρω

ζ

Equation 3.3–54

In deep water, the hydrodynamic mass for 0→ν becomes infinite, because the potential inEquation 3.3–38 becomes:

( )rr ⋅+=∞→νγφ

νlnlim 00

Equation 3.3–55

and the non-dimensional mass of a circle becomes:

( ) ( )

−⋅

+⋅−−⋅=

⋅⋅=

∑∞

=

→→

122

2

"

00

141

ln8

8

limlim

n

V

nnr

B

mC

νγπ

πρνν

and the amplitude ratio in this deep water case becomes:

BdAd V

=

→ νν 0

lim

The hydrodynamic mass for 0→ν in shallow water remains finite. Because the multi-potentials - just as in deep water – provide finite contributions, the radiation potential has to


128

be discussed only, which is decisive (infinite mass) in deep water. The change-over borderline''deep to shallow'' water provides for this radiation potential:

( ) ( ) ( )

+=

⋅−⋅+=+∞→

hr

hrradr

ln

lnlnlim 000

γ

ννγφφν

Equation 3.3–56

It is obvious that Equation 3.3–56 - just as Equation 3.3–55 - provides an infinite value.

When the contributions of the multi-potentials (which disappear here for the borderline case0→ν ) are ignored, it follows from Equation 3.3–27 for the amplitude ratio in shallow water:

[ ][ ] [ ]

[ ][ ] [ ]

[ ][ ] [ ]

[ ] [ ][ ] [ ]hhh

hhA

hhhh

VA

hhhh

zA

hhhh

zgA

zAV

⋅⋅⋅+⋅⋅⋅⋅⋅

⋅⋅=

⋅⋅⋅+⋅⋅

⋅⋅⋅=

⋅⋅⋅+⋅⋅

⋅⋅⋅

⋅=

⋅⋅⋅+⋅⋅

⋅⋅⋅

⋅=

=

000

000'0

000

02

0

000

02

0

000

02

0

coshsinhcoshsinh

coshsinhcosh

coshsinhcosh

coshsinhcosh

ννννννπ

νννννπ

νννν

ωπν

ννννπω

ζ

Because:

1lim'

0

0=

⋅→ B

A πν

follows:[ ] [ ]

[ ] [ ]hhhhh

BAV⋅⋅⋅+⋅

⋅⋅⋅⋅⋅=

→000

000

0 coshsinhcoshsinh

limννν

νννν

and:

[ ] [ ]∞=

⋅⋅⋅+⋅

⋅=

⋅=

→

→→

hhhB

dd

dAd

dAd VV

0000

0

000

coshsinh1

lim2

limlim

0

0

ννν

νν

νν

ν

νν

Thus, ( )2BAV ⋅ν has at 02 =⋅ Bν a vertical tangent.

The fact that the hydrodynamic mass goes to infinity for zero frequency can be explainedphysically as follows. The smaller the frequency becomes, the longer becomes the radiatedwave and the faster travels it away from the cylinder. In the borderline case 0→ν has thewave an infinite length and it travels away - just as the pressure (incompressible fluid) - withan infinite velocity. This means that all fluid particles are in phase with the motions of thebody. This means that the hydrodynamic force is in phase with the motion of the body, whichholds too that:


129

0arctanarctan '

'

=

=

=∞=

Vr

Vj

Vr

ViHT F

FFFε

This condition is fulfilled only when 0' =VjF or ∞== ρ"' mFVj .

However, 'VjF is finite:

2

22

4

2'

νωωρωρV

VVVj

VjA

AgN

VF

F =⋅=⋅

=⋅⋅

=

Because νν

⋅=→

BAV00

lim follows 2'

00

lim BFVj =→ν

. The term ρ"' mFVj = has to be infinite.

The finite value of the hydrodynamic mass at shallow water is physically hard to interpret. Afull explanation is not given here. However, it has been shown here that the result makes somesense. At shallow water can the wave (even in an incompressible fluid) not travel with aninfinite velocity; its maximum velocity is hg ⋅ . In case of long waves at shallow water, theenergy has the same velocity. From that can be concluded that at low decreasing frequenciesthe damping part in the hydrodynamic force will increase. This means that:

0arctan '

'

≠

=∞≠

Vr

ViHT

FFε

So, ρ"' mFVr = has to be finite.

3.3.3.6 Wave Loads

The wave forces EF on the restrained body in waves consist of:• forces 1F in the undisturbed incoming waves (Froude-Krylov hypothesis) and• forces caused by the disturbance of the waves by the body:

• one part 2F in phase with the accelerations of the water particles and• another part 3F in phase with the velocity of the water particles.

Thus:

321 FjFFFE ⋅++=

These forces will be determined from the undisturbed wave potential WΦ and the disturbance

potential SΦ . As mentioned before, for 090≠µ only an approximation will be found.

( )

[ ] [ ] [ ] ( )

( ) ∫

∑∫

∫

⋅

⋅+⋅⋅+⋅−⋅⋅

+⋅⋅⋅⋅⋅⋅−⋅

⋅⋅⋅

⋅⋅⋅−=

⋅Φ+Φ⋅⋅−=

⋅++=

∞

=

⋅⋅⋅−

⋅⋅

S

n Snrnjnjnrnjnjnrnr

xj

tj

SSW

E

dy

AAjAAV

yzhz

e

e

dy

FjFFF

0

''''

0000

cos

321

sincossinhtanhcosh

0

φφφφ

µνννννωζ

ωρ

ωρ

µν

ω

Equation 3.3–57


130

In here:

( )

[ ] [ ] [ ] ( )

dyAAeVF

dyAAeVF

dyyzhz

eF

n S

nrnjnjnrtj

n S

njnjnrnrtj

S

xtj

⋅⋅+⋅⋅⋅⋅⋅−=

⋅⋅−⋅⋅⋅⋅⋅−=

⋅⋅⋅⋅⋅⋅⋅−⋅

⋅⋅⋅⋅

−=

∑∫

∑∫

∫

∞

=

⋅⋅

∞

=

⋅⋅

⋅⋅−⋅⋅

0

''3

0

''2

0000

cos2

1

sincossinhtanhcosh

0

φφωρ

φφωρ

µννννν

ζωρ

ω

ω

µνω

Using ωζ ⋅=V , the non-dimensional amplitudes are:

[ ] [ ] [ ] ( )

dyAAB

BgF

E

dyAAB

BgF

E

dyyzhzB

BgF

E

n S

nrnjnjnr

n S

njnjnrnr

S

⋅⋅+⋅⋅−=

⋅⋅⋅=

⋅⋅−⋅⋅−=

⋅⋅⋅=

⋅⋅⋅⋅⋅⋅⋅−⋅⋅−=

⋅⋅⋅=

∑∫

∑∫

∫

∞

=

∞

=

0

''

33

0

''

22

0000

11

sincossinhtanhcosh1

φφνζρ

φφνζρ

µνννν

ζρ

Equation 3.3–58

In case of 090=µ , so beam waves, the theory of Haskind-Newman – see Haskind [1957] or

Newman [1962] - can be used too to determine the amplitudes 1E , 2E and 3E .When tj

WW e ⋅⋅⋅=Φ ωφ is the potential of the incoming wave and tjSS e ⋅⋅⋅=Φ ωφ is the potential

of the disturbance by the body at a large distance from the body with velocity amplitude1=V , then:

dzyy

eFh

WW

tjE ⋅

∂

∂⋅−

∂∂

⋅⋅⋅⋅−= ∫⋅⋅

0

φφφφωρ ω

Equation 3.3–59

According to Equation 3.3–105 in Appendix I is:

[ ] [ ] [ ] ( )yzhzW ⋅⋅⋅⋅⋅−⋅⋅⋅

= 0000 cossinhtanhcosh νννννωζφ

From the previous subsections follows the asymptotic expression for the disturbance potentialin still water with 1=V :


131

( )[ ]

[ ] [ ][ ] [ ] [ ] ( )

( ) ( ) ( )

⋅⋅+⋅−−⋅+

⋅⋅⋅⋅⋅⋅−⋅

⋅⋅⋅⋅+⋅

⋅⋅=

∑∞

=

−

∞→

1

120

''220

'0

'0

0000

000

02

sinsinhtanhcosh

coshsinhcosh

n

nnjnrjr

y

AjAAjA

yzhz

hhhh

ννν

ννννννν

νπφ

Substituting this in Equation 3.3–59, provides:[ ]

[ ] [ ]

[ ] [ ] [ ]

( ) ( ) ( )

( ) ( ) ( )

⋅⋅+⋅−−⋅+

⋅⋅⋅⋅⋅−=

⋅⋅+⋅−−⋅+

⋅⋅⋅⋅⋅−⋅

⋅⋅⋅⋅+⋅

⋅⋅⋅⋅⋅⋅⋅⋅−=

∑

∑

∫

∞

=

−

⋅⋅

∞

=

−

⋅⋅

1

120

''220

'0

'0

1

120

''220

'0

'0

0

2000

000

02

0

sinhtanhcosh

coshsinhcosh2

n

nnjnrjr

tj

n

nnjnrjr

h

tjE

AjAAjA

eg

AjAAjA

dzzhz

hhhh

egF

ννν

πζρ

ννν

ννν

ννννπνζρ

ω

ω

Non-dimensional:

( ) ( )

( ) ( )

⋅⋅−−⋅−=

⋅⋅⋅=

⋅⋅−−⋅−=

⋅⋅⋅=+

∑

∑

∞

=

−

∞

=

−

1

120

'220

'0

3

1

120

'220

'0

21

Im

Re

n

nnjj

E

n

nnrr

E

AAB

BgF

E

AAB

BgF

EE

νννπ

ζρ

νννπ

ζρ

Equation 3.3–60

3.3.3.7 Solution

The Lewis transformation of a cross section is given by:θθθ 3⋅−⋅−⋅+ ⋅+⋅+=⋅+ iii ebeaeziy

Equation 3.3–61

Then, the co-ordinates of the cross section are:( ) ( ) ( )( ) ( ) ( )θθ

θθ3sinsin1

3coscos1

⋅−⋅−=⋅+⋅+=

baz

bay

Equation 3.3–62

Then:


132

( ) ( )θθθ 3⋅+⋅+⋅− ⋅+⋅+⋅=⋅−⋅=⋅+ iii ebeaeiziyiyiz

Equation 3.3–63

All calculations will be carried out in the Lewis domain. Scale factors are given in the tablebelow.

Ship Lewis formform Lewis

Ship

Breadth BR ( )ba ++⋅ 12 ( ) baBR ++⋅ 12Draught TI ba +−1 ( )baTI +−1Water depth TIHT ⋅ ( )baHTWT +−⋅= 1 ( )baTI +−1Wave number

0ν ( )baTIWF +−⋅= 1/0ν ( ) TIba +−1Acceleration g g 1Forces

GF F ( ) 21 baTI +−

Table 3.3–1: Lewis form parameters

3.3.3.8 Determination of Source Strengths An

The yet unknown complex coefficients nA ( ∞≤≤ n0 ), the source strengths of the by the flowgenerated singularity, can be determined by substituting the stream function in Equation 3.3–36 and the co-ordinates of the cross section in the relevant boundary conditions in Equation3.3–7 through Equation 3.3–9.

tzytzybodybodybody

,,,, Ψ=Ψ

Equation 3.3–64

To determine the unknowns nA , an equal number of equations has to be formulated. Becauseonly Lewis forms are used here, a simple approach is possible.All stream function parts and boundary conditions can be given as a Fourier series:

( ) ( )[ ] ∑∞

=

⋅+⋅+⋅⋅=0

12cos2sinm

mnmnn mdmc θθψ

or with:

( )( ) ( )( )[ ] ( )∑

∞

=

⋅⋅+−⋅

+⋅+⋅−=⋅+1

22

2

2sin124

1222112cosk

kmkk

mm θππ

θθ

in:

( ) ∑∞

=

⋅⋅+=1

0 2sinm

nmnn maa θψ

The solution of the by equating coefficients generated equations provide the unknowns 'nA .


133

3.3.4 Horizontal Motions

3.3.4.1 Boundary Conditions

The first four assumptions for the vertical motions are valid for horizontal motions too. Thepotential must fulfil the motion-dependent boundary conditions, which have been substitutedin Equation 3.3–3 through Equation 3.3–6. However, the fifth boundary condition needs herea new formulation.

Because two motions (a translation and a rotation) are considered here, follows from:

bodynbody

vn

=∂Φ∂

also in still water two boundary conditions:

1. For sway:

body

tj

body

body

bodynbody

dsdz

eU

dsd

dsdz

ydsdy

z

vn

⋅⋅−

Ψ−=

⋅∂Φ∂

−⋅∂Φ∂

=

=∂Φ∂

⋅⋅ω

or:

bodytj

bodydzeUd ⋅⋅=Ψ ⋅⋅ω

from which follows:( ) CzeUtzy

bodytj

body+⋅⋅=Ψ ⋅⋅ω,,

Equation 3.3–65

2. For roll:

body

tj

body

body

bodynbody

rdsdr

e

dsd

dsdz

ydsdy

z

vn

⋅⋅⋅⋅=

Ψ−=

⋅∂Φ∂

−⋅∂Φ∂

=

=∂Φ∂

⋅⋅ωωφ

or:

bodytj

bodydrred ⋅⋅⋅⋅−=Ψ ⋅⋅ωωφ

from which follows:


134

( ) Czyetzybody

tjbody

++⋅⋅⋅

−=Ψ ⋅⋅ 22

2,, ωωφ

Equation 3.3–66

For the restrained body in waves, only the force in the horizontal direction and the momentabout the longitudinal axis of the body will be calculated. One gets in beam waves only the iny -point-symmetric part of the potential and the in y -symmetric part of the stream function ofthe wave (see Appendix I), respectively:

( )

[ ] [ ] [ ] ( )body

tjbodyS

yzhz

etzy

⋅⋅⋅⋅⋅−⋅

⋅⋅⋅

=Ψ ⋅⋅

0000 coscoshtanhsinh

,,

νννννωζ ω

Equation 3.3–67

and in oblique waves:

( ) ( )

( )[ ] [ ] [ ] ( )

( ) [ ] [ ] [ ] body

y

tjbodyS

dyzhzy

zhz

y

xetzyx

∫ ⋅⋅⋅⋅−⋅⋅⋅⋅

⋅−⋅−

⋅⋅⋅−⋅

⋅⋅⋅⋅+

⋅⋅⋅⋅⋅⋅

=Ψ ⋅⋅

1

00000

20

10010

10

011

coshtanhsinhsinsin

sin1

coshtanhsinh

sincossin

cossin,,,

νννµν

µν

νννµνµ

µννωζ ω

Equation 3.3–68

3.3.4.2 Potentials


In a similar way as Equation 3.3–22 for heave, the three-dimensional radiation potential forsway and roll can be derived as:

( ) ( ) ( )( )

( )

( ) ξ

ν

ξξω

ddkdkyk

hkkkkhkk

hzkkkk

xkAetzyx

yxy

yxyxyx

yxyx

x

L

L

tjr

⋅⋅⋅⋅

⋅

⋅+⋅+−

⋅+⋅

−⋅+⋅+

⋅−⋅⋅⋅=Φ

∫

∫∫

∞

∞+

−

⋅⋅

sin

sinhcosh

cosh

cos,,,

0222222

2222

0

2

2

00

Equation 3.3–69

This expression reduces for the two-dimensional case ( 0=xk and kk y = ) into:


135

( )( )[ ]

[ ] [ ] ( ) dkykhkkhk

hzkk

Aetzy tjr

⋅⋅⋅⋅⋅−⋅⋅

−⋅⋅

⋅⋅=Φ

∫∞

⋅⋅

sinsinhcosh

cosh

,,

0

00

ν

ω

Equation 3.3–70

With the Sommerfeld radiation condition in Equation 3.3–6 and Appendix III, the totalradiation potential becomes:

( ) ( )( )[ ]

[ ] [ ] ( )

( )[ ] [ ][ ] [ ] ( )

⋅⋅⋅⋅⋅+⋅

⋅⋅−⋅⋅⋅⋅

+⋅⋅⋅⋅⋅−⋅⋅

−⋅⋅

⋅⋅=Φ⋅+Φ

∫∞

⋅⋅

yhhh

hhzj

dkykhkkhk

hzkk

Aetzyjtzy tjjr

0000

000

0

000

sincoshsinh

coshcosh

sinsinhcosh

cosh

,,,,

νννν

νννπ

ν

ω

Equation 3.3–71

and the stream function becomes:( ) ( )

( )[ ][ ] [ ] ( )

( )[ ] [ ][ ] [ ] ( )

⋅⋅⋅⋅⋅+⋅

⋅⋅−⋅⋅⋅⋅

+⋅⋅⋅⋅⋅−⋅⋅

−⋅⋅

⋅⋅=Ψ⋅+Ψ

∫∞

⋅⋅

yhhh

hhzj

dkykhkkhk

hzkk

Aetzyjtzy tjjr

0000

000

0

000

coscoshsinh

coshsinh

cossinhcosh

sinh

,,,,

νννν

νννπ

ν

ω

Equation 3.3–72

Potential and stream function are divided in:

( )

[ ] [ ][ ] [ ] ( )

( )[ ] [ ][ ] [ ] ( )

⋅⋅⋅⋅⋅+⋅

⋅⋅−⋅⋅⋅⋅

+⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅−

⋅

+⋅⋅⋅−

⋅

⋅⋅=Φ⋅+Φ+Φ=Φ

∫

∫∞ ⋅−

∞ ⋅−

⋅⋅∞

yhhh

hhzj

dkykhkkhkzkkzk

kek

dkykk

ek

Aej

hk

zk

tjjradr

0000

000

0

0

00000

sincoshsinh

coshcosh

sinsinhcoshcoshsinh

sin

νννν

νννπ

νν

ν

ν

ω

Equation 3.3–73


136

( )

[ ] [ ][ ] [ ] ( )

( )[ ] [ ][ ] [ ] ( )

⋅⋅⋅⋅⋅+⋅

⋅⋅−⋅⋅⋅⋅

+⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅−

⋅

+⋅⋅⋅−

⋅

⋅⋅=Ψ⋅+Ψ+Ψ=Ψ

∫

∫∞ ⋅−

∞ ⋅−

⋅⋅∞

yhhh

hhzj

dkykhkkhkzkkzk

kek

dkykk

ek

Aej

hk

zk

tjjradr

0000

000

0

0

00000

coscoshsinh

coshsinh

cossinhcoshsinhcosh

cos

νννν

νννπ

νν

ν

ν

ω

Equation 3.3–74

3.3.4.2.2 2-D Multi-Potential

The two-dimensional multi-pole potential becomes:

( ) ( ) ( )

( ) ( )

[ ] [ ][ ] [ ] ( )

( )

[ ]( )[ ]

[ ] [ ] ( )yhhh

hzh

Aetzy

dkykhkkhkzkkzk

ekkAetzy

dkykekkAetzy

n

ntj

nj

hknn

tjnrad

zknn

tjnr

⋅⋅⋅⋅⋅+⋅

−⋅⋅

⋅⋅

⋅⋅+=Φ

⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅⋅⋅+⋅⋅−=Φ

⋅⋅⋅⋅⋅+⋅⋅−=Φ

+

⋅⋅

⋅−−∞

⋅⋅

⋅−−∞

⋅⋅∞

∫

∫

0000

0

0

120

12

0

12

0

sincoshsinh

coshcosh

,,

sinsinhcoshcoshsinh

,,

sin,,

νννν

νν

νπ

νν

ν

ν

ω

ω

ω

Equation 3.3–75

The related stream function is:

( ) ( ) ( )

( ) ( )

[ ] [ ][ ] [ ] ( )

( )

[ ]( )[ ]

[ ] [ ] ( )yhhh

hzh

Aetzy

dkykhkkhkzkkzk

ekkAetzy

dkykekkAetzy

n

ntj

nj

hknn

tjnrad

zknn

tjnr

⋅⋅⋅⋅⋅+⋅

−⋅⋅

⋅⋅

⋅⋅+=Ψ

⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅

⋅⋅⋅+⋅⋅−=Ψ

⋅⋅⋅⋅⋅+⋅⋅+=Ψ

+

⋅⋅

⋅−−∞

⋅⋅

⋅−−∞

⋅⋅∞

∫

∫

0000

0

0

120

12

0

12

0

coscoshsinh

sinhcosh

,,

cossinhcoshsinhcosh

,,

cos,,

νννν

νν

νπ

νν

ν

ν

ω

ω

ω

Equation 3.3–76


137

3.3.4.2.3 Total Potentials

With exception of the complex constant nA with ∞≤≤ n0 , the potential is known:

( ) ( ) ( )( )( )

[ ] [ ][ ] ( )( )

[ ] [ ]

( )

( ) ( )( ) [ ] [ ]

[ ]( )( )

[ ] [ ]

∑ ∫

∫

∞

=+

∞+

⋅⋅

∞

⋅⋅

⋅⋅⋅+⋅−⋅+⋅

⋅⋅

⋅⋅

−⋅⋅⋅−⋅⋅

−⋅+⋅⋅⋅−

⋅⋅+

⋅−

⋅⋅⋅+⋅−⋅+⋅⋅⋅

⋅⋅⋅

+⋅⋅⋅−⋅⋅

−⋅+⋅⋅

⋅⋅+⋅+=Ψ⋅+Φ

1

000

0

0

120

0

1222

000

000

0

0

coshsinhsin

cosh

sinhcoshsin

coshsinhsincosh

sinhcoshsin

,,,,

nn

n

njnr

tj

ojrtj

hhhhziy

hj

dkhkkhk

hziykkk

AjA

e

hhhhziyh

j

dkhkkhk

hziykk

AjAetzyitzy

νννν

ννπ

νν

ννννννπ

ν

ω

ω

Equation 3.3–77

Writing this in a similar way as for heave provides:

( )( )

( ) ∑∞

= ∞

∞⋅⋅

⋅++⋅⋅−

⋅−+⋅⋅+=Φ

0

,,n njnrnradnrnj

njnjnradnrnrtj

AAj

AAetzy

φφφ

φφφω

Equation 3.3–78

and

( )( )

( ) ∑∞

= ∞

∞⋅⋅

⋅++⋅⋅+

⋅−+⋅⋅+=Ψ

0

,,n njnrnradnrnj

njnjnradnrnrtj

AAj

AAetzy

ψψψ

ψψψω

Equation 3.3–79

with:( )

( ) rollfor

swayfor ''

''

jrjr

jrjr

AjAAjA

AjAUAjA

⋅+⋅⋅=⋅+

⋅+⋅=⋅+

ωφ

'nA has dimension [ ]22 +nL .

nnA φ⋅' and nnA ψ⋅' have for sway dimension [ ]L and for roll dimension [ ]2L .

The determination of the coefficients 'nA follow from the boundary conditions at the body

contour.

3.3.4.3 Expansion of Potential Parts

For 022 →+⋅=⋅ zyr νν :


138

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

⋅⋅

⋅+⋅

⋅+

+⋅⋅

⋅+⋅

⋅+⋅+

⋅⋅−+

+=

⋅⋅

⋅+⋅

⋅+

−⋅⋅

⋅+⋅

⋅+⋅+

⋅⋅++

−=

∑

∑

∑

∑

∞

=

∞

=⋅−∞

∞

=

∞

=⋅−∞

yyizmmz

y

yyizmm

r

ezy

z

yyizmmz

y

yyizmm

r

ezy

y

m

mm

m

m

m

zr

m

mm

m

m

m

zr

νν

νν

νγνψ

νν

νννγ

νφ

ν

ν

sinIm!

arctan

cosRe!

ln

cosIm!

arctan

sinRe!

ln

1

1

220

1

1

220

Equation 3.3–80

with the Euler constant: 57722.0=γ .

For ∞→+⋅=⋅ 22 zyr νν :

( ) ( ) ( )

( ) ( ) ( )yyy

eyizr

mzy

z

yyy

eyizr

mzy

y

zM

m

mmmr

zM

m

m

mmr

⋅⋅⋅⋅⋅−

⋅+⋅

⋅−

⋅−+

+=

⋅⋅⋅⋅⋅−

⋅+⋅

⋅−

⋅++

−=

⋅−

=∞

⋅−

=∞

∑

∑

ννπν

νψ

ννπν

νφ

ν

ν

sinRe!1

cosIm!1

12220

12220

Equation 3.3–81

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

⋅+⋅−⋅+⋅⋅−=

⋅+⋅+⋅+⋅⋅−=

−+−+∞

−+−+∞

nnnnr

nnnnr

ziyn

ziyn

ziyn

ziyn

2121

2121

Re2

Im!21

Im2

Re!21

νψ

νφ

Equation 3.3–82

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ∑

∑∞

=

+

∞

=

+

⋅+⋅+

⋅−⋅+⋅++

=

⋅+⋅+⋅−⋅+⋅++=

0

2120

0

2120

Re!212

Re!1232

Im!212Im

!1232

t

ttrad

t

ttrad

yizttG

yizttG

yizttGyiz

ttG

νψ

νφ

Equation 3.3–83

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

∑

∑

∞

=

+

∞

=

+

⋅+⋅−+⋅−++

⋅−

⋅+⋅+

++⋅−+++

=

⋅+⋅−+⋅−++

⋅−

⋅+⋅+

++⋅−+++

=

0 22

122

0 22

122

Re!2

122122

Re!12

122322

Im!2

122122

Im!12

122322

t t

t

nrad

t t

t

nrad

yizt

ntGntG

yizt

ntGntG

yizt

ntGntG

yizt

ntGntG

νν

ν

ψ

νν

ν

φ

Equation 3.3–84


139

[ ][ ] [ ]

( ) ( )

[ ] ( ) ( ) [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )

⋅+⋅⋅⋅−

⋅+⋅+

⋅⋅⋅⋅+⋅

⋅⋅⋅+=

⋅+⋅⋅⋅−

⋅+⋅+

⋅⋅⋅⋅+⋅

⋅⋅⋅+=

∑

∑

∑

∑

∞

=

∞

=

++

∞

=

∞

=

++

0

22

00

0

1212

0

000

02

0

0

22

00

0

1212

0

000

02

0

Re!2

tanh

Re!12

coshsinhcosh

Im!2

tanh

Im!12

coshsinhcosh

t

tt

t

tt

oj

t

tt

t

tt

oj

yizt

h

yizt

hhhh

yizt

h

yizt

hhhh

νν

ν

νννννπψ

νν

ν

νννννπφ

Equation 3.3–85( ) ( )( ) ( ) jn

nj

jn

nj

022

012

0

022

012

0

ψνννψ

φνννφ

⋅−⋅=

⋅−⋅=−

−

Equation 3.3–86

3.3.4.4 Zero-Frequency Potential

Grim [1956, 1957] gives for the horizontal motions at zero frequency the complex potential:

( ) ( )

( ) ( ) ( ) ( )

⋅−⋅++⋅+⋅+

+⋅+⋅=⋅+∑∑ ∞

=

+−+−

+−∞

=

1

1212

12

0 22m

nn

n

nn mhiziymhiziy

ziyAi ψφ

Equation 3.3–87

For Lewis forms this becomes:


140

( ) ( ) ( )

( ) ( )

( )

( ) ( )

( ) ( ) ( )

( )

⋅⋅⋅

⋅

+⋅

+

++

⋅⋅⋅−⋅

+

⋅⋅⋅

⋅

+⋅−

⋅=

⋅

⋅+⋅+

⋅

+

++⋅−

⋅⋅⋅⋅⋅

+

⋅+⋅⋅

+⋅−⋅

⋅=⋅+

∑∑ ∑∑

∑ ∑

∑

∑ ∑

∑

∑

∞

=

∞

=

+

= =

⋅+−−−

++−+

∞

= =

⋅+++⋅−−

∞

=

∞

=

∞

=+⋅−⋅−⋅+

+−

∞

=

⋅−⋅−⋅+⋅−

∞

=

1 0

12

0 0

1222

12

0 0

1222

0

1 0123

12

0

4212

0

12

12

122

21

2

21

2

12

1221

22

21

m p

p

l

l

k

klpikkl

npnp

p

p

l

lpnillpp

nn

m ppiii

p

n

p

piipni

nn

ebak

l

l

p

p

pn

Hm

ebal

p

p

pn

A

Hmebeae

p

np

Hmii

ebeap

pne

Ai

θ

θ

θθθ

θθθ

ψφ

Equation 3.3–88

These sums converge as long as:

( ) 1122

33

<+−⋅⋅⋅+⋅+

=⋅

⋅+⋅+ ⋅−⋅−⋅+⋅−⋅−⋅+

baHTmebeae

Hmebeae iiiiii θθθθθθ

Equation 3.3–89

Because 1≥m , it follows the condition:

baebea

HTii

+−⋅+⋅+

≥⋅⋅−⋅−

11

242! θθ

Equation 3.3–90

The potential converges too when:

TB

baba

HT⋅

=+−++

≥⋅21

12 or hB ⋅≤ 4

Equation 3.3–91

3.3.4.5 Hydrodynamic Loads

The hydrodynamic force at sway oscillations in still water becomes:

( ) dzAAjAA

eU

FjFF

n Snrnjnjnrnjnjnrnr

tj

QjQrQ

⋅⋅+⋅⋅+⋅−⋅

⋅⋅⋅⋅−=

⋅+=

∑∫∞

=

⋅⋅

0

'''' φφφφ

ωρ ω

Equation 3.3–92

and at roll oscillations:


141

( ) dzAAjAA

e

FjFF

n Snrnjnjnrnjnjnrnr

tj

RjRrR

⋅⋅+⋅⋅+⋅−⋅

⋅⋅⋅⋅−=

⋅+=

∑∫∞

=

⋅⋅

0

''''

2

φφφφ

φωρ ω

Equation 3.3–93

The hydrodynamic moment at sway oscillations in still water becomes:

( ) ( )dzzdyyAAjAA

eU

MjMM

n Snrnjnjnrnjnjnrnr

tj

QjQrQ

⋅+⋅⋅⋅+⋅⋅+⋅−⋅

⋅⋅⋅⋅−=

⋅+=

∑∫∞

=

⋅⋅

0

'''' φφφφ

ωρ ω

Equation 3.3–94

and at roll oscillations:

( ) ( )dzzdyyAAjAA

e

MjMM

n Snrnjnjnrnjnjnrnr

tj

RjRrR

⋅+⋅⋅⋅+⋅⋅+⋅−⋅

⋅⋅⋅⋅−=

⋅+=

∑∫∞

=

⋅⋅

0

''''

2

φφφφ

φωρ ω

Equation 3.3–95

Of course, the coefficients 'nrA and '

njA of sway and roll will differ.

Fictive moment levers are defined by:

RrRr

QrQr

FI

H

mUM

H

φωω

⋅⋅=

⋅⋅=

2"

"

Rj

RRj

Q

QjQj

F

BNH

NUM

H

⋅⋅⋅⋅

=

⋅=

2

φω

Non-dimensional values for the sway motions are:

[ ][ ] [ ]

Qj

QjQj

Qr

QrQr

QjQ

QrH

FTM

TH

FTM

TH

Fhhh

hUy

A

TU

F

T

mC

⋅=

⋅=

⋅⋅⋅⋅+⋅

⋅⋅

⋅⋅==

⋅⋅⋅⋅=

⋅⋅=

000

022

2

22

22

"

coshsinhcosh

22

νννν

ωρνζ

πωρπρ

Equation 3.3–96

and for roll motions:


142

[ ][ ] [ ]

Rj

RjRj

Rr

RrRr

RjR

RrR

FT

MT

H

FT

MT

H

Mhhh

h

BBA

T

M

T

IC

⋅=

⋅=

⋅⋅⋅⋅+⋅

⋅⋅

⋅⋅⋅⋅

=⋅

=

⋅⋅⋅⋅=

⋅⋅=

000

02

22

2

22

22

424

"

coshsinhcosh4

4

88

νννν

φωρν

φ

ζ

φπωρπρ

Equation 3.3–97

3.3.4.6 Wave Loads

The wave loads are separated in contributions of the undisturbed wave and diffraction:

[ ] [ ] [ ]( ) ( )

( ) ∫

∑

∫

⋅

⋅+⋅⋅+⋅−⋅⋅+

⋅⋅⋅⋅⋅⋅−⋅

⋅⋅⋅

−

⋅⋅⋅−=

⋅Φ+Φ⋅⋅−=

⋅++=

∞

=

⋅⋅⋅−

⋅⋅

S

nnrnjnjnrnjnjnrnr

xj

tj

SSW

E

dz

AAjAAU

yzhz

e

e

dz

FjFFF

0

''''

0000

cos

321

sinsinsinhtanhcosh

0

φφφφ

µνννννωζ

ωρ

ωρ

µν

ω

Equation 3.3–98

[ ] [ ] [ ]( ) ( )

( ) ( )∫

∑

⋅+⋅⋅

⋅+⋅⋅+⋅−⋅⋅+

⋅⋅⋅⋅⋅⋅−⋅

⋅⋅⋅

−

⋅⋅⋅−=

⋅++=

∞

=

⋅⋅⋅−

⋅⋅

S

nnrnjnjnrnjnjnrnr

xj

tj

E

dzzdyy

AAjAAU

yzhz

e

e

MjMMM

0

''''

0000

cos

321

sinsinsinhtanhcosh

0

φφφφ

µνννννωζ

ωρ

µν

ω

Equation 3.3–99

The separate parts are:


143

( )

[ ] [ ] [ ]( ) ( )

dzAAeUF

dzAAeUF

dzyzhz

eF

n Snrnjnjnr

tj

n S

njnjnrnrtj

S

xtj

⋅⋅+⋅⋅⋅⋅⋅−=

⋅⋅−⋅⋅⋅⋅⋅−=

⋅⋅⋅⋅⋅⋅⋅−⋅

⋅⋅⋅⋅

+=

∑∫

∑∫

∫

∞

=

⋅⋅

∞

=

⋅⋅

⋅⋅−⋅⋅

0

''3

0

''2

0000

cos2

1

sinsinsinhtanhcosh

0

φφωρ

φφωρ

µννννν

ζωρ

ω

ω

µνω

( )

[ ] [ ] [ ]( ) ( ) ( )

( )

( )dzzdyyAAeUM

dzzdyyAAeUM

dzzdyyyzhz

eM

n Snrnjnjnr

tj

n S

njnjnrnrtj

S

xtj

⋅+⋅⋅⋅+⋅⋅⋅⋅⋅−=

⋅+⋅⋅⋅−⋅⋅⋅⋅⋅−=

⋅+⋅⋅⋅⋅⋅⋅⋅⋅−⋅

⋅⋅⋅⋅

+=

∑∫

∑∫

∫

∞

=

⋅⋅

∞

=

⋅⋅

⋅⋅−⋅⋅

0

''3

0

''2

0000

cos2

1

sinsinsinhtanhcosh

0

φφωρ

φφωρ

µννννν

ζωρ

ω

ω

µνω

Dimensionless:

[ ]

[ ] [ ] [ ]( ) ( )

[ ]

[ ] dzAAA

h

AgF

E

dzAAA

h

AgF

E

dzyzhz

Ah

AgF

E

n Snrnjnjnr

x

x

n Snjnjnrnr

x

x

S

x

x

⋅⋅+⋅⋅⋅

−=

⋅⋅⋅⋅=

⋅⋅−⋅⋅⋅

−=

⋅⋅⋅⋅=

⋅⋅⋅⋅⋅⋅⋅−⋅

⋅⋅

⋅+=

⋅⋅⋅⋅=

∑∫

∑∫

∫

∞

=

∞

=

0

''0

0

33

0

''0

0

22

0000

0

0

11

tanh

tanh

sinsinsinhtanhcosh

tanh

φφν

νζρ

φφν

νζρ

µνννν

νν

νζρ

( )21

21

FFTMM

TH W r

+⋅+

= 3

3

FTM

TH W j

⋅=

Equation 3.3–100

The Haskind-Newman relations – see Newman [1962] - are valid too here:


144

( ) ( )

( ) ( )

⋅⋅−+⋅=

⋅⋅⋅⋅=

⋅⋅−+⋅=

⋅⋅⋅⋅=+

∑

∑

∞

=

−

∞

=

−

1

120

'220

'0

0

3

1

120

'220

'0

0

21

Im

Re

n

nnjj

x

x

E

n

nnrr

x

x

E

AAA

AgF

E

AAA

AgF

EE

νννπ

νζρ

νννπ

νζρ

Equation 3.3–101

3.3.4.7 Solution

To determine the unknowns nA , an equal number of equations have to be formulated. BecauseLewis forms are used only here, a simple approach is possible.All stream function parts and boundary conditions can be given as a Fourier series:

( )( ) ( ) ∑∞

=

⋅⋅+⋅+⋅=0

2cos12sinm

nmnmn mdmc θθψ

or with

( ) ( ) ( ) ( ) ( )( )∑∞

=

⋅+⋅+⋅+−⋅++

⋅+=⋅0

2

12sin12122122

1612cos

k

kkmkmk

mm θ

πθ

in:

( )( ) ∑∞

=

⋅+⋅=0

12sinm

nmn ma θψ

The solution of the by equating coefficients generated equations provide the unknowns 'nA .

3.3.5 Appendices

3.3.5.1 Appendix I: Undisturbed Wave Potential

The general expression of the complex potential of a shallow water wave, travelling in thenegative y -direction, is:

( )( ) [ ]

[ ]( )[ ] ( )

( )[ ] ( )

[ ]( )[ ] ( )

( )[ ] ( )

⋅+⋅⋅−⋅⋅−

⋅+⋅⋅−⋅⋅

⋅⋅⋅

=

⋅+⋅⋅−⋅⋅−⋅+⋅⋅−⋅

⋅⋅⋅

⋅=

=⋅

⋅+−⋅+⋅⋅⋅=Ψ⋅+Φ

tyhzi

tyhz

h

tyhzi

tyhz

h

ch

thziyci WW

ωννωνν

ννωζ

ωννωνν

ννωζ

νω

νων

ζ

00

00

0

00

00

00

00

0

sinsinh

coscosh

cosh

sinsinh

coscosh

sinh

:ith wsinh

cos

Equation 3.3–102


145

[ ] ( )[ ] ( )

[ ] ( )[ ]

[ ] [ ] [ ] yjtj

yjtj

W

ezhze

ehzeh

tyhzh

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅−⋅⋅⋅⋅

=

⋅−⋅⋅⋅⋅⋅

⋅=

⋅+⋅⋅−⋅⋅⋅⋅

⋅+=Φ

0

0

000

00

000

sinhtanhcosh

coshcosh

coscoshcosh

νω

νω

ννννωζ

νννωζ

ωννννωζ

[ ] ( )[ ] ( )

[ ] ( )[ ]

[ ] [ ] [ ] yjtj

yjtj

W

ezhzej

ehzeh

j

tyhzh

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅−⋅⋅⋅⋅

⋅=

⋅−⋅⋅⋅⋅⋅

⋅⋅=

⋅+⋅⋅−⋅⋅⋅⋅

⋅−=Ψ

0

0

000

00

000

coshtanhsinh

sinhcosh

sinsinhcosh

νω

νω

ννννωζ

νννωζ

ωννννωζ

For the vertical motions is the in y -symmetrical part of the potential significant. For thehorizontal motions is the in y -point-symmetrical part (multiplied with j , so a phase shift of

090 ) important.

[ ] [ ] [ ] ( )

[ ] [ ] [ ] ( )yzhze

yzhze

tjW V

tjW V

⋅⋅⋅⋅⋅−⋅⋅⋅⋅−=Ψ

⋅⋅⋅⋅⋅−⋅⋅⋅⋅

+=Φ

⋅⋅

⋅⋅

0000

0000

sincoshtanhsinh

cossinhtanhcosh

νννννωζ

νννννωζ

ω

ω

[ ] [ ] [ ] ( )

[ ] [ ] [ ] ( )yzhze

yzhze

tjW H

tjW H

⋅⋅⋅⋅⋅−⋅⋅⋅⋅

−=Ψ

⋅⋅⋅⋅⋅−⋅⋅⋅⋅

−=Φ

⋅⋅

⋅⋅

0000

0000

coscoshtanhsinh

sinsinhtanhcosh

νννννωζ

νννννωζ

ω

ω

Equation 3.3–103

When the wave travels in the wx -direction, the potential becomes:

( ) [ ] [ ] [ ] zhze wxtjW ⋅⋅⋅−⋅⋅⋅

⋅=Φ ⋅−⋅⋅

000 sinhtanhcosh0 ννννωζ νω

With:

µµµµ

cossin

sincos

⋅+⋅=⋅−⋅=

yxy

yxx

w

w

Equation 3.3–104

the potential becomes:

( ) [ ] [ ] [ ] zhzee xyjtjW V ⋅⋅⋅−⋅⋅⋅⋅

⋅=Φ ⋅−⋅⋅⋅⋅⋅

000cossin sinhtanhcosh0 ννν

νωζ µµνω

Equation 3.3–105

This results for the vertical motions in:


146

( ) [ ] [ ] [ ] ( )µνννννωζ µνω sincossinhtanhcosh 0000

cos0 ⋅⋅⋅⋅⋅⋅−⋅⋅⋅⋅

+=Φ ⋅⋅−⋅⋅ yzhze xtjW V

Equation 3.3–106

and for the horizontal motions in:

( ) [ ] [ ] [ ] ( )µνννννωζ µνω sinsinsinhtanhcosh 0000

cos0 ⋅⋅⋅⋅⋅⋅−⋅⋅⋅⋅

−=Φ ⋅⋅−⋅⋅ yzhze xtjW H

Equation 3.3–107

3.3.5.2 Appendix II: Series Expansions of Hyperbolic Functions

With:

β⋅±

⋅±

⋅=

⋅+=⋅±i

zy

i

er

ezyyizarctan

22

the following series expansions can be found.

[ ] ( ) [ ] [ ]( )[ ] ( )[ ]

[ ] [ ] ( )

( )( )

( )( )

( ) ( )

( ) ( ) ∑

∑

∑∑

∞

=

∞

=

∞

=

⋅⋅−∞

=

⋅⋅+

⋅−⋅+

⋅+⋅=

⋅⋅⋅

=

⋅⋅

+⋅⋅

⋅=

⋅⋅+⋅⋅⋅=

⋅−⋅+⋅+⋅⋅=

⋅⋅⋅⋅=⋅⋅⋅

0

22

0

2

0

22

0

22

Re!2

2cos!2

!2!221

coshcosh21

coshcosh21

coshcoshcoscosh

t

tt

t

t

t

tit

t

tit

ii

yizt

k

ttrk

etrk

etrk

erkerk

yizkyizk

ykizkykzk

β

ββ

ββ

[ ] ( ) [ ] [ ]( )[ ] ( )[ ]

[ ] [ ] ( )( )

( ) ( )( )

( )

( )( ) ( )( )

( ) ( ) ∑

∑

∑∑

∞

=

++

∞

=

+

∞

=

⋅+⋅−+∞

=

⋅+⋅++

⋅−⋅+

⋅+⋅+

=

⋅+⋅+

⋅=

⋅+

⋅+⋅

+⋅

⋅=

⋅⋅+⋅⋅⋅=

⋅−⋅+⋅+⋅⋅=

⋅⋅⋅⋅=⋅⋅⋅

0

1212

0

12

0

1212

0

1212

Re!12

12cos!12

!12!1221

sinhsinh21

sinhsinh21

coshsinhcossinh

t

tt

t

t

t

tit

t

tit

ii

yiztk

ttrk

etrk

etrk

erkerk

yizkyizk

ykizkykzk

β

ββ

ββ


147

[ ] ( ) [ ] [ ]( )[ ] ( )[ ]

[ ] [ ] ( )

( )( )

( )( )

( ) ( )

( ) ( ) ∑

∑

∑∑

∞

=

∞

=

∞

=

⋅⋅−∞

=

⋅⋅+

⋅−⋅+

⋅+⋅=

⋅⋅⋅

=

⋅⋅

−⋅⋅

⋅−=

⋅⋅−⋅⋅⋅−=

⋅−⋅−⋅+⋅⋅−=

⋅⋅⋅⋅⋅−=⋅⋅⋅

0

22

0

2

0

22

0

22

Im!2

2sin!2

!2!22

coshcosh2

coshcosh2

sinhsinhsinsinh

t

tt

t

t

t

tit

t

tit

ii

yizt

k

ttrk

etrk

etrki

erkerki

yizkyizki

ykizkiykzk

β

ββ

ββ

[ ] ( ) [ ] [ ]( )[ ] ( )[ ]

[ ] [ ] ( )( )

( ) ( )( )

( )

( )( ) ( )( )

( ) ( ) ∑

∑

∑∑

∞

=

++

∞

=

+

∞

=

⋅+⋅−+∞

=

⋅+⋅++

⋅−⋅+

⋅+⋅+

=

⋅+⋅+

⋅=

⋅+

⋅−⋅

+⋅

⋅−=

⋅⋅−⋅⋅⋅−=

⋅−⋅−⋅+⋅⋅−=

⋅⋅⋅⋅⋅−=⋅⋅⋅

0

1212

0

12

0

1212

0

1212

Im!12

12sin!12

!12!122

sinhsinh2

sinhsinh2

sinhcoshsincosh

t

tt

t

t

t

tit

t

tit

ii

yiztk

ttrk

etrk

etrki

erkerki

yizkyizki

ykizkiykzk

β

ββ

ββ

3.3.5.3 Appendix III: Treatment of Singular Points

The determination of j0Φ and its terms - which can be added to r0Φ in Equation 3.3–23 with

which the by jr j 00 Φ⋅+Φ described flow of the waves (travelling from both sides of the body

away) is given - is also possible in another way. This approach is based on work carried out byRayleigh and is given in the literature by Lamb [1932] for an infinite water depth.In this approach, a viscous force w⋅⋅ µρ will be included in the Euler equations, µ is thedynamic viscosity and w is the velocity. Because the fluid is assumed to be non-viscous, in alater stage this dynamic viscosity µ will be set to zero.From the Euler equation follows with this viscosity force the with time changing pressurechange:

∂Φ∂

−∂Φ∂

⋅−∂Φ∂

⋅⋅=∂∂

→ 2

2

0lim

ttzg

tp µρ

µ

From this follows the approach as given in a subsection before for the two-dimensionalradiation potential:


148

( ) ( )

( )[ ][ ] [ ]

( )

( )

[ ] [ ]

[ ] [ ]( )

( ) ( ) jadradjrtj

hk

zk

tj

tjjr

jjAe

dkykhkkhk

gj

zkkzkg

j

kg

j

e

dkykk

gj

e

Ae

dkykhkkhk

gj

hzk

Aetzyjtzy

00000

0

0

0

0

00

000

cossinhcosh

coshsinh

cos

lim

cossinhcosh

coshlim

,,,,

φφφφ

µων

µων

µων

µων

µων

ω

µ

ω

µ

ω

⋅++⋅+⋅⋅=

⋅⋅⋅⋅⋅−⋅⋅

⋅⋅−

⋅⋅−⋅⋅

⋅⋅−

⋅−⋅⋅−

+⋅⋅⋅−

⋅⋅−

⋅⋅=

⋅⋅⋅⋅⋅−⋅⋅

⋅⋅−

−⋅

⋅⋅=Φ⋅+Φ

∞∞⋅⋅

∞ ⋅−

∞ ⋅−

→

⋅⋅

∞

→

⋅⋅

∫

∫

∫

The first integral leads to the potential in Equation 3.3–29 and the second integral can beexpanded as follows:

[ ] [ ]

[ ] [ ]( )

( ) ( )

( ) ( )

[ ] [ ]

( ) ( )

( ) ( )

( ) ( )

+⋅++

⋅

⋅+

−+⋅+

⋅

⋅⋅−

−=

⋅

⋅⋅−⋅⋅

⋅⋅−⋅

−

⋅⋅−

⋅

⋅

⋅+

−+⋅+

⋅

⋅⋅−

=

⋅⋅⋅⋅⋅−⋅⋅

⋅⋅−

⋅⋅−⋅⋅

⋅⋅−

⋅−

⋅⋅−

=

⋅+

∑

∑∫

∫

∞

=

+

→

∞

=

∞ +⋅−

+

→

∞ ⋅−

→

0

212

0

0

0

12

212

0

00

00

1212

!2Re

!12Re

lim

sinhcosh

!2Re

!12Re

lim

cossinhcosh

coshsinh

lim

t

tt

t

thk

tt

hk

jadrad

tHjtG

tyiz

tyiz

gj

dk

hkkhkg

jkg

j

ke

tyiz

tyiz

gj

dkykhkkhk

gj

zkkzkg

j

kg

j

e

j

µων

µωνµων

µων

µων

µων

µων

φφ

µ

µ

µ

The function:


149

( ) ( )

[ ] [ ]

⋅⋅⋅−⋅⋅

⋅⋅−

⋅

⋅⋅−−

=⋅+ ∫∞

⋅−

→0

0

sinhcosh

lim

dkhkkhk

gi

k

gik

e

tHitGt

hk

µων

µων

µ

will be normalised as done before:

( ) ( ) ( ) ( )( )

[ ] [ ]

⋅

⋅−⋅

⋅⋅⋅−⋅

⋅

⋅⋅⋅−⋅−

⋅=

=⋅+=⋅+

∫∞ −

→

−

00

1''

sinhcosh

lim du

uuug

hih

gh

ihu

eu

tHitGhtHitG

ut

t

µων

µωνµ

This is a complex integral and must be solved in the complex domain with viuw ⋅+= .The integrand has a singularity for:

gh

ihw⋅⋅

⋅−⋅=µων1

and 2w is the solution of the equation:

[ ] [ ] 0sinhcosh =⋅−⋅

⋅⋅⋅−⋅ www

gh

ihµων

see Figure 3.3–5-a.

Figure 3.3–5: Treatment of singularities


150

( ) [ ] [ ]( )

0

...lim

.........limsinhcosh

lim

00

00

110

=

++=

++=⋅⋅−⋅⋅−

⋅

∫

∫∫ ∫∫

→

→

−

→

III

R

II

R

I

tw

JJdu

dwdwdudwwwwwww

we

µ

µµ

Also:

( ) ( )

[ ] [ ]

IIIR

ut

JJ

du

uuug

hih

gh

ihu

eutHitG

+=

⋅

⋅−⋅

⋅⋅⋅−⋅

⋅

⋅⋅⋅−⋅−

⋅=⋅+

∞→→

∞ −

→ ∫

and 0

00

''

lim

sinhcosh

lim

µ

µ

µων

µων

Because:0lim =

∞→ IRJ

follows:

( ) ( ) ( ) [ ] [ ]( )

⋅⋅−⋅⋅−

⋅−=⋅+ ∫−

→II

wt

dwwwwwww

ewtHitGsinhcosh

lim11

0

''

µ

The real part of this integral:

( ) ( ) [ ] [ ]( )

⋅⋅−⋅⋅−

⋅−= ∫

−

→II

wt

dwwwwwww

ewtG

sinhcoshlimRe

110

'

µ

will be calculated as done before.This integral has no singularity and the boundary 0→µ can be passed before integration; seeFigure 3.3–5-b.

The imaginary part of this integral:

( ) ( ) [ ] [ ]( )

⋅⋅−⋅⋅−

⋅−= ∫

−

→II

wt

dwwwwwww

ewtH

sinhcoshlimIm

110

'

µ

can be calculated numerically in a similar way.

It is also possible to solve this integral independently by using another integral path:


151

( ) [ ] [ ]( )

( ) ( ) 210

0

110

ResidueResiduelim2

lim

sinhcoshlim

wwi

JJJ

dwwwwwww

ew

IVIIIII

wt

+⋅⋅⋅=

++=

⋅⋅−⋅⋅−

⋅

→

→

−

→ ∫

µ

µ

µ

π

IVJ disappears for ∞→R .

It can be found that: IIIII JJ ReRe −= and IIIII JJ ImIm +=

Then it follows:( )

( ) ( )

( ) [ ][ ] [ ] [ ]

⋅−⋅⋅⋅+⋅

⋅⋅⋅⋅=

+⋅−=

−=

−−

→

→

hhhh

hh

ww

JtH

tt

II

01

000

02

10

210

0

'

tanhcoshsinh

cosh

ResidueResiduelim

limIm

νννν

ννπ

πµ

µ

and the imaginary additional potential becomes:

( ) ( ) ( )

( ) ( )

[ ][ ] [ ] [ ]( )

( )( ) ( )

[ ][ ] [ ] [ ]

( ) ( ) [ ] ( ) ( ) ( )ye

yizt

hyizt

hhhh

tyiz

tyiz

hhhh

tyiz

tyiz

tH

z

t

tt

t

tt

ttt

tt

t

tt

jad

⋅⋅⋅−

⋅+⋅

+⋅⋅−⋅+⋅

⋅⋅⋅⋅+⋅

⋅⋅=

+⋅+

⋅−⋅+

⋅

−

⋅⋅⋅+⋅⋅⋅

⋅

=

+⋅+

⋅−⋅+

⋅+=

⋅−

∞

=

++∞

=

∞

=+

∞

=

+

∑∑

∑

∑

νπ

ννν

ννννπ

ν

νννν

ννπ

νφ

ν cos

Re!12

tanhRe!2

coshsinhcosh

!12Re

!2Re

coshsinhcosh

!12Re

!2Re

12

0

1212

00

0

22

0

000

02

0122

2

000

022

0

0

122

0

The same will be found as a difference between Equation 3.3–45 and Equation 3.3–29.


152


153

3.4 Potential Theory of Frank

As a consequence of conformal mapping of a cross section to the unit circle, the cross sectionneeds to have a certain breadth at the water surface. Fully submersed cross sections - such asat the bulbous bow - cannot be mapped. Mapping problems can also appear for cross sectionswith a very high or low area coefficient. These cases require another approach: the pulsatingsource method of Frank or the so-called Frank Close-Fit Method.For explaining this method as it has been used in the computer code SEAWAY, relevant partsof the report of Frank [1967] have been copied to this Section, supplemented with somenumerical improvements.

Hydrodynamic research of horizontal cylinders oscillating in or below the free surface of adeep fluid has increased in importance in the last decades and has been studied by a number ofinvestigators. The history of this subject began with Ursell [1949], who formulated and solvedthe boundary-value problem for the semi-immersed heaving circular cylinder within theframework of linearised free-surface theory. He represented the velocity potential as the sumof an infinite set of multi-poles, each satisfying the linear free-surface condition and eachbeing multiplied by a coefficient determined by requiring the series to satisfy the kinematicboundary condition at a number of points on the cylinder.Grim [1953] used a variation of the Ursell method to solve the problem for two-parameterLewis form cylinders by conformal mapping onto a circle. Tasai [1959] and Porter [1960],using the Ursell approach obtained the added mass and damping for oscillating contoursmappable onto a circle by the more general Theodorsen transformation. Ogilvie [1963]calculated the hydrodynamic forces on completely submerged heaving circular cylinders.

Despite the success of the multi-pole expansion-mapping methods, Frank [1967] discussed theproblem from a different point of view. The velocity potential is represented by a distributionof sources over the submerged cross section. The density of the sources is an unknownfunction (of position along the contour) to be determined from integral equations found byapplying the kinematic boundary condition on the submerged part of the cylinder. Thehydrodynamic pressures are obtained from the velocity potential by means of the linearisedBernoulli equation. Integration of these pressures over the immersed portion of the cylinderyields the hydrodynamic forces or moments.

3.4.1 Notations of Frank

Frank’s notations have been maintained here as far as possible:

)(mA oscillation amplitude in the m -th modeB beam of cross section 0C

0C submerged part of cross sectional contour in rest positiong acceleration of gravity

)(mijI influence coefficient in-phase with displacement on the i -th midpoint

due to the j -th segment in the m -th mode of oscillation)(m

ijJ influence coefficient in-phase with velocity on the i -th midpoint due tothe j -th segment in the m -th mode of oscillation


154

( )( )mM ω added mass force or moment for the m -th mode of oscillation at

frequency ωN number of line segments defining submerged portion of half section in

rest position

( )( )mN ω damping force or moment for the m -th mode of oscillation at

frequency ω( )m

in direction cosine of the normal velocity at i -th midpoint for the m -thmode of oscillation

PV Cauchy pricipal value of integral( )m

ap hydrodynamic pressure in-phase with displacement for the m -th modeof oscillation

( )mvp hydrodynamic pressure in-phase with velocity for the m -th mode of

oscillation( )m

jQ source strength in-phase with displacement along the j -th segment forthe m -th mode of oscillation

( )mNjQ + source strength in-phase with velocity along the j -th segment for the

m -th mode of oscillations length variable along 0C

js j -th line segmentT draft of cross sectiont time

( )miv normal velocity component at the i -th midpoint for the m -th mode of

oscillation

1x abscissa of the i -th midpoint

iy ordinate of the i -th midpoint

0y ordinate of the center of rollyixz ⋅+= complex field point in region of fluid domain

iii yixz ⋅+= complex midpoint of i -th segment

iα angle between i -th segment and positive x -axis

ζ complex variable along 0C

jζ j -th complex input point along 0C

jη ordinate of j -th input point

g2ων = wave number

kν k -th irregular wave number for adjoint interior problem

jξ abscissa of the j -th input pointρ density of fluid

( )mΦ velocity potential for th m -th mode of oscillationω radian frequency of oscillation

kω k -th irregular frequency for adjoint interior problem(k -th eigen-frequency)


155

3.4.2 Formulation of the Problem

Consider a cylinder, whose cross section is a simply connected region, which is fully orpartially immersed horizontally in a previously undisturbed fluid of infinite depth. The body isforced into simple harmonic motion and it is assumed that steady state conditions have beenattained.The two-dimensional nature of the problem implies three degrees of freedom of motion.Therefore, consider the following three types of oscillatory motions: vertical or heave,horizontal or sway and rotational about a horizontal axis or roll.To use linearised free-surface theory, the following assumptions are made:1. the fluid is incompressible and inviscid,2. the effects of surface tension are negligible,3. the fluid is irrotational and4. the motion amplitudes and velocities are small enough that all but the linear terms of the

free-surface condition, the kinematic boundary condition on the cylinder and the Bernoulliequation may be neglected.

For complete discussions of linearised free-surface theory, Frank refers the reader to Stoker[1957] and Wehausen and Laitone [1960].

Given the above conditions and assumptions, the problem reduces to the following boundary-value problem of potential theory. The cylinder is forced into simple harmonic motion

( ) ( )tA m ⋅⋅ ωcos with a prescribed radian frequency of oscillation ω , where the superscript mmay take on the values 2, 3 and 4, denoting swaying, heaving and rolling motions,respectively.It is required to find a velocity potential:

( )( ) ( )( ) timm eyxtyx ⋅⋅−⋅=Φ ωφ ,Re,,

Equation 3.4–1

satisfying the following conditions:

1. The Laplace equation:

( )( ) ( )

02

2

2

22 =

∂Φ∂

+∂Φ∂

=Φ∇yx

mmm

Equation 3.4–2

in the fluid domain, i.e., for 0<y and outside the cylinder.

2. The free surface condition:( ) ( )

02

2

=∂Φ∂

⋅+∂Φ∂

yg

t

mm

Equation 3.4–3

on the free surface 0=y outside the cylinder, while g is the acceleration of gravity.

3. The seabed boundary condition for deep water:


156

( )( )

0limlim =∂Φ∂

=Φ∇−∞→−∞→ y

m

y

m

y

Equation 3.4–4

4. The condition of the normal velocity component of the fluid at the surface of theoscillating cylinder being equal to the normal component of the forced velocity of thecylinder. i.e., if nv is the component of the forced velocity of the cylinder in the direction

of the outgoing unit normal vector n , then( )

nm vn =Φ∇⋅

Equation 3.4–5

This is the kinematic boundary condition on the oscillating body surface, being satisfied atthe mean (rest) position of the cylindrical surface.

5. The radiation condition that the disturbed surface of the fluid takes the form of regularprogressive outgoing gravity waves at large distances from the cylinder.

According to Wehausen and Laitone [1960], the complex potential at z of a pulsating pointsource of unit strength at the point ζ in the lower half plane is:

( ) ( ) ( )( )

( )

( ) ( )te

tdkk

ePVzztzG

zi

zki

⋅⋅−

⋅⋅

−⋅+−−−⋅

⋅=

−⋅⋅−

∞ −⋅⋅−

∫

ω

ων

ζζπ

ζ

ζν

ζ

sin

cos2lnln2

1,,

0

*

Equation 3.4–6

so that the real point-source potential is:

( ) ( ) tzGtyxH ,,Re,,,, * ζηξ =

Equation 3.4–7

where:yixz ⋅+= ηξζ ⋅+= i ηξζ ⋅−= i g2ων =

Letting:

( ) ( ) ( ) ( )

( ) ζν

ζ

νζζ

πζ

−⋅⋅−

∞ −⋅⋅−

⋅−

−⋅+−−−⋅

⋅= ∫

zi

zki

ei

dkk

ePVzzzG

Re

2lnlnRe2

1,

0

Equation 3.4–8

then:

( ) ( ) tietzGtyxH ⋅⋅−⋅= ωζηξ ,,Re,,,,

Equation 3.4–9

Equation 3.4–9 satisfies the radiation condition and also Equation 3.4–1 through Equation3.4–4.Another expression satisfying all these conditions is:


157

( ) tiezGityxH ⋅⋅−⋅⋅=

⋅− ωζ

ωπηξ ,Re

2,,,,

Equation 3.4–10

Since the problem is linear, a superposition of Equation 3.4–9 and Equation 3.4–10 results inthe velocity potential:

( )( ) ( ) ( )

⋅⋅⋅=Φ ∫ ⋅⋅−

0

,Re,,C

tim dsezGsQtyx ωζ

Equation 3.4–11

where 0C is the submerged contour of the cylindrical cross section at its mean (rest) position

and ( )sQ represents the complex source density as a function of the position along 0C .Application of the kinematic boundary condition on the oscillating cylinder at z yields:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )mm

C

C

nAdszGsQn

dszGsQn

⋅⋅=

⋅⋅⋅∇⋅

=

⋅⋅⋅∇⋅

∫

∫

ωζ

ζ

0

0

,Im

0,Re

Equation 3.4–12

where ( )mA denotes the amplitude of oscillation and ( )mn the direction cosine of the normalvelocity at z on the cylinder. Both ( )mA and ( )mn depend on the mode of motion of thecylinder, as will be shown in the following section.The fact that ( )sQ is complex implies that Equation 3.4–12 represent a set of two coupledintegral equations for the real functions ( ) sQRe and ( ) sQIm . The solution of these integralequations and the evaluation of the kernel and potential integrals are described in thefollowing section and in Appendices II and III, respectively.

3.4.3 Solution of the Problem

Since ship sections are symmetrical, this investigation is confined to bodies with right and leftsymmetry.

Take the x -axis to be coincident with the undisturbed free surface of a conventional two-dimensional Cartesian co-ordinate system. Let the cross sectional contour 0C of thesubmerged portion of the cylinder be in the lower half plane, the y -axis being the axis of

symmetry of 0C ; see Figure 3.4–1.


158

Figure 3.4–1: Axes system and notations, as used by Frank [1967]

Select 1+N points ( )ii ηξ , of 0C to lie in the fourth quadrant so that ( )11,ηξ is located on the

negative y -axis. For partially immersed cylinders, ( )11, ++ NN ηξ is on the positive x -axis. For

fully submerged bodies, 11 ξξ =+N and 01 <+Nη .Connecting these 1+N points by successive straight lines, N straight line segments areobtained which, together with their reflected images in the third quadrant, yield anapproximation to the given contour as shown in Figure 3.4–1.The co-ordinates, length and angle associated with the j -th segment are identified by thesubscript j , whereas the corresponding quantities for the reflected image in the third quadrant

are denoted by the subscript j− , so that by symmetry jj +− −= ξξ and jj +− += ηη for11 +≤≤ Nj .

Potentials and pressures are to be evaluated at the midpoint of each segment. The co-ordinatesof the midpoint of the i -th segment are:

21++

= iiix

ξξ and

21++

= iiiy

ηη for: Ni ≤≤1

Equation 3.4–13

The length of the i -th segment is:

( ) ( )21

21 iiiiis ηηξξ −+−= ++

Equation 3.4–14

while the angle made by the i -th segment with the positive x -axis is given by:

−−

=+

+

ii

iii ξξ

ηηα1

1arctan

Equation 3.4–15

The outgoing unit vector normal to the cross section at the i -th midpoint ( )ii yx , is:

iii jin αα cossin ⋅−⋅=

Equation 3.4–16

where i and j are unit vectors in the directions of increasing x and y , respectively.


159

The cylinder is forced into simple harmonic motion with radian frequency ω , according to thedisplacement equation:

( ) ( ) ( )tAS mm ⋅⋅= ωcos

Equation 3.4–17

for 4,3,2=m corresponding to sway, heave or roll, respectively.

The rolling oscillations are about an axis through a point ( )0,0 y in the symmetry plane of thecylinder.In the translation modes, any point on the cylinder moves with the velocity:

( ) ( ) ( )tAiv ⋅⋅⋅⋅−= ωω sin :sway 22

Equation 3.4–18( ) ( ) ( )tAjv ⋅⋅⋅⋅−= ωω sin :heave 33

Equation 3.4–19

The rolling motion about ( )0,0 y is illustrated in Figure 3.4–1.

Considering a point ( )ii yx , on 0C , an inspection of this figure yields:

( )20

2 yyxR iii −+= and

=

−=

−=

i

i

i

i

i

ii

Rx

Ryy

xyy

arccos

arcsin

arctan

0

0θ

Therefore, by elementary two-dimensional kinematics, the unit vector in the direction ofincreasing θ is:

jRx

iR

yy

ji

i

i

i

i

iii

⋅+⋅−

−=

⋅+⋅−=

0

cossin θθτ

so that:( ) ( )

( ) ( ) ( )tjxiyyA

SRv

ii

ii

⋅⋅⋅−⋅−⋅⋅=

⋅⋅=

ωω

τ

sin

:roll

04

44

Equation 3.4–20

The normal components of the velocity ( ) ( )mi

mi vnv ⋅= at the midpoint of the i -th segment

( )ii yx , are:( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )txyyAv

tAv

tAv

iiiii

ii

ii

⋅⋅⋅+⋅−⋅⋅+=

⋅⋅⋅⋅+=

⋅⋅⋅⋅−=

ωααω

ωαω

ωαω

sincossin :roll

sincos :heave

sinsin :sway

044

33

22

Equation 3.4–21

Defining:


160

( )( )

( ) ( )tAv

nm

mim

i ⋅⋅⋅=

ωω sinthen - consistent with the previously mentioned notation - the direction cosines for the threemodes of motion are:

( )

( )

( ) ( ) iiiii

ii

ii

xyyn

n

n

αα

α

α

cossin :roll

cos :heave

sin :sway

04

3

2

⋅+⋅−+=

+=

−=

Equation 3.4–22

Equation 3.4–22 illustrates that heaving is symmetrical, i.e., ( ) ( )33ii nn +− = . Swaying and

rolling, however, are anti-symmetrical modes, i.e., ( ) ( )22ii nn +− −= and ( ) ( )44

ii nn +− −= .Equation 3.4–12 is applied at the midpoints of each of the N segments and it is assumed thatover an individual segment the complex source strength ( )sQ remains constant, although itvaries from segment to segment. With these stipulations, the set of coupled integral equations(Equation 3.4–12) becomes a set of N2 linear algebraic equations in the unknowns:

( )( ) ( )mjj

m QsQ =Re and ( )( ) ( )mjNj

m QsQ +=Im

Thus, for Ni ,...,2,1= :

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )m

im

N

j

N

j

mij

mjN

mij

mj

N

j

N

j

mij

mjN

mij

mj

nAIQJQ

JQIQ

⋅⋅=⋅+⋅−

=⋅+⋅+

∑ ∑

∑ ∑

= =+

= =+

ω1 1

1 1

0

Equation 3.4–23

where the superscript ( )m denotes the mode of motion.

The ''influence coefficients'' ( )mijI and ( )m

ijJ and the potential ( )( )tyx iim ,,Φ are evaluated in

Appendix II. The resulting velocity potential consists of a term in-phase with the displacementand a term in-phase with the velocity.Note: Most authors refer to this first term as a component in phase with the acceleration.

However due to the displacement Equation 3.4–17, Frank deemed it more appropriateto refer to this term as being 180 degrees out-of-phase with the acceleration or in-phase with the displacement.

The hydrodynamic pressure at ( )ii yx , along the cylinder is obtained from the velocitypotential by means of the linearized Bernoulli equation:

( )( )( )

( )tyxt

tyxp ii

m

iim ,,,,,, ωρω

∂Φ∂

⋅−=

Equation 3.4–24

as:( )( ) ( )( ) ( ) ( )( ) ( )tyxptyxptyxp ii

mvii

maii

m ⋅⋅+⋅⋅= ωωωωω sin,,cos,,,,,

Equation 3.4–25


161

where ( )map and ( )m

vp are the hydrodynamic pressures in-phase with the displacement and in-phase with the velocity, respectively and ρ denotes the density of the fluid.As indicated by the notation of Equation 3.4–24 and Equation 3.4–25, the pressure as well asthe potential is a function of the oscillation frequency ω .The hydrodynamic force or moment (when 4=m ) per unit length along the cylinder,necessary to sustain the oscillations, is the integral of ( ) ( )mm np ⋅ over the submerged contourof the cross section 0C . It is assumed that the pressure at the i -th midpoint is the meanpressure for the i -th segment, so that the integration reduces to a summation, whence:

( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ∑

∑

=

=

⋅⋅⋅=

⋅⋅⋅=

N

ii

miii

mv

m

N

ii

miii

ma

m

snyxpN

snyxpM

1

1

,,2

,,2

ωω

ωω

Equation 3.4–26

for the added mass and damping forces or moments, respectively.The velocity potentials for very small and very large frequencies are derived and discussed inthe next section.

3.4.4 Low and High Frequencies

For very small frequencies, i.e., as 0→ω , the free-surface condition in Equation 3.4–3 of thesection formulating the problem degenerates into the wall-boundary condition:

0=∂Φ∂y

Equation 3.4–27

on the surface of the fluid outside the cylinder, whereas for extremely large frequencies, i.e.,when ∞→ω , the free-surface condition becomes the ''impulsive'' surface condition:

0=Φ

Equation 3.4–28

on 0=y and outside the cylinder.Equation 3.4–2, Equation 3.4–4 and Equation 3.4–5 remain valid for both asymptotic cases.The radiation condition is replaced by a condition of boundedness at infinity.Therefore, there is a Neumann problem for the case 0→ω and a mixed problem when

∞→ω . The appropriate complex potentials for a source of unit strength at a point ζ in thelower half plane are:

( ) ( ) ( ) 00 lnln2

1, KzzzG +−−−⋅

⋅= ζζ

πζ

Equation 3.4–29

and:

( ) ( ) ( ) ∞∞ +−−−⋅⋅

= KzzzG ζζπ

ζ lnln2

1,

Equation 3.4–30


162

for the Neuman and mixed problems, respectively, where 0K and ∞K are not yet specifiedconstants.Let:

( ) ( ) ζηξφ ,Re,,, zGyx aa =so that the velocity potentials for the m -th mode of motion are:

( )( ) ( )( ) ( )∫ ⋅⋅=Φ0

,,,,C

am

am

a dsyxsQyx ηξφ

Equation 3.4–31

for 0=a , and ∞=a , where ( )maQ is the expression for the source strength as a function of

position along the submerged contour of the cross section 0C .An analysis similar to the one in the section on formulating the problem leads to the integralequation:

( ) ( )( ) ( ) ( ) ( )mm

C

am

a nAdsyxsQn ⋅=⋅⋅⋅∇⋅ ∫0

,,, ηξφ

Equation 3.4–32

which - after application at the N segmental midpoint - yields a set of N linear algebraicequations in the N unknown source strengths jQ .It remains to be shown whether these two problems are, in the language of potential theory,well posed, i.e., whether the solutions to these problems lead to unique forces or moments.The mixed problem raises no difficulty, since as ∞→z , ( ) 0, →∞ ζzG . In fact 0=∞K , whichcan be inferred from the pulsating source-potential Equation 3.4–8 by letting ∞→ν .Considering the Neumann problem, note that the constant 0K in the Green's function equation(Equation 3.4–29) yields by integration an additive constant K to the potential. However, fora completely submerged cylinder the cross sectional contour 0C is a simply closed curve, sothat the contribution of K in integrating the product of the pressure with the direction cosineof the body-surface velocity vanishes. For partially submerged bodies 0C is no longer closed.

But since ( ) ( )mi

mi nn +− −= for m being even,

( ) 00

=⋅⋅∫C

m dsnK

so that the swaying force and rolling moment are unique.The heaving force on a partially submerged cylinder is not unique for, in this case,

( ) ( )33ii nn +− = , so that:

( ) 00

3 ≠⋅⋅∫C

dsnK

The constant 0K may be obtained by letting 0→ν in the pulsating source-potential Equation3.4–8.

3.4.5 Irregular Frequencies

John [1950] proved the existence and uniqueness of the solutions to the three- and two-dimensional potential problems pertaining to oscillations of rigid bodies in a free surface. The


163

solutions were subject to the provisions that no point of the immersed surface of the bodywould be outside a cylinder drawn vertically downward from the intersection of the body withthe free surface and that the free surface would be intersected orthogonally by the body in itsmean or rest position.John [1950] also showed that for a set of discrete ''irregular'' frequencies the Green's function-integral equation method failed to give a solution. He demonstrated that the irregularfrequencies occurred when the following adjoint interior-potential problem had eigen-frequencies.

Let ( )yx,ψ be such that:

1. 02

2

2

2

=∂∂

+∂∂

yxψψ

inside the cylinder in the region bounded by the immersed surface of the

body and the extension of the free surface inside the cylinder.

2. 0=⋅−∂∂ ψνψ

ky on the extension of the free surface inside the cylinder, kν being the wave

number corresponding to the irregular frequency kω , =k 1,2,3,…etc.

3. 0=ψ on the surface of the cylinder below the free surface.

For a rectangular cylinder with beam B and draft T , the irregular wave numbers may beeasily obtained by separation of variables in the Laplace equation. Separating variables givesthe eigen-functions:

⋅⋅⋅

⋅⋅⋅=

Byk

BxkBkk

ππψ sinhsin for: =k 1,2,3,…etc.

where kB are Fourier coefficients to be determined from an appropriate boundary condition.Applying the free surface condition (Equation 3.4–2) on Ty = for Bx <<0 , the eigen-wavenumbers (or irregular wave numbers):

⋅⋅⋅⋅=

BTk

Bk

kππν coth

Equation 3.4–33

are obtained for k = 1, 2, 3, ..., etc.In particular, the lowest such irregular wave number is given by:

⋅⋅=

BT

Bππν coth1

Equation 3.4–34

Keeping T fixed in Equation 3.4–34 but letting B vary and setting Bb π= , then from theTaylor expansion:

[ ] ( )

+⋅−⋅+

⋅⋅=⋅⋅ ......

4531coth

3TbTbTb

bTbb

it is seen that as 0→b , which is equivalent to ∞→B , T11 →ν .Therefore, for rectangular cylinders of draft T ,


164

T11 ≥ν

Equation 3.4–35

is a relation that John proved for general shapes complying with the restrictions previouslyoutlined.For a beam-to-draft ratio of 5.2=TB : 48.11 =ν , while for 0.2=TB : 71.11 =ν .

At an irregular frequency the matrix of influence coefficients of Equation 3.4–23 becomessingular as the number of defining points per cross section increases without limit, i,e., as

∞→N . In practice, with finite N , the determinant of this matrix becomes very small, notonly at the irregular frequency but also at an interval about this frequency. This interval can bereduced by increasing the number of defining points N for the cross section.

Most surface vessels have nearly constant draft over the length of the ship and the maximumbeam occurs at or near amidships, where the cross section is usually almost rectangular, sothat for most surface ships the first irregular frequency 1ω is less for the midsection than forany other cross section.As an example, for a ship with a 7:1 length-to-beam and a 5:2 beam-to-draft, the first irregularwave encounter frequency - in non-dimensional form with L denoting the ship length -occurs at 09.51 ≈⋅ gLω , which is beyond the range of practical interest for ship-motionanalysis.Therefore, for slender surface vessels, the phenomenon of the first irregular frequency ofwave encounter is not too important.

Increasing the number of contour line elements (or panels in 3-D) does not remove theirregular frequency, but tends to restrict the effects to a narrower band around it; see forinstance Huijsmans [1996]. It should be mentioned too that irregular frequencies appear forfree surface piercing bodies only; fully submerged bodies do not display these characteristics.An effective method to reduce the effects of irregular frequencies is ''closing'' the body bymeans of a discretisation of the free surface inside the body, i.e. putting a ''lid'' on the freesurface inside the body.See here the computed added mass and damping of a hemisphere in Figure 3.4–2. The solidline in this figure results from including the ''lid''.

0

5 00

10 00

15 00

20 00

0 1 2 3

h ea ve

su rge /swa y

Fre qu e ncy (ra d/ s)

Ma

ss (

ton

)

0

25 0

50 0

75 0

1 00 0

1 25 0

0 1 2 3

h ea ve

su rge /swa y

Fre qu en cy (ra d /s)

Da

mp

ing

(to

n/s

)

Figure 3.4–2: Effect of use of ''Lid-Method'' on irregular frequencies


165

3.4.6 Appendices

3.4.6.1 Appendix I: Evaluation of Principle Value Integrals

The real and imaginary parts of the principle value integral:( )

dkk

ePV

zki

⋅−∫

∞ −⋅⋅−

0 ν

ζ

are used in evaluating some of the kernel and potential integrals.The residue of the integrand at ν=k is ( )ζν −⋅⋅− zie , so that:

( ) ( ) ( )ζνζζ

πνν

−⋅⋅−∞ −⋅⋅−∞ −⋅⋅−

⋅−⋅−

=⋅− ∫∫ zi

zkizki

eidkk

edk

ke

PV00

Equation 3.4–36

where the path of integration is the positive real axis indented into the upper half plane aboutν=k .

Notice hereby that:( ) 02 >= gων 0Im <z 0Im ≤ζ

The transformation ( ) ( )ζνω −⋅−⋅= zki converts the contour integral on the right hand sideof Equation 3.4–36 to:

( ) ( )( )

( ) ( )( )

( )

( ) ( ) ( )

<−>−⋅

+

⋅−⋅⋅−⋅−

+−⋅⋅−+

⋅−=

<−>−⋅

+−⋅⋅−⋅−=

⋅⋅−=⋅−

∑

∫∫

∞

=

−⋅⋅−

−⋅⋅−

∞

−⋅⋅−

−−⋅⋅−

∞ −⋅⋅−

0

0 :for

0

2

!1

ln

0

0 :for

0

2

1

1

0

ξξπ

ζν

ζνγ

ξξπ

ζν

ν

ζν

ζν

ζν

ζνζ

x

xi

nnzi

zi

e

x

xiziEe

dww

eedk

ke

n

nnzi

zi

zi

wzi

zki

where 57722.0=γ is the well-known Euler-Mascheroni constant.The definition of 1E has been given by Abramovitz and Stegun [1964].Setting:

( )ζν −⋅−= zir and ( ) ( ) π

ζνζνθ +

−⋅⋅−−⋅⋅−

=zizi

ReIm

arctan

the following expression is obtained for Equation 3.4–36:


166

( )( ) ( )( ) ( )( )

( ) ( )

( )

⋅⋅⋅

+

<−>−

−⋅

+

⋅⋅⋅

++

⋅−⋅⋅−−⋅⋅=⋅−

∑

∑

∫

∞

=

∞

=

+⋅∞ −⋅⋅−

1

1

0

!sin

0

0 :for

2

!cos

ln

sincos

n

n

n

n

yzki

nnnr

x

xi

nnnr

r

xixedkk

ePV

θξξ

πθθ

θγ

ξνξνν

ηνζ

Equation 3.4–37

Separating Equation 3.4–37 into its real and imaginary parts yields:( ) ( )( ) ( ) ( ) ( )( )

( ) ( )( )( ) ( )( ) ( ) ( ) ( )( )

( ) ( )( )

−⋅⋅−−⋅⋅

⋅=⋅−

−⋅⋅

−⋅⋅+−⋅⋅

⋅=⋅−

−⋅⋅

+⋅∞ +⋅

+⋅∞ +⋅

∫

∫

ξνθξνθ

νξ

ξνθξνθ

νξ

ηνη

ηνη

xrS

xrCedk

kxke

PV

xrS

xrCedk

kxke

PV

yyk

yyk

cos,

sin,sin

sin,

cos,cos

0

0

Equation 3.4–38

provided that:

( ) ( ) ( )

( ) ( )

<−>−

−+

⋅⋅⋅

+=

⋅⋅⋅

++=

∑

∑∞

=

∞

=

0

0 :for

2!sin

,

!cos

ln,

1

1

ξξ

πθθθθθ

θγθ

x

x

nnnr

rS

nnnr

rrC

n

n

n

n

3.4.6.2 Appendix II: Evaluation of Kernel Integrals

The influence coefficients of Equation 3.4–23 are:

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

⋅

⋅−

⋅

++−+⋅⋅

⋅−−⋅

⋅−

⋅

+−−−⋅⋅

⋅∇⋅=

=

∞ +⋅⋅−

∞ −⋅⋅−

∫∫

∫∫

−

+

i

j

j

zz

szki

m

szki

im

ij

dsdk

kePV

zz

dsdk

ke

PV

zz

nI

0

0

1

lnln2

1

11

lnln2

1

Re

νπ

ζζπ

νπ

ζζπ

ζ

ζ

Equation 3.4–39

and:

( ) ( ) ( ) ( ) ( )

⋅−−⋅⋅∇⋅=

=

+⋅⋅−−⋅⋅− ∫∫−+

ijj zz

s

zim

s

zii

mij dsedsenJ ζνζν 1Re

Equation 3.4–40


167

Notice that in the complex plane with iz on is :

( ) ( ) ( )i

i

izz

izzi

dzzdF

eizFn=

⋅=

⋅⋅−=⋅∇⋅ αReRe

Considering the term containing ( )ζ−zln , it is evident that the kernel integral is singularwhen ji = , so that the indicated differentiation cannot be performed under the integral sign.However, in that case one may proceed as follows.Since:

ηξζ ⋅+= iand

dse

dsids

didd

ji

jj

⋅=

⋅⋅+⋅=⋅+=

⋅α

ααηξζ

sincos

for ζ along the j -th segment.Therefore:

ζα deds ji ⋅= ⋅−

and:

( ) ( ) ( )

( )

⋅−

⋅−

=

−⋅⋅

⋅⋅−

=

⋅−⋅∇⋅

=

=

⋅−

⋅

=

∫

∫∫

+

+

j

j

j

j

j

j

j

j

jj

zz

zz

i

i

zzs

i

dzdzd

i

zdedzd

ei

dszn

1

1

lnRe

lnRelnRe

ζ

ζ

ζ

ζ

α

α

ζζ

ζζζ

Setting ζζ −= z' , the last integral becomes:

( ) ( ) ( )

π

ζζζζζ

ζ

=

−−−=

⋅⋅− +

=

−

−∫

+

1'' argarglnRe

1

jjjj

zz

z

z

zzddzd

i

j

j

j

Equation 3.4–41

If ji ≠ , differentiation under the integral sign may be performed, so that:

( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )

−−

−

−−

⋅++

−+−

−+−⋅−=

⋅−⋅∇⋅=

+

+

++

=

∫

1

1

21

21

22

1

arctanarctancos

lnsin

lnRe

ji

ji

ji

jiji

jiji

jijiji

zzs

i

x

y

x

y

yx

yx

dsznL

ij

ξη

ξη

αα

ηξηξ

αα

ζ


168

Equation 3.4–42

For the integral containing the ( )ζ−zln term, ζα deds ji ⋅= ⋅ , so that:

( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )

−+

−

−+

⋅++

++−

++−⋅+=

⋅−⋅∇⋅=

+

+

++

=

∫

1

1

21

21

22

2

arctanarctancos

lnsin

lnRe

ji

ji

ji

jiji

jiji

jijiji

zzs

i

x

y

x

y

yx

yx

dsznL

ij

ξη

ξη

αα

ηξηξ

αα

ζ

Equation 3.4–43

The kernel integral, containing the principal value integrals, is:

( ) ( ) ( )

( ) ( )

( )

( ) ( )( )

( ) ( )( )

( )

( ) ( )( )

( ) ( )( )

⋅−

−⋅⋅−

⋅−

−⋅⋅+

⋅+−

⋅−

−⋅⋅−

⋅−

−⋅⋅+

⋅+=

⋅−

⋅⋅⋅−=

⋅

−⋅⋅∇⋅=

∫

∫

∫

∫

∫ ∫

∫ ∫

∞+

+⋅

∞ +⋅

∞+

+⋅

∞ +⋅

∞ −⋅⋅−+⋅

=

∞ −⋅⋅−

+

+

+

0

1

0

0

1

0

0

05

sin

sin

cos

cos

cos

sin

Re

Re

1

1

1

dkk

xkePV

dkk

xkePV

dkk

xkePV

dkk

xkePV

dkk

ePV

d

ddei

dkk

ePVdsnL

jiyk

jiyk

ji

jiyk

jiyk

ji

zkii

zzs

zki

i

ji

ji

ji

ji

j

j

iji

ij

νξ

νξ

αα

νξ

νξ

αα

νζζ

ν

η

η

η

η

ζ

ζ

ζαα

ζ

Equation 3.4–44

The first integral on the right hand side of Equation 3.4–40 becomes:


169

( ) ( ) ( )

( )( ) ( )( )( ) ( )( )

( )( ) ( )( )( ) ( )( )

−⋅⋅−

−⋅⋅+⋅++

−⋅⋅−

−⋅⋅+⋅+−=

⋅⋅∇⋅=

++⋅

+⋅

++⋅

+⋅

=

−⋅⋅−

+

+

∫

1

1

7

sin

sincos

cos

cossin

Re

1

1

jiy

jiy

ji

jiy

jiy

ji

zzs

zii

xe

xe

xe

xe

dsenL

ji

ji

ji

ji

ij

ξν

ξναα

ξν

ξναα

ην

ην

ην

ην

ζν

Equation 3.4–45

The kernel integrals over the image segments are obtained from Equation 3.4–43 throughEquation 3.4–45 by replacing jξ , 1+jξ and jα with jj ξξ −=− , ( ) 11 ++− −= jj ξξ and jj αα −=− ,respectively.

3.4.6.3 Appendix III: Potential Integrals

The velocity potential of the m -th mode of oscillation at the i -th midpoint ( )ii yx , is:( )( )

( ) ( )( )

( )

( ) ( )( )

( ) ( ) ( ) ( )( )∑ ∫∫

∑

∫ ∫

∫ ∫

=

+⋅⋅−−⋅⋅−+

= ∞ +⋅⋅−

∞ −⋅⋅−

⋅⋅

⋅

⋅−−⋅⋅

⋅

⋅

−⋅++−+

⋅−−

⋅

⋅

−⋅+−−−

⋅⋅⋅

=Φ

−

N

j s

zim

s

zijN

N

j

s

zki

ii

m

s

zki

ii

j

iim

t

tdsedseQ

dsdkk

ePVzz

dsdkk

ePVzz

Q

tyx

j

i

j

i

j

i

j

i

1

1

0

0

sin

cos1Re

2lnln

1

2lnln

Re2

1

,,

ωω

νζζ

νζζ

π

ζνζν

ζ

ζ

m

Equation 3.4–46

The integration of the ( )ζ−izln term is straight forward, yielding:


170

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

−

−⋅−−

−

−⋅−+

−+−⋅−−

−+−+−⋅−+

⋅+

−

−⋅−+

−

−⋅−−

−+−⋅−−

−+−+−⋅−+

⋅+=

⋅−

+

++

+++

+

+

++

+++

+

∫

1

11

21

211

122

1

11

21

211

122

arctanarctan

ln

ln

sin

arctanarctan

ln

ln

coslnRe

ji

jiji

ji

jiji

jijiji

jjjijiji

j

ji

jiji

ji

jiji

jijiji

jjjijiji

js

i

x

yx

x

yx

yxy

yxy

x

yy

x

yy

yxx

yxx

dszj

ξη

ξξη

ξ

ηξη

ηηηξη

α

ξη

ηξη

η

ηξξ

ξξηξξ

αζ

Equation 3.4–47

In the integration of the ( )ζ−zln term, note that jη and 1+jη are replaced by jη− and 1+− jη ,

respectively.

To evaluate the potential integral containing the principal value integral, proceed in thefollowing manner.For an arbitrary z in the fluid domain:

( ) ( )

dkk

eek

ePVe

dedkk

ePVe

dek

dkPVedk

ke

PVds

jkikizki

i

kizki

i

zkiizki

jj

j

j

j

j

j

j

j

j

⋅−

⋅−

⋅−=

⋅⋅⋅−

⋅=

⋅⋅−

⋅=⋅−

⋅

⋅⋅⋅⋅∞ ⋅⋅−⋅

⋅⋅−∞ ⋅⋅−

⋅

−⋅⋅−∞

⋅∞ −⋅⋅−

+

+

++

∫

∫∫

∫∫∫ ∫

ζζα

ζ

ζ

ζα

ζ

ζ

ζαζ

ζ

ζ

ν

ζν

ζνν

1

1

11

0

0

00

where the change of integration is permissible since only one integral requires a principlevalue interpretation.

After dividing by ν and multiplying by kk +−ν under the integral sign, the last expressionbecomes:

( ) ( )

⋅−

−⋅−

+⋅−

⋅⋅⋅

− ∫∫∫∞ −⋅⋅−∞ −⋅⋅−⋅⋅⋅⋅∞

⋅⋅−⋅ ++

000

11

dkk

ePVdk

ke

PVdkk

eee

ei jjjj zkizkij

kikizki

i

ννν

ζζζζα

Equation 3.4–48

Regarding the first integral in Equation 3.4–48 as a function of z :


171

( ) dkk

eeezF j

kikizki

j

⋅−

⋅=⋅⋅⋅⋅∞

⋅⋅−+

∫ζζ 1

0

Equation 3.4–49

Differentiating Equation 3.4–49 with respect to z gives:

( ) ( ) ( )

11

00

11

' 1

+

∞−⋅⋅−

∞−⋅⋅−

−−

−=

⋅−⋅⋅−= ∫∫ +

j

zkizki

zz

dkedkeizF jj

ζζ

ζζ

So:( ) ( ) ( ) KzzzF jj +−−−= +1lnln ζζ

Equation 3.4–50

where K is a constant of integration to be determined presently. Since ( )zF is defined and

analytic for all z in the lower half plane and since by Equation 3.4–49, ( ) 0lim =−∞→

zFz

, it

follows from Equation 3.4–50 that 0=K .Therefore:

( )( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

⋅−

−⋅⋅−⋅

−

−⋅⋅

−

+−

−

++

⋅+

⋅−

−⋅⋅−⋅

−

−⋅⋅

++−

++−

⋅+

⋅=

−⋅−

−⋅−

+

−−−

⋅⋅

−=

⋅−

⋅=

∫∫

∫∫

∫∫

∫ ∫

∞+

+⋅∞ +⋅

+

+

∞ +⋅∞+

+⋅

++

∞ −⋅⋅−∞ −⋅⋅−

+⋅

∞ −⋅⋅−

+

+

+

dkk

xkePVdk

k

xkePV

x

y

x

y

dkk

xkePVdk

k

xkePV

yx

yx

dkk

ePVdk

ke

PV

zzei

dkk

ePVdsK

jiyk

jiyk

ji

ji

ji

ji

j

jiyk

jiyk

jiji

jiji

j

zkizki

jijii

s

zki

jiji

jiji

jiji

j

j

i

0

1

0

1

1

00

1

21

21

22

00

1

0

5

sinsin

arctanarctan

cos

coscos

ln

sin

1

lnln

Re

Re

1

1

1

νξ

νξ

ξη

ξη

α

νξ

νξ

ηξ

ηξ

α

ν

νν

ζζ

ν

ν

ηη

ηη

ζζ

α

ζ

Equation 3.4–51

The integration of the potential component in-phase with the velocity over js gives:


172

( ) ( )

( ) ( )( )( ) ( )( )

−−⋅⋅−

−−⋅⋅+⋅=

⋅=

++⋅

+⋅

−⋅⋅−

+

∫

jjiy

jjiy

s

zi

xe

xe

dseK

ji

ji

j

i

αξν

αξν

ν ην

ην

ζν

1

7

sin

sin1

Re

1

Equation 3.4–52


173

3.5 Comparisons between Calculated Potential Data

Figure 3.5–1 compares the calculated coefficients for an amidships cross section of acontainer vessel by the three previous methods:1. Ursell-Tasai's method with 2-parameter Lewis conformal mapping.2. Ursell-Tasai's method with 10-parameter close-fit conformal mapping.3. Frank's pulsating source method.

0

100

200

300

400

500

0 0.5 1.0 1.5 2.0 2 .5

H eave

M'

3 3

0

1 000

2 000

3 000

4 000

5 000

0 0.5 1.0 1.5 2 .0 2.5

R oll

M'

4 4

0

50

100

150

0 0 .5 1 .0 1.5 2.0 2.5

He ave

N'

33

f re qu ency (rad/s)

0

5 0

10 0

15 0

20 0

0 0. 5 1.0 1.5 2.0 2.5

Sway

N'

22

f requ ency (ra d/s)

Dam

ping

Coe

ffic

ient

0

100

200

300

0 0.5 1.0 1.5 2.0 2.5

M id ship s ectionof a c onta inersh ip

S way

M'

22

Mas

s C

oeff

icie

nt

0

5 0

10 0

15 0

20 0

0 0 .5 1.0 1.5 2.0 2 .5

Sw ay - Ro llRoll - Sway

N'

24 = N

'

42

f requ en cy (rad /s)

3000

3250

3500

3750

4000

4250

0 0.5 1.0 1.5 2.0 2.5

Sway - Ro llRoll - Sway

M'

2 4 = M

'

42

0

1 00

2 00

3 00

0 0 .5 1.0 1 .5 2.0 2 .5

C lo se-fit

L ew is

Fra nkRo ll

N'

44

f req ue ncy (ra d/s)

Figure 3.5–1: Comparison of various calculated potential coefficients

With the exception of the roll motions, the three results are very close. The roll motiondeviation, predicted with the Lewis conformal mapping method, is caused by a too muchrounded description of the ''bilge'' by the simple Lewis transformation.

A disadvantage of Frank's method could be the relatively large computing time, whencompared with Ursell-Tasai's method. However - because of the significantly increasedcomputing speed of nowadays computers - this should not be a problem anymore.

Generally, it is advised to use the very robust Ursell-Tasai's method with 10 parameter close-fit conformal mapping.For submerged sections, bulbous sections and sections with an area coefficient sσ less than0.4 however, Frank's pulsating source method should be used.


174


175

3.6 Estimated Potential Surge Coefficients

An equivalent longitudinal section, being constant over the ship's breadth B , is defined by:sectional breadth xB = ship length L

sectional draught xd = amidships draught d

sectional area coefficient MxC = block coefficient BCBy using a Lewis transformation of this equivalent longitudinal section to the unit circle, thetwo-dimensional potential mass *

11M and damping *11N can be calculated in a similar manner

as has been described for the two-dimensional potential mass and damping of sway, '22M and

'22N .

With these two-dimensional values, the total potential mass and damping of surge are definedby:

*1111

*1111

NBN

MBM

⋅=

⋅=

Equation 3.6–1

in which B is the breadth of the ship.

These frequency-dependent hydrodynamic coefficients do not include three-dimensionaleffects. Only the hydrodynamic mass coefficient - of which a large three-dimensional effect isexpected - will be adapted here empirically.

According to Tasai [1961], the zero-frequency potential mass for sway can be expressed inLewis-coefficients:

( ) ( ) 23

21

2

31

'22 31

120 aa

aad

M x ⋅+−⋅

+−

⋅⋅=→πρω

Equation 3.6–2

When using this formula for surge, the total potential mass of surge is defined by:( ) ( )00 *

1111 →⋅=→ ωω MBM

Equation 3.6–3

A frequency-independent total hydrodynamic mass coefficient is estimated empirically bySargent and Kaplan [1974] as a proportion of the total mass of the ship ∇⋅ρ :

( ) ∇⋅⋅−

= ρa

aKSM

2&11

with:

⋅−

−+

⋅−

= bbb

bb

a 211

ln1

3

2

where 2

1

−=

LB

b

Equation 3.6–4


176

Figure 3.6–1: Hydrodynamic mass for surge

With this hydrodynamic mass value, a correction factor β for three-dimensional effects hasbeen determined:

( )( )0

&

11

11

==

ωβ

MKSM

Equation 3.6–5

The three-dimensional effects for the potential damping of surge are ignored.So, the potential mass and damping of surge are defined by:

*1111

*1111

NBN

MBM

⋅=

⋅⋅= β

Equation 3.6–6

To obtain a uniform approach during all ship motions calculations, the cross sectional two-dimensional values of the hydrodynamic mass and damping have to be obtained.Based on results of numerical 3-D studies with a Wigley hull form by Adegeest [1994], aproportionality of both the two-dimensional hydrodynamic mass and damping with theabsolute values of the derivatives of the cross sectional areas xA in the bx -direction has beenfound:

11'

11 Mdx

dxdA

dxdA

M

L

bb

x

b

x

⋅⋅

=

∫ and 11

'11 N

dxdxdA

dxdA

N

L

bb

x

b

x

⋅⋅

=

∫

Equation 3.6–7


177

4 Viscous Damping

The strip theory is based on the potential flow theory. This holds that viscous effects areneglected, which can deliver serious problems when predicting roll motions at resonancefrequencies. In practice, viscous roll damping effects can be accounted for by empiricalformulas. For surge and roll, additional damping coefficients have to be introduced. Becauseof these additional contributions to the damping are from a viscous origin mainly, it is notpossible to calculate the total damping in a pure theoretical way.


178

4.1 Surge Damping

The total damping for surge vt BBB 111111 += consists of a potential part, 11B , and an

additional viscous part, vB11 . At forward ship speed V , the total damping coefficient, tB11 ,

can be determined simply from the resistance-speed curve of the ship in still water, ( )VRsw :( )

dVVRd

BBB swvt =+= 111111

Equation 4.1–1

4.1.1 Total Surge Damping

For a rough estimation of the still water resistance use can be made of a somewhat modifiedempirical formula of Troost [1955], in principle valid at the ship's service speed for hull formswith a block coefficient BC between about 0.60 and 0.80:

232 VCR tsw ⋅∇⋅⋅= ρ with: 60.0log0152.0

0036.010 +

+≈L

Ct

Equation 4.1–2

in which:∇ volume of displacement of the ship in m3,L length of the ship in m,V forward ship speed in m/s.

This total resistance coefficient tC is given in Figure 4.1–1 as a function of the ship length.

Figure 4.1–1: Total Still Water Resistance Coefficient of Troost

Then the total surge damping coefficient at forward ship speed V becomes:VCB tt ⋅∇⋅⋅⋅= 32

11 2 ρ


179

4.1.2 Viscous Surge Damping

This total damping coefficient includes a viscous part, which can be derived from thefrictional part of the ship's resistance, defined by the 1957 ITTC-line:

( ) ( ) 22

2ln075.0

21

−⋅⋅⋅⋅=

RnSVVR f ρ with:

νLV

Rn⋅

=

Equation 4.1–3

in which:ν kinematic viscosity of seawaterS wetted surface of the hull of the shipRn Reynolds number

From this empirical formula follows the pure viscous part of the additional dampingcoefficient at forward ship speed V :

( ) dV

VRdB f

v =11

Equation 4.1–4

which can be obtained numerically.


180

4.2 Roll Damping

In case of pure free rolling in still water (free decay test), the uncoupled linear equation of theroll motion about the centre of gravity G is given by:

( ) ( ) 044444444 =⋅+⋅++⋅+ φφφ CBBAI vxx&&&

with:

GMgC

bB

bOGbOGbOGbB

aOGaOGaOGaA

vv

⋅∇⋅⋅=

=⋅+⋅+⋅+=

⋅+⋅+⋅+=

ρ44

4444

22

2

24424444

22

2

24424444

Equation 4.2–1

For zero forward speed: 2442 aa = and 2442 bb = .

Equation 4.2–1 can be rewritten as:02 2

0 =⋅+⋅+ φωφνφ &&&

with:

44

44442AIBB

xx

v

++

=ν (quotient of damping and moment of inertia)

44

4420 AI

C

xx +=ω (natural roll frequency squared)

Equation 4.2–2

The non-dimensional roll damping coefficient, κ , is given by:

( ) 4444

4444

0

2 CAIBB

xx

v

⋅+⋅+=

=ωνκ

Equation 4.2–3

This damping coefficient is written as a fraction between the actual damping coefficient,

vBB 4444 + , and the critical damping coefficient, ( ) 444444 2 CAIB xxcr ⋅+⋅= ; so for critical

damping: 1=crκ .

Herewith, the equation of motion can be re-written as:02 2

00 =⋅+⋅⋅⋅+ φωφωκφ &&&

Equation 4.2–4

Suppose the vessel is deflected to an initial heel angle, aφ , in still water and then released. Thesolution of the equation of motion of this decay becomes:

( ) ( )

⋅⋅+⋅⋅⋅= ⋅− tte ta 0

00 sincos ω

ωνωφφ ν

Equation 4.2–5


181

Then, the logarithmic decrement of the motion is:

( )( )

+=

⋅⋅=⋅

φ

φφ

φφ

ωκν

Ttt

TT

ln

0

Equation 4.2–6

Because 220

2 νωωφ −= for the natural frequency oscillation and the damping is small so that2

02 ων << , one can neglect 2ν here and use 0ωωφ ≈ ; this leads to:

πωω φφφ 20 =⋅≈⋅ TT

Equation 4.2–7

The non-dimensional total roll damping is given now by:

( )( )

( )44

04444 2

ln2

1

CBB

Ttt

v ⋅⋅+=

+⋅

⋅=

ω

φφ

πκ

φ

Equation 4.2–8

The non-potential part of the total roll damping coefficient follows from the average value ofκ by:

440

4444

2B

CB v −

⋅⋅=

ωκ

Equation 4.2–9

4.2.1 Experimental Determination

The κ -values can easily been found when results of free rolling experiments with a model instill water are available, see Figure 4.2–1.

Figure 4.2–1: Time History of a Roll Decay Test

The results of free decay tests can be presented in different ways:


182

1. Generally they are presented by plotting the non-dimensional damping coefficient,obtained from two successive positive or negative maximum roll angles

iaφ and 2+iaφ , by:

⋅⋅

=+2

ln2

1

i

i

a

a

φφ

πκ versus:

22+

+= ii aa

a

φφφ

Equation 4.2–10

2. To avoid spreading in the successively determined κ -values, caused by a possible zero-shift of the measuring signal, double amplitudes can be used instead:

−−

⋅⋅

=++

+

32

1ln2

1

ii

ii

aa

aa

φφφφ

πκ versus:

( ) ( )4

321 +++−+−

= iiii aaaaa

φφφφφ

Equation 4.2–11

3. Sometimes the results of free rolling tests are presented by:

a

a

φφ∆

versus: aφ

with the absolute value of the average of two successive positive or negativemaximum roll angles, given by:

21+

+= ii aa

a

φφφ

and the absolute value of the difference of the average of two successive positive ornegative maximum roll angles, given by:

1+−=∆

ii aaa φφφThen the total non-dimensional roll damping coefficient becomes:

∆−

∆+

⋅⋅

=

a

a

a

a

φφφφ

πκ

2

2ln

21

Equation 4.2–12

The decay coefficient κ can therefore be estimated from the decaying oscillation bydetermining the ratio between any pair of successive (double) amplitudes. When the dampingis very small and the oscillation decays very slowly, several estimates of the decay can beobtained from a single record. It is obvious that for a linear system a constant κ -value shouldbe found in relation to aφ .Note that these decay tests provide no information about the relation between the potentialcoefficients and the frequency of oscillation. Indeed, this is impossible since decay tests arecarried out at only one frequency: the natural frequency. These experiments deliver noinformation on the relation with the frequency of oscillation.The method is not really practical when ν is much greater than about 0.2 and is in any casestrictly valid for small values of ν only. Luckily, this is generally the case.Be aware that this damping coefficient is determined by assuming an uncoupled roll motion(no other motions involved). Strictly, this damping coefficient is not valid for the actualcoupled motions of a ship that will be moving in all directions simultaneously.


183

The successively found values for κ , plotted on base of the average roll amplitude, will oftenhave a non-linear behaviour as illustrated in Figure 4.2–2.

0

0.01

0.02

0.03

0.04

0 1 2 3 4 5 6

mean linear and cubic dampingmean linear and quadrat ic dampingsecond experiment, negat ive anglessecond experiment, posit ive anglesfirst experiment, negative anglesfirst experiment, pos itive angles

Produc t carrier, V = 0 knots

mean roll amplitude (deg)

roll

dam

ping

coe

ffici

ent κ

(-)

Figure 4.2–2: Roll Damping Coefficients

For a behaviour like this, it will be found:

aφκκκ ⋅+= 21

Equation 4.2–13

while sometimes even a cubic roll damping coefficient, 23 aφκ ⋅ , has to be added to this

formula.This non-linear behaviour holds that during frequency domain calculations, the damping termis depending on the - so far unknown - solution for the transfer function of roll: aa ζφ . With a

known wave amplitude, aζ , this problem can be solved in an iterative manner. A less accurate

method is to use a fixed aφ .

4.2.2 Empirical Formula for Barges

From model experiments with rectangular barges - with its center of gravity, G , in the waterline - it has been found by Journee [1991]:

aφκκκ ⋅+= 21 with:

50.0

0013.0

2

2

1

=

⋅=

κ

κdB

Equation 4.2–14

in which B is the breadth and d is the draft of the barge.

4.2.3 Empirical Method of Miller


184

According to Miller [1974], the non-dimensional total roll damping coefficient, κ , can beobtained by:

aφκκκ ⋅+= 21

with:

b

b

b

bkbk

bbbV

CdBLr

BLrl

A

CFn

CFn

CFn

GML

BL

C

⋅⋅⋅⋅

⋅⋅+⋅⋅=

⋅+

+

⋅⋅⋅⋅=

3

3

2

32

1

0024.025.19

200085.0

κ

κ

Equation 4.2–15

where:

bkbkbk hlA ⋅= one sided area of bilge keel (m2)

bkl length of bilge keel (m)

bkh height of bilge keel (m)

br distance center line of water plane to turn of bilge (m)(first point at which turn of bilge starts, relative to water plane)

L length of ship (m)B breadth of ship (m)d draft of ship (m)

bC block coefficient (-)

GM initial metacentric height (m)Fn Froude number (-)

aφ amplitude of roll (rad)

VC correction factor on 1κ for speed effect (-)

(in the original formulation of Miller: 0.1=VC )

Generally 0.1=VC , but (according to an experienced user of computer code SEAWAY) for

slender ships, like frigates, a suitable value for VC seems to be:

GMCV ⋅−= 00.385.4

Equation 4.2–16

4.2.4 Semi-Empirical Method of Ikeda

Because the viscous part of the roll damping acts upon the viscosity of the fluid significantly,it is not possible to calculate the total roll damping coefficient in a pure theoretical way.Besides this, also experiments showed a non-linear behaviour of viscous parts of the rolldamping.Sometimes, for applications in frequency domain, an equivalent linear roll dampingcoefficient, ( )1

44vB , has to be determined. This coefficient can be obtained by stipulating that


185

an equivalent linear roll damping dissipates an identical amount of energy as the non-linearroll damping. This results for a linearised quadratic roll damping coefficient, ( )2

44vB , into:

( ) ( ) ∫∫ ⋅⋅⋅⋅=⋅⋅⋅φφ

φφφφφT

v

T

v dtBdtB0

244

0

144

&&&&&

or with some algebra:( ) ( )2

441

44 38

vav BB ⋅

⋅⋅

⋅= ωφ

π

Equation 4.2–17

For the estimation of the non-potential parts of the roll damping, use has been made of workpublished by Ikeda, Himeno and Tanaka [1978]. A few sub-ordinate parts have been modifiedand this empirical method is called here the ''Ikeda method''.

This Ikeda method estimates the following linear components of the roll damping coefficientof a ship:

SB44 a correction on the potential roll damping coefficient due to forward speed,

FB44 the frictional roll damping coefficient,

EB44 the eddy making roll damping coefficient,

LB44 the lift roll damping coefficient and

KB44 the bilge keel roll damping coefficient.So, the additional - mainly viscous - roll damping coefficient VB44 is given by:

KLEFSV BBBBBB 444444444444 ++++=

Equation 4.2–18

Ikeda, Himeno and Tanaka [1978] claim fairly good agreements between their predictionmethod and experimental results. They conclude that the method can be used safely forordinary ship forms, which conclusion has been confirmed by the author too. But for unusualship forms, for very full ship forms and for ships with a very large breadth to draft ratio themethod is not always accurate sufficiently.For numerical reasons three restrictions have to be made:• if, locally, 999.0>sσ then: 999.0=sσ .

• if, locally, ssDOG σ⋅−< then: ssDOG σ⋅−= .• if a calculated component of the viscous roll damping coefficient becomes less than zero,

this component has to be set to zero.

4.2.4.1 Notations of Ikeda et.al.

In this description of the Ikeda method, the notation of the authors (Ikeda, Himeno andTanaka) is maintained as far as could be possible here:

ρ density of waterν kinematic viscosity of waterg acceleration of gravityV forward ship speedRn Reynolds numberω circular roll frequency


186

aφ roll amplitudeL length of the shipB breadthD amidships draught

MC amidships section coefficient

BC block coefficientDLSL ⋅≈ lateral area

fS wetted hull surface area

OG distance of centre of gravity above still water level, positive upwards(this sign convention deviates from that in the paper of Ikeda)

sB sectional breadth water line

sD sectional draft

sA sectional area

sσ sectional area coefficient

0H sectional half breadth to draft ratio

1a sectional Lewis coefficient

3a sectional Lewis coefficient

sM sectional Lewis scale factor

fr average distance between roll axis and hull surface

OL distance point of taking representative angle of attack to roll axis,

approximated by DLO ⋅= 3.0

RL distance of centre of action of lift force in roll motion to roll axis,approximated by DLR ⋅= 5.0

kh height of the bilge keels

kL length of the bilge keels

kr distance between roll axis and bilge keel

kf correction for increase of flow velocity at the bilge

pC pressure coefficient

ml lever of the moment

br local radius of the bilge circle

4.2.4.2 Effect of Forward Speed, SB44

Ikeda obtained an empirical formula for the three-dimensional forward speed correction onthe zero speed potential damping by making use of the general characteristics of a doubletflow model. Two doublets have represented the rolling ship: one at the stern and one at thebow of the ship.With this, semi-theoretically the forward speed effect on the linear potential dampingcoefficient has been approximated as a fraction of the potential damping coefficient by:


187

( ) ( )[ ]( ) ( )

−

⋅−−⋅+

−Ω⋅⋅−++⋅⋅=

−⋅−0.1

12

3.020tanh115.0 225.0150

21

22

4444 ωeAA

AABB S

Equation 4.2–19

with:

44B potential roll damping coefficient of the ship (about G )gV⋅=Ω ω non-dimensional circular roll frequency

gDD ⋅= 2ωξ non-dimensional circular roll frequency squaredDeA D

ξξ ⋅−− ⋅+= 22.11 0.1 maximum value of 44B at 25.0=ω

DeA Dξξ ⋅−− ⋅+= 20.1

2 5.0 minimum value of 44B at large ω

4.2.4.3 Frictional Roll Damping, FB44

Kato deduced semi-empirical formulas for the frictional roll damping from experimentalresults of circular cylinders, wholly immersed in the fluid. An effective Reynolds number ofthe roll motion was defined by:

ν

ωφ

⋅

⋅

=

2

512.0a

fr

Rn

Equation 4.2–20

In here, for ship forms the average distance between the roll axis and the hull surface can beapproximated by:

( )π

OGLS

Cr

fB

f

⋅+⋅⋅+=

2145.0887.0

Equation 4.2–21

with a wetted hull surface area fS , approximated by:

( )BCDLS Bf ⋅+⋅⋅= 7.1

Equation 4.2–22

The relation between the density, kinematic viscosity and temperture of fresh water and seawater are given in Figure 4.2–3.


188

990

1000

1010

1020

1030

0 10 20 30

Fresh Water

Sea Water

Temperature (0C)

De

nsi

ty

(kg

/m3)

0.5

1.0

1.5

2.0

0 10 20 30

Sea Water

Fresh Water

Temperature (0C)

Kin

em

ati

c V

isc

osi

ty

(m

2 s)

Figure 4.2–3: Relation between density, kinematic viscosity and temperature of water

When eliminating the temperature of water, the following relation can express the kinematicviscosity into the density of water in the kg-m-s system:

( ) ( )( ) ( )26

26

102502602.010251039.0063.110 : watersea

100007424.010003924.0442.110 :rfresh wate

−⋅+−⋅+=⋅

−⋅+−⋅+=⋅

ρρν

ρρν

Equation 4.2–23

as given in Figure 4.2–4.

0

5

10

15

20

25

997 998 999 1000

Viscosity Actual Viscosity Polynomial Temperature

Density Fresh Water (kg/m3)

Kin

em

ati

c V

isc

osi

ty *

10

7

(m

2 s)

Te

mp

era

ture

(

0 C)

Fresh Water

0

5

10

15

20

25

1023 1024 1025 1026 1027 1028

Viscosity Actual Viscosity Polynomial Temperature

Density Salt Water (kg/m3)

Kin

em

ati

c V

isc

osi

ty *

10

7

(m

2 s)

Te

mp

era

ture

(

0 C)

Salt Water

Figure 4.2–4: Kinematic viscosity as a function of density

Kato expressed the skin friction coefficient as:114.05.0 014.0328.1 −− ⋅+⋅= RnRnC f

Equation 4.2–24

The first part in this expression represents the laminar flow case. The second part has beenignored by Ikeda, but has been included here.Using this, the quadratic roll damping coefficient due to skin friction at zero forward shipspeed is expressed as:


189

( )fffF CSrB ⋅⋅⋅⋅= 32

44 21

0ρ

Equation 4.2–25

This frictional roll damping component increases slightly with forward speed.Semi-theoretically, Tamiya deduced a modification coefficient for the effect of forward speedon the friction component. Accurate enough from a practical point of view, this results into thefollowing formula for the speed dependent frictional damping coefficient:

( )

⋅⋅+⋅⋅⋅⋅⋅=

LVCSrB fffF ω

ρ 1.40.121 32

44

Equation 4.2–26

Then, the equivalent linear roll damping coefficient due to skin friction is expressed as:

⋅⋅+⋅⋅⋅⋅⋅⋅

⋅=

LVCSrB fffaF ω

ρωφπ

1.40.121

38 3

44

Equation 4.2–27

Ikeda confirmed the use of his formula for the three-dimensional turbulent boundary layerover the hull of an oscillating ellipsoid in roll motion.

4.2.4.4 Eddy Making Damping, EB44

At zero forward speed the eddy making roll damping for the naked hull is mainly caused byvortices, generated by a two-dimensional separation. From a number of experiments with two-dimensional cylinders it was found that for a naked hull this component of the roll moment isproportional to the roll frequency squared and the roll amplitude squared. This means that thecorresponding quadratic roll damping coefficient does not depend on theperiod parameter but on the hull form only.When using a simple form for the pressure distribution on the hull surface it appears that thepressure coefficient pC is a function of the ratio γ of the maximum relative velocity maxU to

the mean velocity meanU on the hull surface:

meanUUmax=γ

Equation 4.2–28

The γ−pC relation was obtained from experimental roll damping data of two-dimensional

models. These experimental results are fitted by:50.10.235.0 187.0 +⋅−⋅= ⋅−− γγ eeCp

Equation 4.2–29

The value of γ around a cross section is approximated by the potential flow theory for arotating Lewis form cylinder in an infinite fluid.An estimation of the local maximum distance between the roll axis and the hull surface, maxr ,has to be made.Values of ( )ψmaxr have to be calculated for:


190

0.01 ==ψψ and ( )

⋅+⋅

==

3

31

2

41

cos

5.0

aaa

ψψ

Equation 4.2–30

The values of ( )ψmaxr follow from:

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

⋅⋅+⋅−

+⋅⋅−⋅+⋅=

231

231

max3coscos1

3sinsin1

ψψ

ψψψ

aa

aaMr s

Equation 4.2–31

With these two results a value maxr and a value ψ follow from the conditions:

• For ( ) ( )2max1max ψψ rr > : ( )1maxmax ψrr = and 1ψψ =• For ( ) ( )2max1max ψψ rr < : ( )2maxmax ψrr = and 2ψψ =

Equation 4.2–32

The relative velocity ratio γ on a cross section is obtained by:

+⋅⋅+⋅

+⋅⋅⋅

⋅= 22max

0

3 2

2

baHMr

DOG

HD

f s

sss σ

πγ

with:

( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( )25 11065.13

2131

21

231

313

2131

21

231

313

3312

32

1

3131

0

41

sin336

3sin15sin2

cos336

3cos15cos2

4cos62cos31291

112

2

sef

aaaaaa

aaab

aaaaaa

aaaa

aaaaaH

aaD

aaB

M

DBA

DB

H

sss

ss

ss

s

s

σ

ψ

ψψψ

ψψψψ

σ

−⋅⋅−⋅+=

⋅+⋅⋅++⋅⋅+

+⋅⋅−⋅+⋅⋅⋅−=⋅+⋅⋅−+⋅⋅−

+⋅⋅−⋅+⋅⋅⋅−=⋅⋅⋅−⋅⋅⋅−⋅⋅+⋅++=

+−=

++⋅=

⋅=

⋅=

Equation 4.2–33

With this a quadratic sectional eddy making damping coefficient for zero forward speedfollows from:


191

( )

⋅−⋅+

⋅−+⋅

⋅−

⋅⋅

⋅⋅⋅=

2

102

11

2

max4'244

11

21

0

s

b

s

b

ss

b

ps

sE

Drf

HfD

rfDOG

Drf

CDr

DB ρ

with:[ ]

( )( ) ( ) ( )ss

s

sef

f

σπσπ

σσ ⋅⋅−⋅−⋅−⋅=

−⋅+⋅=⋅− 255

2

1

sin15.1cos15.0

1420tanh15.0

Equation 4.2–34

The approximations of the local radius of the bilge circle br are given as:

( )

2 : and 1for

: and 1for 4

12 :

2 and for

00

0

0

sbsb

sbsb

ssb

sbsb

BrDHrH

DrDrH

HDr

BrDr

=⋅><

=>>−

−⋅⋅⋅=<<

πσ

Equation 4.2–35

For three-dimensional ship forms the zero forward speed eddy making quadratic roll dampingcoefficient is found by an integration over the ship length:

( ) ( )∫ ⋅=L

bEE dxBB '244

244 00

Equation 4.2–36

This eddy making roll damping decreases rapidly with the forward speed to a non-linearcorrection for the lift force on a ship with a small angle of attack. Ikeda has analysed thisforward speed effect by experiments and the result has been given in an empirical formula.With this the equivalent linear eddy making damping coefficient at forward speed is given by:

( )2

24444 1

13

80 K

BB EaE +⋅⋅

⋅⋅

⋅= ωφ

π with:

LV

K⋅⋅

=ω04.0

Equation 4.2–37

4.2.4.5 Lift Damping, LB44

The roll damping coefficient due to the lift force is described by a modified formula ofYumuro:

⋅⋅+⋅+⋅⋅⋅⋅⋅⋅⋅=

RORRONLL LL

OGL

OGLLkVSB

2

44 7.04.10.121 ρ

Equation 4.2–38

The slope of the lift curve αLC is defined by:


192

−⋅⋅+

⋅⋅=

=

045.01.42

LB

LD

Ck L

N

χπα

Equation 4.2–39

in which the coefficient χ is given by Ikeda in relation to the amidships section coefficient

MC :

30.0 :99.097.0

10.0 :97.092.0

00.0 :92.0

=<<=<<=<

χχχ

M

M

M

C

C

C

Equation 4.2–40

These data are fitted here by:( ) ( )32 91.070091.0106 −⋅−−⋅= MM CCχ

with the restrictions:• if 91.0<MC then 00.0=χ• if 00.1>MC then 35.0=χ

Equation 4.2–41

4.2.4.6 Bilge Keel Damping, KB44

The quadratic bilge keel roll damping coefficient has been into two components:• a component ( )2

44 NKB due to the normal force on the bilge keels

• a component ( )244 SKB due to the pressure on the hull surface, created by the bilge keels.

The coefficient of the normal force component ( )244 NKB of the bilge keel damping can be

deduced from experimental results of oscillating flat plates. The drag coefficient DC dependson the period parameter or the Keulegan-Carpenter number. Ikeda measured this non-lineardrag also by carrying out free rolling experiments with an ellipsoid with and without bilgekeels. This resulted in a quadratic sectional damping coefficient:

( )DkkkK CfhrB

N⋅⋅⋅⋅= 23'2

44 ρ with: ( )sef

frh

C

k

kak

kD

σ

φπ−⋅−⋅+=

+⋅⋅⋅

⋅=

0.11603.00.1

40.25.22

Equation 4.2–42

The approximation of the local distance between the roll axis and the bilge keel kr is given as:22

0 293.00.1293.0

⋅−++

⋅−⋅=

s

b

ss

bsk D

rDOG

Dr

HDr

Equation 4.2–43


193

The approximation of the local radius of the bilge circle br in here is given before.Assuming a pressure distribution on the hull caused by the bilge keels, a quadratic sectionalroll damping coefficient can be defined:

( ) ∫ ⋅⋅⋅⋅⋅⋅=k

S

h

mpkkK dhlCfrB0

22'244 2

1 ρ

Equation 4.2–44

Ikeda carried out experiments to measure the pressure on the hull surface created by bilgekeels. He found that the coefficient +

pC the pressure on the front face of the bilge keel does

not depend on the period parameter, while the coefficient −pC of the pressure on the back face

of the bilge keel and the length of the negative pressure region depend on the periodparameter.Ikeda defines an equivalent length of a constant negative pressure region 0S over the height ofthe bilge keels, which is fitted to the following empirical formula:

kakk hrfS ⋅+⋅⋅⋅⋅= 95.13.00 φπ

Equation 4.2–45

The pressure coefficients on the front face of the bilge keel, +pC , and on the back face of the

bilge keel, −pC , are given by:

20.1=+pC and 20.15.22 −

⋅⋅⋅⋅−=−

akk

kp rf

hC

φπ

Equation 4.2–46

and the sectional pressure moment is given by:

( )∫ +− ⋅+⋅−⋅=⋅⋅kh

ppsmp CBCADdhlC0

2

with:


194

( )

( )( ) ( )

( ) ( )

( )( ) ( )

( )( ) ( )

bb

b

b

bs

s

s

b

rSrS

mm

rSmmm

rS

rSmDS

m

mmHmHmH

m

mmHmHmH

m

mHm

mmm

DOG

m

Drm

mmmmmm

mmmmH

mB

mmmmA

⋅⋅<

−⋅⋅+=

⋅⋅>⋅+=

⋅⋅<=

⋅⋅>⋅⋅−=

⋅−⋅⋅−⋅⋅+−⋅+⋅

=

⋅−⋅⋅−⋅+⋅−⋅+⋅

=

−=

−−=

−=

=

⋅+⋅⋅+⋅−⋅−⋅⋅−

+⋅−⋅

=

−⋅+=

π

ππ

ππ

25.0 :for cos1414.1

25.0 :for 414.0

25.0 :for 0.0

25.0 :for 25.0

215.01215.00106.0382.00651.0414.0

215.01215.00106.0382.00651.0414.0

0.1

215.01621

215.03

00

17

0178

0

010

7

110

102

106

110

102

105

104

213

2

1

645311

232

1

10

34

27843

Equation 4.2–47

The equivalent linear total bilge keel damping coefficient can be obtained now by integratingthe two sectional roll damping coefficients over the length of the bilge keels and linearizingthe result:

( )∫ ⋅+⋅⋅⋅⋅

=k

SN

L

bKKaK dxBBB '44

'4444 3

8 ωφπ

Equation 4.2–48

Experiments of Ikeda have shown that the effect of forward ship speed on this roll dampingcoefficient can be ignored.

4.2.4.7 Calculated Roll Damping Components

In Figure 4.2–5 an example is given of the several roll damping components, as derived withIkeda's method, for the S-175 container ship design.


195

Figure 4.2–5: Roll damping coefficients of Ikeda, Himeno and Tanaka

It may be noted that for full-scale ships, because of the higher Reynolds number, the frictionalpart of the roll damping is expected to be smaller than showed above.


196


197

5 Hydromechanical Loads

With the approach as mentioned before, a description will be given here of the determinationof the hydromechanical forces and moments for all six modes of motions.

In the ''Ordinary Strip Theory'', as published by Korvin-Kroukovsky and Jacobs [1957] andothers, the uncoupled two-dimensional potential hydromechanical loads in the direction j aredefined by:

'*'*''RSjhjjjhjjjhj XNM

DtD

X +⋅+⋅= ζζ && (Ordinary Strip Theory, OST)

In the ''Modified Strip Theory'', as has been published later by for instance Tasai [1969] andothers, these loads become:

'*'''RSjhjjj

ejjhj XNiM

DtDX +

⋅

⋅−= ζ

ω& (Modified Strip Theory, MST)

In these definitions of the two-dimensional hydromechanical load, *hjζ& is the harmonic

oscillatory motion, 'jjM and '

jjN are the two-dimensional potential mass and damping and

the non-diffraction part 'RSjX is the two-dimensional quasi-static restoring spring term.

At all following pages, the hydromechanical load has been calculated in the ( )bbb zyxG ,, axes

system with the centre of gravity G in the still water level, so 0=OG .

Some of the terms in the hydromechanical loads have been outlined there:• the ''Modified Strip Theory'' (OST) includes these outlined terms, but• when ignoring these outlined terms, the ''Ordinary Strip Theory'' (MST) has been

presented.


198

5.1 Hydromechanical Forces for Surge

The hydromechanical forces for surge are found by integration over the ship length of thetwo-dimensional values:

∫ ⋅=L

bhh dxXX '11

Equation 5.1–1

When assuming that the cross sectional hydromechanical force hold at a plane through thelocal centroid of the cross section, b , parallel to ( )bb yx , , equivalent longitudinal motions ofthe water particles, relative to the cross section of an oscillating ship in still water, are definedby:

θ

θθθζ

θ

θθζ

θζ

&&&&

&&&&&&&

&&

&&&

⋅+−≈

⋅∂

∂⋅+⋅

∂∂

⋅⋅−⋅+−=

⋅+−≈

⋅∂∂

⋅−⋅+−=

⋅+−=

bGx

xbG

VxbG

VbGx

bGx

xbG

VbGx

bGx

bbh

bh

h

2

22*

1

*1

*1

2

Equation 5.1–2In here, bG is the vertical distance of the centre of gravity of the ship G above the centroidb of the local submerged sectional area.

According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalforce on a surging cross section in still water is defined by:

*1

'11'

11*

1'

11

*1

'11

*1

'11

'1

hb

h

hhh

dxdM

VNM

NMDtD

X

ζζ

ζζ

&&&

&&

⋅

⋅−+⋅=

⋅+⋅=

Equation 5.1–3

According to the ''Modified Strip Theory'' this hydromechanical force becomes:

*1

'11'

11*

1

'11

2'

11

*1

'11

'11

'1

hb

hbe

he

h

dxdM

VNdx

dNVM

Ni

MDtD

X

ζζω

ζω

&&&

&

⋅

⋅−+⋅

⋅+=

⋅

⋅−=

Equation 5.1–4


199

This results into the following coupled surge equation:

( )

1151515

131313

1111111

w

h

Xcba

zczbza

xcxbxaXx

=⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+∇⋅+=−⋅∇⋅

θθθ

ρρ

&&&&&&

&&&&&

Equation 5.1–5

with:

0

0

0

0

0

15

11

'11'

1115

'11

2'

1115

13

13

13

11

11

'11'

1111

'11

2'

1111

=

⋅−⋅⋅

⋅−−=

⋅⋅⋅−

+⋅⋅−=

==

=

=

+⋅

−+=

⋅⋅+⋅+=

∫

∫∫

∫

∫∫

c

BGbdxbGdx

dMVNb

dxbGdx

dNVdxbGMa

c

b

a

c

bdxdx

dMNb

dxdx

dNVdxMa

VL

bb

Lb

beLb

VL

bb

Lb

beLb

ω

ω

Equation 5.1–6

The ''Modified Strip Theory'' includes the outlined terms. When ignoring these outlined termsthe ''Ordinary Strip Theory'' is presented.

A small viscous surge damping coefficient Vb11 , derived from the still water resistanceapproximation of Troost [1955], has been added here.


200

After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled surge equation become:

0

0

0

0

0

15

11

'11'

1115

'11

2'

1115

13

13

13

11

11'

1111

'1111

=

⋅−⋅⋅

⋅−−=

⋅⋅⋅−

+⋅⋅−=

==

=

=

+⋅+=

⋅+=

∫

∫∫

∫

∫

c

BGbdxbGdx

dMVNb

dxbGdx

dNVdxbGMa

c

b

a

c

bdxNb

dxMa

VL

bb

Lb

beLb

V

L

b

Lb

ω


201

5.2 Hydromechanical Forces for Sway

The hydromechanical forces for sway are found by integration over the ship length of the two-dimensional values:

∫ ⋅=L

bhh dxXX '22

Equation 5.2–1

The lateral and roll motions of the water particles, relative to the cross section of an oscillatingship in still water, are defined by:

φψψζ

φψψζ

φψζ

&&&&&&&&&

&&&&

⋅−⋅⋅+⋅−−=

⋅−⋅+⋅−−=

⋅−⋅−−=

OGVxy

OGVxy

OGxy

bh

bh

bh

2*2

*2

*2

φζ

φζ

φζ

&&&&

&&

−=

−=

−=

*4

*4

*4

h

h

h

Equation 5.2–2

According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalforce on a swaying cross section in still water is defined by:

*4

'24'

24*

4'

24*

2

'22'

22*

2'

22

*4

'24

*4

'24

*2

'22

*2

'22

'2

hb

hhb

h

hhhhh

dxdM

VNMdx

dMVNM

NMDtD

NMDtD

X

ζζζζ

ζζζζ

&&&&&&

&&&&

⋅

⋅−+⋅+⋅

⋅−+⋅=

⋅+⋅+⋅+⋅=

Equation 5.2–3


*4

'24'

24*

4

'24

2'

24

*2

'22'

22*

2

'22

2'

22

*2

'22

'22

*2

'22

'22

'2

hb

hbe

hb

hbe

he

he

h

dxdM

VNdx

dNVM

dxdMVN

dxdNVM

Ni

MDtD

Ni

MDtD

X

ζζω

ζζω

ζω

ζω

&&&

&&&

&&

⋅

⋅−+⋅

⋅++

⋅

⋅−+⋅

⋅+=

⋅

⋅−+

⋅

⋅−=

Equation 5.2–4


202

This results into the following coupled sway equation:

( )

2262626

242424

2222222

w

h

Xcba

cba

ycybyaXy

=⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+∇⋅+=−⋅∇⋅

ψψψφφφ

ρρ

&&&

&&&

&&&&&

with:

0

2

0

0

26

'22

2

2

'22

'22'

2226

'22

2

'222

'22'

222

'2226

24

'22'

22

'24'

2424

'22

2

'24

2

'22

'2424

22

'22'

2222

'22

2'

2222

=

⋅⋅+

⋅⋅⋅−⋅⋅

⋅−+=

⋅⋅⋅+⋅⋅+

⋅

⋅−⋅+⋅⋅+=

=

⋅

⋅−⋅+⋅

⋅−+=

⋅⋅⋅+⋅⋅+

⋅⋅+⋅+=

=

⋅

⋅−+=

⋅⋅+⋅+=

∫

∫∫

∫ ∫

∫∫

∫∫

∫∫

∫∫

∫

∫∫

c

dxdx

dNV

dxMVdxxdx

dMVNb

dxxdx

dNVdxN

V

dxdx

dMVN

VdxxMa

c

dxdx

dMVNOGdx

dxdM

VNb

dxdx

dNOG

Vdx

dxdNV

dxMOGdxMa

c

dxdx

dMVNb

dxdx

dNVdxMa

Lb

be

Lb

Lbb

b

L Lbb

beb

e

Lb

beLbb

Lb

bLb

b

Lb

beLb

be

L

b

L

b

Lb

b

Lb

beLb

ω

ωω

ω

ωω

ω

Equation 5.2–5

The ''Modified Strip Theory'' includes the outlined terms. When ignoring these terms the''Ordinary Strip Theory'' is presented.


203

After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled sway equation become:

0

0

0

26

'22

'2226

'222

'2226

24

'22

'2424

'22

'2424

22

'2222

'2222

=

⋅⋅−⋅⋅+=

⋅⋅+⋅⋅+=

=

⋅⋅+⋅+=

⋅⋅+⋅+=

=

⋅+=

⋅+=

∫∫

∫∫

∫∫

∫∫

∫

∫

c

dxMVdxxNb

dxNV

dxxMa

c

dxNOGdxNb

dxMOGdxMa

c

dxNb

dxMa

Lb

Lbb

Lb

eLbb

L

b

L

b

Lb

Lb

Lb

Lb

ω

Equation 5.2–6

So no terms have been added for the ''Modified Strip Theory''.


204

5.3 Hydromechanical Forces for Heave

The hydromechanical forces for heave are found by integration over the ship length of thetwo-dimensional values:

∫ ⋅=L

bhh dxXX '33

Equation 5.3–1

The vertical motions of the water particles, relative to the cross section of an oscillating shipin still water, are defined by:

θθζ

θθζ

θζ

&&&&&&&

&&&

⋅⋅−⋅+−=

⋅−⋅+−=

⋅+−=

Vxz

Vxz

xz

bh

bh

bh

2*3

*3

*3

Equation 5.3–2

According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalforce on a heaving cross section in still water is defined by:

*3

*3

'33'

33*

3'

33

*3

*3

'33

*3

'33

'3

2

2

hwhb

h

hwhhh

ygdx

dMVNM

ygNMDtD

X

ζρζζ

ζρζζ

⋅⋅⋅⋅+⋅

⋅−+⋅=

⋅⋅⋅⋅+⋅+⋅=

&&&

&&

Equation 5.3–3


*3

*3

'33'

33*

3

'33

2'

33

*3

*3

'33

'33

'3

2

2

hwhb

hbe

hwhe

h

ygdx

dMVN

dxdNV

M

ygNi

MDtD

X

ζρζζω

ζρζω

⋅⋅⋅⋅+⋅

⋅−+⋅

⋅+=

⋅⋅⋅⋅+

⋅

⋅−=

&&&

&

Equation 5.3–4


205

This results into the following coupled heave equation:

( )3353535

333333

3131313

w

h

Xcba

zczbza

xcxbxaXz

=⋅+⋅+⋅+⋅+⋅+⋅+∇⋅+⋅+⋅+⋅+=−⋅∇⋅

θθθρ

ρ

&&&&&&

&&&&&

Equation 5.3–5

with:

∫

∫

∫∫

∫ ∫

∫∫

∫

∫

∫∫

⋅⋅⋅⋅⋅−=

⋅⋅+

⋅⋅⋅+⋅⋅

⋅−−=

⋅⋅⋅−

+⋅⋅−

+

⋅

−⋅−⋅⋅−=

⋅⋅⋅⋅+=

⋅

⋅−+=

⋅⋅+⋅+=

===

Lbbw

Lb

be

Lb

Lbb

b

L Lbb

be

b

e

Lb

beLbb

Lbw

L

bb

L

bbeL

b

dxxygc

dxdx

dNV

dxMVdxxdx

dMVNb

dxxdx

dNVdxN

V

dxdx

dMN

VdxxMa

dxygc

dxdx

dMVNb

dxdx

dNVdxMa

c

b

a

ρ

ω

ωω

ω

ρ

ω

2

2

2

0

0

0

35

'33

2

2

'33

'33'

3335

'33

2'

332

'33'

332'

3335

33

'33'

3333

'33

2

'3333

31

31

31

Equation 5.3–6

The ''Modified Strip Theory Method'' includes the outlined terms. When ignoring these termsthe ''Ordinary Strip Theory'' is presented.


206

After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled heave equation become:

∫

∫∫

∫∫

∫

∫

∫

⋅⋅⋅⋅⋅−=

⋅⋅+⋅⋅−=

⋅⋅−⋅⋅−=

⋅⋅⋅⋅+=

⋅+=

⋅+=

==

=

Lbbw

L

b

L

bb

L

b

eL

bb

Lbw

L

b

Lb

dxxygc

dxMVdxxNb

dxNV

dxxMa

dxygc

dxNb

dxMa

c

b

a

ρ

ω

ρ

2

2

0

0

0

35

'33

'3335

'332

'3335

33

'3333

'3333

31

31

31

Equation 5.3–7



207

5.4 Hydromechanical Moments for Roll

The hydromechanical moments for roll are found by integration over the ship length of thetwo-dimensional values:

∫ ⋅=L

bhh dxXX '44

Equation 5.4–1

The roll and lateral motions of the water particles, relative to the cross section of an oscillatingship in still water, are defined by:

φζ

φζ

φζ

&&&&

&&

−=

−=

−=

*4

*4

*4

h

h

h

φψψζ

φψψζ

φψζ

&&&&&&&&&

&&&&

⋅−⋅⋅+⋅−−=

⋅−⋅+⋅−−=

⋅−⋅−−=

OGVxy

OGVxy

OGxy

bh

bh

bh

2*2

*2

*2

Equation 5.4–2

According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalmoment on a rolling cross section in still water is defined by:

*2

'42'

42*

2'

42

*4

3*

4

'44'

44*

4'

44

*2

'42

*2

'42

*4

3*

4'

44*

4'

44'

2

232

232

hb

h

hsw

hb

h

hh

hsw

hhh

dxdM

VNM

bGAy

gdx

dMVNM

NMDtD

bGAy

gNMDtD

X

ζζ

ζρζζ

ζζ

ζρζζ

&&&

&&&

&&

&&

⋅

⋅−+⋅+

⋅

⋅−⋅⋅⋅+⋅

⋅−+⋅=

⋅+⋅+

⋅

⋅−⋅⋅⋅+⋅+⋅=

Equation 5.4–3

According to the ''Modified Strip Theory'' this hydromechanical moment becomes:

*2

'42'

42*

2

'42

2

'42

*4

3

*4

'44'

44*

4

'44

2

'44

*2

'42

'42

*4

3*

4'

44'

44'

2

232

232

hb

hbe

hsw

hb

hbe

he

hsw

he

h

dxdM

VNdx

dNVM

bGAy

g

dxdM

VNdx

dNVM

Ni

MDtD

bGAy

gNi

MDtD

X

ζζω

ζρ

ζζω

ζω

ζρζω

&&&

&&&

&

&

⋅

⋅−+⋅

⋅++

+⋅

⋅−⋅⋅⋅+

⋅

⋅−+⋅

⋅+=

⋅

⋅−+

⋅

⋅−⋅⋅⋅+

⋅

⋅−=

Equation 5.4–4


208

This results into the following coupled roll equation:

( )( ) 4464646

444444

4242424

wxz

xx

hxzxx

XcbaI

cbaI

ycybyaXII

=⋅+⋅+⋅+−+⋅+⋅+⋅+++⋅+⋅+⋅+=−⋅−⋅

ψψψφφφ

ψφ

&&&

&&&

&&&&&&&

Equation 5.4–5

with:

2646

'42

2

2

26'

42

'42'

4246

'42

2

'422

26

'42'

422

'4246

24

3

44

2444

'42'

42

'44'

4444

'42

2

'44

2

24'

42'

4444

2242

22

'42'

4242

'24

222'

4242

0

2

232

0

cOGc

dxdx

dNV

bOGdxMVdxxdx

dMVNb

dxxdx

dNVdxN

V

aOGdxdx

dMVN

VdxxMa

GMg

cOGdxbGAy

gc

bOGbdxdx

dMVNOGdx

dxdM

VNb

dxdx

dNOG

Vdx

dxdNV

aOGdxMOGdxMa

cOGc

bOGdxdx

dMVNb

dxdx

dNVaOGdxMa

Lb

be

Lb

Lbb

b

L Lbb

beb

e

Lb

beLbb

Lb

sw

LVb

bLb

b

L Lb

beb

be

L

b

L

b

L

bb

L

bbeL

b

⋅+=

⋅⋅+

⋅+⋅⋅⋅−⋅⋅

⋅−+=

⋅⋅⋅+⋅⋅+

⋅+⋅

⋅−⋅+⋅⋅+=

⋅∇⋅⋅+=

⋅+⋅

⋅+⋅⋅⋅+=

⋅++⋅

⋅−⋅+⋅

⋅−+=

⋅⋅⋅+⋅⋅+

⋅+⋅⋅+⋅+=

⋅+=

⋅+⋅

⋅−+=

⋅⋅+⋅+⋅+=

∫

∫∫

∫ ∫

∫∫

∫

∫∫

∫ ∫

∫∫

∫

∫∫

ω

ωω

ω

ρ

ρ

ωω

ω

Equation 5.4–6

The ''Modified Strip Theory'' includes the outlined terms. When ignoring these terms the''Ordinary Strip Theory Method'' is presented.


209

A viscous roll damping coefficient Vb44 , derived for instance with the empirical method ofIkeda [1978], has been added here.

After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled roll equation become:

0

0

46

26'

42'

4246

26'

422'

4246

44

2444'

42'

4444

24'

42'

4444

24

22'

4242

22'

4242

=

⋅+⋅⋅−⋅⋅+=

⋅+⋅⋅+⋅⋅+=

⋅∇⋅⋅+=

⋅++⋅⋅+⋅+=

⋅+⋅+⋅+=

=

⋅+⋅+=

⋅+⋅+=

∫∫

∫∫

∫∫

∫∫

∫

∫

c

bOGdxMVdxxNb

aOGdxNV

dxxMa

GMgc

bOGbdxNOGdxNb

aOGdxMdxMa

c

bOGdxNb

aOGdxMa

Lb

Lbb

Lb

eLbb

LVb

Lb

Lb

Lb

Lb

Lb

ω

ρ

Equation 5.4–7



210

5.5 Hydromechanical Moments for Pitch

The hydromechanical moments for pitch are found by integration over the ship length of thetwo-dimensional contributions of surge and heave into the pitch moment:

∫ ⋅=L

bhh dxXX '55 with: bhhh xXbGXX ⋅−⋅−= '

3'

1'

5

Equation 5.5–1

According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalmoment on a pitching cross section in still water is defined by surge and heave contributions:

*3

*3

'33'

33*

3'

33

*1

'11'

11*

1'

11'

5

2 hbwhbb

hb

hb

hh

xygxdx

dMVNxM

bGdx

dMVNbGMX

ζρζζ

ζζ

⋅⋅⋅⋅⋅−⋅⋅

⋅−−⋅⋅−

⋅⋅

⋅−−⋅⋅−=

&&&

&&&

Equation 5.5–2

According to the ''Modified Strip Theory'' this hydromechanical moment becomes:

*3

*3

'33'

33*

3

'33

2'

33

*1

'11'

11*

1

'11

2'

11'

5

2 hbw

hbb

hbbe

hb

hbe

h

xyg

xdx

dMVNx

dxdNV

M

bGdx

dMVNbG

dxdNV

MX

ζρ

ζζω

ζζω

⋅⋅⋅⋅⋅−

⋅⋅

⋅−−⋅⋅

⋅+−

⋅⋅

⋅−−⋅⋅

⋅+−=

&&&

&&&

Equation 5.5–3


211

This results into the following coupled pitch equation:

( ) 5555555

535353

5151515

wyy

hyy

XcbaI

zczbza

xcxbxaXI

=⋅+⋅+⋅++

⋅+⋅+⋅+⋅+⋅+⋅+=−⋅

θθθ

θ

&&&&&&

&&&&&

Equation 5.5–4

with:

∫

∫

∫∫

∫

∫ ∫

∫∫

∫∫

∫

∫

∫∫

∫

∫∫

⋅⋅⋅⋅⋅+=

⋅⋅⋅−

+

⋅⋅⋅⋅−⋅⋅

⋅−+

⋅+⋅⋅

⋅−+=

⋅⋅⋅+⋅⋅⋅+

⋅⋅

−⋅+⋅⋅+

⋅⋅⋅+⋅⋅+=

⋅⋅⋅⋅⋅−=

⋅⋅

⋅−−=

⋅⋅⋅−

+⋅⋅−=

=

⋅−⋅⋅

⋅−−=

⋅⋅⋅−

+⋅⋅−=

Lbbw

Lbb

be

Lbb

Lbb

b

VL

bb

L Lbb

bebb

e

L

bbbeL

bb

Lb

beLb

Lbbw

Lbb

b

Lbb

beLbb

VL

bb

L

bbeL

b

dxxygc

dxxdx

dNV

dxxMVdxxdx

dMVN

BGbdxbGdx

dMVNb

dxxdx

dNVdxxN

V

dxxdx

dMN

VdxxM

dxbGdx

dNVdxbGMa

dxxygc

dxxdx

dMVNb

dxxdx

dNVdxxMa

c

BGbdxbGdx

dMVNb

dxbGdx

dNVdxbGMa

255

'33

2

2

'33

2'

33'33

2

11

2'

11'1155

2'

332

'332

'33'

3322'

33

2'

112

2'1155

53

'33'

3353

'33

2

'3353

51

11

'11'

1151

'11

2

'1151

2

2

2

0

ρ

ω

ωω

ω

ω

ρ

ω

ω


212

Equation 5.5–5

The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.

After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled pitch equation become:

∫

∫∫

∫

∫

∫∫∫

∫∫

∫

∫∫

∫∫

∫

∫∫

⋅⋅⋅⋅⋅+=

⋅⋅+⋅⋅+

⋅+⋅⋅

⋅−+=

⋅⋅⋅−

+

⋅⋅+⋅⋅⋅+⋅⋅+

⋅⋅⋅+⋅⋅+=

⋅⋅⋅⋅⋅−=

⋅⋅−⋅⋅−=

⋅⋅+⋅⋅−=

=

⋅−⋅⋅

⋅−−=

⋅⋅⋅−

+⋅⋅−=

Lbbw

Lb

eLbb

V

L

bb

Lbb

e

L

b

eL

bb

eL

bb

L

bbeL

b

Lbbw

L

b

L

bb

L

b

eL

bb

VL

bb

L

bbeL

b

dxxygc

dxNV

dxxN

BGbdxbGdx

dMVNb

dxxNV

dxMV

dxxNV

dxxM

dxbGdx

dNVdxbGMa

dxxygc

dxMVdxxNb

dxNV

dxxMa

c

bGbdxbGdx

dMVNb

dxbGdx

dNVdxbGMa

255

'332

22'

33

2

11

2'

11'1155

'332

'332

'332

2'33

2'

112

2'1155

53

'33

'3353

'332

'3353

51

11

'11'

1151

'11

2'

1151

2

2

0

ρ

ω

ω

ωω

ω

ρ

ω

ω

Equation 5.5–6


213

5.6 Hydromechanical Moments for Yaw

The hydromechanical moments for yaw are found by integration over the ship length of thetwo-dimensional contributions of sway into the yaw moment:

∫ ⋅=L

bhh dxXX '66 with: bhh xXX ⋅+= '

2'

6

Equation 5.6–1

According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalforce on a yawing cross section in still water is defined by sway contributions:

*4

'24'

24*

4'

24

*2

'22'

22*

2'

22'

2

hbb

hb

hbb

hbh

xdx

dMVNxM

xdx

dMVNxMX

ζζ

ζζ

&&&

&&&

⋅⋅

⋅−+⋅⋅+

⋅⋅

⋅−+⋅⋅=

Equation 5.6–2


*4

'24'

24*

4

'24

2'

24

*2

'22'

22*

2

'22

2'

22'

2

hbb

hbbe

hbb

hbbe

h

xdx

dMVNx

dxdNV

M

xdx

dMVNx

dxdNV

MX

ζζω

ζζω

&&&

&&&

⋅⋅

⋅−+⋅⋅

⋅++

⋅⋅

⋅−+⋅⋅

⋅+=

Equation 5.6–3


214

This results into the following coupled yaw equation:

( )( ) 6666666

646464

6262626

wzz

zx

hzxzz

XcbaI

cbaI

ycybyaXII

=⋅+⋅+⋅+++⋅+⋅+⋅+−+⋅+⋅+⋅+=−⋅−⋅

ψψψφφφ

φψ

&&&

&&&

&&&&&&&

Equation 5.6–4

with:

0

2

0

0

66

'22

2

2

'22

2'

22'2266

2'

222

'222

'22'

2222'

2266

64

'22'

22

'24'

2464

'22

2

'24

2

'22

'2464

62

'22'

2262

'22

2'

2262

=

⋅⋅⋅+

⋅⋅⋅⋅−⋅⋅

⋅−+=

⋅⋅⋅+⋅⋅⋅+

⋅⋅

⋅−⋅+⋅⋅+=

=

⋅⋅

⋅−⋅+⋅⋅

⋅−+=

⋅⋅⋅⋅+⋅⋅⋅+

⋅⋅⋅+⋅⋅+=

=

⋅⋅

⋅−+=

⋅⋅⋅+⋅⋅+=

∫

∫∫

∫ ∫

∫∫

∫∫

∫∫

∫∫

∫

∫∫

c

dxxdx

dNV

dxxMVdxxdx

dMVNb

dxxdx

dNVdxxN

V

dxxdx

dMVN

VdxxMa

c

dxxdx

dMVNOGdxx

dxdM

VNb

dxxdx

dNOG

Vdxx

dxdNV

dxxMOGdxxMa

c

dxxdx

dMVNb

dxxdx

dNVdxxMa

Lbb

be

Lbb

Lbb

b

L Lbb

bebb

e

Lbb

beLbb

Lbb

bLbb

b

Lbb

beLbb

be

L

bb

L

bb

Lbb

b

Lbb

beLbb

ω

ωω

ω

ωω

ω

Equation 5.6–5



215

After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled yaw equation become:

0

0

0

66

'222

22'

2266

'222

'222

2'

2222'

2266

64

'22

'22

'24

'2464

'222

'242

'22

'2464

62

'22

'2262

'222

'2262

=

⋅⋅−

+⋅⋅+=

⋅⋅⋅−

+

⋅⋅+⋅⋅⋅+⋅⋅+=

=

⋅⋅⋅+⋅⋅⋅+

⋅⋅+⋅⋅+=

⋅⋅⋅−

+⋅⋅−

+

⋅⋅⋅+⋅⋅+=

=

⋅⋅+⋅⋅+=

⋅⋅−

+⋅⋅+=

∫∫

∫

∫∫∫

∫∫

∫∫

∫∫

∫∫

∫∫

∫∫

c

dxNV

dxxNb

dxxNV

dxMV

dxxNV

dxxMa

c

dxMOGVdxxNOG

dxMVdxxNb

dxNOGV

dxNV

dxxMOGdxxMa

c

dxMVdxxNb

dxNV

dxxMa

Lb

eLbb

Lbb

e

Lb

eLbb

eLbb

Lb

Lbb

Lb

Lbb

Lb

eLb

e

L

bb

L

bb

Lb

Lbb

Lb

eLbb

ω

ω

ωω

ωω

ω

Equation 5.6–6


216


217

6 Exciting Wave Loads

6.1 Wave Potential

The first order wave potential in a fluid - with any arbitrary water depth h - is given by:( )[ ][ ] ( )µµωζ

ωsincossin

coshcosh

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅−

=Φ bbeab

w ykxkthk

zhkg

Equation 6.1–1

in an axes system with the centre of gravity in the waterline.The velocities and accelerations in the direction j of the water particles have to be defined.The local relative orbital velocities of the water particles in a certain direction follow from thederivative in that direction of the wave potential. The orbital accelerations of the waterparticles can be obtained from these velocities by:

''wjwj Dt

D ζζ &&& = with:

∂∂

⋅−∂∂

=bx

VtDt

D for: 4,3,2,1=j

Equation 6.1–2

With this, the relative velocities and accelerations in the different directions can be found:

• Surge direction:

( )[ ][ ] ( )

( )[ ][ ] ( )µµωζµζ

µµωζω

µ

ζ

sincossincosh

coshcos

sincoscoscosh

coshcos

'1

'1

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅⋅−=

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅⋅+

=

∂Φ∂

=

bbeab

w

bbeab

b

ww

ykxkthk

zhkgk

ykxkthk

zhkgk

x

&&

&

Equation 6.1–3

• Sway direction:

( )[ ][ ] ( )

( )[ ][ ] ( )µµωζµζ

µµωζω

µ

ζ

sincossincosh

coshsin

sincoscoscosh

coshsin

'2

'2

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅⋅−=

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅⋅+

=

∂Φ∂

=

bbeab

w

bbeab

b

ww

ykxkthk

zhkgk

ykxkthk

zhkgk

y

&&

&

Equation 6.1–4

• Heave direction:


218

( )[ ][ ] ( )

( )[ ][ ] ( )µµωζζ

µµωζω

ζ

sincoscoscosh

sinh

sincossincosh

sinh

'3

'3

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅−=

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅−

=

∂Φ∂

=

bbeab

w

bbeab

b

ww

ykxkthkzhk

gk

ykxkthkzhkgk

z

&&

&

Equation 6.1–5

• Roll direction:

0

0

'4

'3

'2'

4

=

=∂

∂−

∂∂

=

w

b

w

b

ww yz

ζ

ζζζ

&&

&&&

Equation 6.1–6

These zero solutions are obvious, because the potential fluid is free of rotation.

The pressure in the fluid follows from the linearised equation of Bernoulli:( )[ ][ ] ( )

bb

bb

bb

bbeab

b

dzzp

dyyp

dxxp

p

ykxkthk

zhkgzgp

⋅∂∂

+⋅∂∂

+⋅∂∂

+=

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅⋅⋅+⋅⋅−=

0

sincoscoscosh

cosh µµωζρρ

Equation 6.1–7

with the following expressions for the pressure gradients:( )[ ][ ] ( )

( )[ ][ ] ( )

( )[ ][ ] ( )µµωζρρ

µµωζµρ

µµωζµρ

sincoscoscosh

cosh

sincossincosh

coshsin

sincossincosh

coshcos

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅⋅+⋅−=∂∂

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅⋅⋅+=∂∂

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅⋅⋅+=∂∂

bbeab

b

bbeab

b

bbeab

b

ykxkthk

zhkkgg

zp

ykxkthk

zhkkg

yp

ykxkthk

zhkkgxp

Equation 6.1–8

These pressure gradients can be expressed in the orbital accelerations too:

( )'3

'2

'1

wb

wb

wb

gzp

yp

xp

ζρ

ζρ

ζρ

&&

&&

&&

+⋅−=∂∂

⋅+=∂∂

⋅−=∂∂

Equation 6.1–9


219

6.2 Classical Approach

First the classical approach to obtain the wave loads - according to the relative motionprinciple - is given here.

6.2.1 Exciting Wave Forces for Surge

The exciting wave forces for surge on a ship are found by integration over the ship length ofthe two-dimensional values:

∫ ⋅=L

bww dxXX '11

Equation 6.2–1

According to the ''Ordinary Strip Theory'' the exciting wave forces for surge on a restrainedcross section of a ship in waves are defined by:

'

1*

1

'11'

11*

1'

11

'1

*1

'11

*1

'11

'1

FKwb

w

FKwww

Xdx

dMVNM

XNMDtD

X

+⋅

⋅−+⋅=

+⋅+⋅=

ζζ

ζζ

&&&

&&

Equation 6.2–2

According to the ''Modified Strip Theory'' these forces become:

'1

*1

'11'

11*

1

'11

2'

11

'1

*1

'11

'11

'1

FKwb

wbe

FKwe

w

Xdx

dMVN

dxdNV

M

XNi

MDtD

X

+⋅

⋅−+⋅

⋅+=

+

⋅

⋅+=

ζζω

ζω

&&&

&

Equation 6.2–3

Figure 6.2–1: Wave pressure distribution on a cross section for surge

The Froude-Krilov force in the surge direction - so the longitudinal force due to the pressurein the undisturbed fluid, see Figure 6.2–1 - is given by:


220

∫ ∫

∫ ∫

−

+

−

−

+

−

⋅⋅⋅=

⋅⋅∂∂

−=

ζ

ζ

ζρT

y

ybbw

T

y

ybb

bFK

b

b

b

b

dzdy

dzdyxp

X

'1

1

&&

Equation 6.2–4

After neglecting the second order terms, the Froude-Krilov force can be written as:( ) ( )µωζµρ cossincos1 ⋅⋅−⋅⋅⋅⋅⋅−⋅⋅= beachFK xktgkAX

with:( ) ( )[ ]

[ ]∫−

⋅⋅⋅+⋅

⋅⋅⋅−

⋅⋅−⋅=

0

coshcosh

sinsinsin

2T

bbb

b

bch dzy

hkzhk

ykyk

Aµµ

Equation 6.2–5

When expanding the Froude-Krilov force in deep water with wy⋅⋅>> πλ 2 and T⋅⋅>> πλ 2in series, it is found:

( ) ( )µωζµρ cossincos...21 2

1 ⋅⋅−⋅⋅⋅⋅⋅−⋅

+⋅⋅+⋅+⋅= beayyFK xktgkIkSkAX

with:

∫−

⋅⋅=0

2T

bb dzyA ∫−

⋅⋅⋅=0

2T

bbby dzzyS ∫−

⋅⋅⋅=0

22T

bbby dzzyI

Equation 6.2–6

The acceleration term agk ζµ ⋅⋅⋅ cos in here is the amplitude of the longitudinal component

of the relative orbital acceleration in deep water at 0=bz .The dominating first term in this series consists of a mass and this acceleration. The massterm A⋅ρ is used to obtain from the total Froude-Krilov force an equivalent longitudinalcomponent of the orbital acceleration of the water particles:

*11 wFK AX ζρ &&⋅⋅=

Equation 6.2–7

This holds that the equivalent longitudinal components of the orbital acceleration and velocityare equal to the values at 0=bz in a wave with reduced amplitude *

1aζ :

( )

( )µωζω

µζ

µωζµζ

coscoscos

cossincos

*1

*1

*1

*1

⋅⋅−⋅⋅⋅⋅⋅+

=

⋅⋅−⋅⋅⋅⋅⋅−=

beaw

beaw

xktgk

xktgk

&

&&

with:

ach

a AA ζζ ⋅=*

1

Equation 6.2–8

This equivalent acceleration and velocity will be used in the diffraction part of the wave forcefor surge.

From the foregoing follows the total wave loads for surge:


221

∫

∫

∫∫

⋅+

⋅⋅

⋅−⋅+

⋅⋅⋅⋅

+⋅⋅+=

LbFK

bw

L be

Lbw

beLbww

dxX

dxdx

dMVN

dxdx

dNVdxMX

'1

*1

'11'

11

*1

'11*

1'

111

ζωω

ζωω

ζ

&

&&&&

Equation 6.2–9


6.2.2 Exciting Wave Forces for Sway

The exciting wave forces for sway on a ship are found by integration over the ship length ofthe two-dimensional values:

∫ ⋅=L

bww dxXX '22

Equation 6.2–10

According to the ''Ordinary Strip Theory'' the exciting wave forces for sway on a restrainedcross section of a ship in waves are defined by:

'

2*

2

'22'

22*

2'

22

'2

*2

'22

*2

'22

'2

FKwb

w

FKwww

Xdx

dMVNM

XNMDtD

X

+⋅

⋅−+⋅=

+⋅+⋅=

ζζ

ζζ

&&&

&&

Equation 6.2–11


'2

*2

'22'

22*

2

'22

2

'22

'2

*2

'22

'22

'2

FKwb

wbe

FKwe

w

Xdx

dMVN

dxdNV

M

XNi

MDtD

X

+⋅

⋅−+⋅

⋅+=

+

⋅

⋅+=

ζζω

ζω

&&&

&

Equation 6.2–12


222

Figure 6.2–2: Wave pressure distribution on a cross section for sway

The Froude-Krilov force in the sway direction - so the lateral force due to the pressure in theundisturbed fluid - is given by:

∫ ∫

∫ ∫

−

+

−

−

+

−

⋅⋅⋅=

⋅⋅∂∂

−=

ζ

ζ

ζρT

y

ybbw

T

y

ybb

bFK

b

b

b

b

dzdy

dzdyyp

X

'2

2

&&

Equation 6.2–13

After neglecting the second order terms, the Froude-Krilov force can be written as:( ) ( )µωζµρ cossinsin2 ⋅⋅−⋅⋅⋅⋅⋅−⋅⋅= beachFK xktgkAX

with:( ) ( )[ ]

[ ]∫−

⋅⋅⋅+⋅

⋅⋅⋅−

⋅⋅−⋅=

0

coshcosh

sinsinsin

2T

bbb

b

bch dzy

hkzhk

ykyk

Aµµ

Equation 6.2–14


( ) ( )µωζµρ cossinsin...21 2

2 ⋅⋅−⋅⋅⋅⋅⋅−⋅

+⋅⋅+⋅+⋅= beayyFK xktgkIkSkAX

with:

∫−

⋅⋅=0

2T

bb dzyA ∫−

⋅⋅⋅=0

2T

bbby dzzyS ∫−

⋅⋅⋅=0

22T

bbby dzzyI

Equation 6.2–15

The acceleration term agk ζµ⋅⋅⋅ sin in here is the amplitude of the lateral component of the

relative orbital acceleration in deep water at 0=bz .The dominating first term in this series consists of a mass and this acceleration.This mass term A⋅ρ is used to obtain from the total Froude-Krilov force an equivalent lateralcomponent of the orbital acceleration of the water particles:

*22 wFK AX ζρ &&⋅⋅=


223

This holds that the equivalent lateral components of the orbital acceleration and velocity areequal to the values at 0=bz in a wave with reduced amplitude *

2aζ :

( )

( )µωζω

µζ

µωζµζ

coscossin

cossinsin

*2

*1

*2

*2

⋅⋅−⋅⋅⋅⋅⋅+

=

⋅⋅−⋅⋅⋅⋅⋅−=

beaw

beaw

xktgk

xktgk

&

&&

with:

ach

a AA ζζ ⋅=*

2

Equation 6.2–16

This equivalent acceleration and velocity will be used in the diffraction part of the wave forcefor sway.

From the foregoing follows the total wave loads for sway:

∫

∫

∫∫

⋅+

⋅⋅

⋅−⋅+

⋅⋅⋅⋅

+⋅⋅+=

LbFK

bw

L be

Lbw

beLbww

dxX

dxdx

dMVN

dxdx

dNVdxMX

'2

*2

'22'

22

*2

'22*

2'

222

ζωω

ζωω

ζ

&

&&&&

Equation 6.2–17


6.2.3 Exciting Wave Forces for Heave

The exciting wave forces for heave on a ship are found by integration over the ship length ofthe two-dimensional values:

∫ ⋅=L

bww dxXX '33

Equation 6.2–18

According to the ''Ordinary Strip Theory'' the exciting wave forces for heave on a restrainedcross section of a ship in waves are defined by:

'

3*

3

'33'

33*

3'

33

'3

*3

'33

*3

'33

'3

FKwb

w

FKwww

Xdx

dMVNM

XNMDtD

X

+⋅

⋅−+⋅=

+⋅+⋅=

ζζ

ζζ

&&&

&&

Equation 6.2–19



224

'3

*3

'33'

22*

3

'33

2

'33

'3

*3

'33

'33

'3

FKwb

wbe

FKwe

w

Xdx

dMVN

dxdNV

M

XNi

MDtD

X

+⋅

⋅−+⋅

⋅+=

+

⋅

⋅+=

ζζω

ζω

&&&

&

Equation 6.2–20

Figure 6.2–3: Wave pressure distribution on a cross section for heave

The Froude-Krilov force in the heave direction - so the vertical force due to the pressure in theundisturbed fluid - is given by:

( )∫ ∫

∫ ∫

−

+

−

−

+

−

⋅⋅+⋅=

⋅⋅∂∂

−=

ζ

ζ

ζρT

y

y

bbw

T

y

ybb

bFK

b

b

b

b

dzdyg

dzdyzp

X

'3

3

&&

Equation 6.2–21

After neglecting the second order terms, the Froude-Krilov force can be written as:( ) ( ) ( )µωζ

µµρ coscos

sinsinsin2

3 ⋅⋅−⋅⋅⋅⋅−⋅

+

⋅⋅−⋅⋅−

⋅⋅−

⋅= beashb

bwFK xktgkA

ykyk

ky

X

with:( ) ( )[ ]

[ ]∫−

⋅⋅⋅+⋅

⋅⋅⋅−

⋅⋅−⋅=

0

coshsinh

sinsinsin

2T

bbb

b

bsh dzy

hkzhk

ykyk

Aµµ

Equation 6.2–22


( ) ( )µωζρ coscos...212 2

3 ⋅⋅−⋅⋅⋅⋅−⋅

+⋅⋅+⋅++

⋅−⋅= beayy

wFK xktgkIkSkA

kyX

with:


225

∫−

⋅⋅=0

2T

bb dzyA ∫−

⋅⋅⋅=0

2T

bbby dzzyS ∫−

⋅⋅⋅=0

22T

bbby dzzyI

Equation 6.2–23

??????*

3T can be considered as the draft at which the pressure in the vertical direction is equal to theaverage vertical pressure on the cross section in the fluid and can be obtained by.

???*3 =T

This holds that the equivalent vertical components of the orbital acceleration and velocity areequal to the values at *

3Tzb −= :???When expanding the Froude-Krilov force in shallow water with 0→⋅ hk and in long waveswith ??? in series, it is found:

???3 =Cwith:???So in shallow water, *

3T can be obtained by.

???*3 =T

This holds that the equivalent vertical components of the orbital acceleration and velocity areequal to the values at *

3Tzb −= :???It may be noted that this shallow water definition for *

3T is valid in deep water too, because:???These equivalent accelerations and velocities will be used to determine the diffraction part ofthe wave forces for heave.

From the foregoing follows the total wave loads for heave:

∫

∫

∫∫

⋅+

⋅⋅

⋅−⋅+

⋅⋅⋅⋅

+⋅⋅+=

LbFK

bw

L be

Lbw

beLbww

dxX

dxdx

dMVN

dxdx

dNVdxMX

'3

*3

'33'

33

*3

'33*

3'

333

ζωω

ζωω

ζ

&

&&&&

Equation 6.2–24


6.2.4 Exciting Wave Moments for Roll

The exciting wave moments for roll on a ship are found by integration over the ship length oftwo-dimensional values:


226

∫ ⋅=L

bww dxXX '44

Equation 6.2–25

According to the ''Ordinary Strip Theory'' the exciting wave moments for roll on a restrainedcross section of a ship in waves are defined by:

'

2'

4*

2

'42'

42*

2'

42

'2

'4

*2

'42

*2

'42

'4

wFKwb

w

wFKwww

XOGXdx

dMVNM

XOGXNMDtD

X

⋅++⋅

⋅−+⋅=

⋅++⋅+⋅=

ζζ

ζζ

&&&

&&

Equation 6.2–26

According to the ''Modified Strip Theory'' these moments become:

'2

'4

*2

'42'

24*

2

'42

2

'42

'2

'4

*2

'42

'42

'4

wFKwb

wbe

wFKwe

w

XOGXdx

dMVN

dxdNV

M

XOGXNi

MDtD

X

⋅++⋅

⋅−+⋅

⋅+=

⋅++

⋅

⋅+=

ζζω

ζω

&&&

&

Equation 6.2–27

Figure 6.2–4: Wave pressure distribution on a cross section for roll

The Froude-Krilov moment in the roll direction - so the roll moment due to the pressure in theundisturbed fluid - is given by:

( ) ∫ ∫

∫ ∫

−

+

−

−

+

−

⋅⋅⋅++⋅−⋅=

⋅⋅

⋅

∂∂

+⋅∂∂

−−=

ζ

ζ

ζζρT

y

ybbbwbw

T

y

ybbb

bb

bFK

b

b

b

b

dzdyygz

dzdyyzp

zyp

X

'3

'2

4

&&&&

Equation 6.2–28

After neglecting the second order terms, the Froude-Krylov moment can be written as:

( ) ( )µωζµρ cossinsin24 ⋅⋅−⋅⋅⋅⋅⋅−⋅

++−⋅= beazsh

ychywFK xktgkI

k

S

k

CX


227

with:( ) ( )

( )( ) ( )[ ]

[ ]( ) ( )

( )( )[ ][ ]∫

∫

−

−

⋅⋅⋅+⋅

⋅⋅⋅−

⋅⋅−−⋅⋅−

⋅⋅−

⋅=

⋅⋅⋅⋅+⋅⋅

⋅⋅−⋅⋅−⋅=

⋅⋅⋅−

⋅⋅−−⋅⋅−

⋅⋅−

⋅=

03

2

0

32

coshcosh

sin

sincossin

sinsin

2

coshcosh

sinsinsin2

sin

sincossin

sinsin

2

Tbb

b

b

bb

b

zsh

T

bbbb

b

bych

wb

ww

w

yw

dzyhk

zhkyk

ykyk

yk

I

dzzyhk

zhkyk

ykS

yyk

ykyk

yk

C

µ

µµµ

µµ

µ

µµµ

Equation 6.2–29

For deep water, the cosine-hyperbolic expressions in here reduce to exponential expressions.

From the foregoing follows the total wave loads for roll:

2'

4

*2

'42'

42

*2

'42*

2'

424

wL

bFK

bwL be

Lbw

beLbww

XOGdxX

dxdx

dMVN

dxdx

dNVdxMX

⋅+⋅+

⋅⋅

⋅−⋅+

⋅⋅⋅⋅

+⋅⋅+=

∫

∫

∫∫

ζωω

ζωω

ζ

&

&&&&

Equation 6.2–30


6.2.5 Exciting Wave Moments for Pitch

The exciting wave moments for pitch are found by integration over the ship length of the two-dimensional contributions of surge and heave into the pitch moment:

∫ ⋅=L

bww dxXX '55

with:

bwww xXbGXX ⋅−⋅−= '3

'1

'5

Equation 6.2–31

In here, bG is the vertical distance of the centre of gravity of the ship G above the centroidb of the local submerged sectional area.From this follows the total wave loads for pitch:


228

∫

∫

∫∫

∫

∫

∫∫

⋅⋅−

⋅⋅⋅

⋅−⋅−

⋅⋅⋅⋅⋅

−+⋅⋅⋅−

⋅⋅−

⋅⋅⋅

⋅−⋅−

⋅⋅⋅⋅⋅

−+⋅⋅⋅−=

LbbFK

bwbL be

Lbwb

beLbwb

L

bFK

bw

L be

L

bwbeL

bww

dxxX

dxxdx

dMVN

dxxdx

dNVdxxM

dxbGX

dxbGdx

dMVN

dxbGdx

dNVdxbGMX

'3

*3

'33'

33

*3

'33*

3'

33

'1

*1

'11'

11

*1

'11*

1'

115

ζωω

ζωω

ζ

ζωω

ζωω

ζ

&

&&&&

&

&&&&

Equation 6.2–32


6.2.6 Exciting Wave Moments for Yaw

The exciting wave moments for yaw are found by integration over the ship length of the two-dimensional contributions of sway into the yaw moment:

∫ ⋅=L

bww dxXX '66

with:

bww xXX ⋅+= '2

'6

Equation 6.2–33

From this follows the total wave loads for yaw:

∫

∫

∫∫

⋅⋅+

⋅⋅⋅

⋅−⋅+

⋅⋅⋅⋅⋅

+⋅⋅⋅+=

LbbFK

bwb

L be

Lbwb

beLbwbw

dxxX

dxxdx

dMVN

dxxdx

dNVdxxMX

'2

*2

'22'

22

*2

'22*

2'

226

ζωω

ζωω

ζ

&

&&&&

Equation 6.2–34



229

6.3 Approximating 2-D Diffraction Approach

In the classic relative motion theory, the average (or equivalent) motions of the water particlesaround the cross section are calculated from the pressure distribution in the undisturbed waveson this cross section. An alternative approach - based on diffraction of waves - to determinethe equivalent accelerations and velocities of the water particles around the cross section, asgiven by Journee and van ‘t Veer [1995], is described now.

6.3.1 Hydromechanical Loads

Suppose an infinite long cylinder in the still water surface of a fluid. The cylinder is forced tocarry out a simple harmonic oscillation about its initial position with frequency of oscillationω and small amplitude of displacement jax :

( )txx jaj ⋅⋅= ωcos for: 4,3,2=j

Equation 6.3–1

The 2-D hydrodynamic loads 'hiX in the sway, heave and roll directions i , exercised by the

fluid on a cross section of the cylinder, can be obtained from the 2-D velocity potentials andthe linearised equations of Bernoulli. The velocity potentials have been obtained by using thework of Ursell [1949] and N -parameter conformal mapping. These hydrodynamic loads are:

( ) ( ) jj xijxij

jawlhi tBtA

gygX ΦΦ +⋅⋅++⋅⋅⋅

⋅⋅⋅⋅⋅= εωεω

πζ

ρ sincos2'

Equation 6.3–2

in where j is the mode of oscillation and i is the direction of the load. The phase lag jxΦε is

defined as the phase lag between the velocity potential of the fluid Φ and the forced motion

jx . The radiated damping waves have an amplitude jaζ and wly is half the breadth of the

cross section at the waterline. The potential coefficients ijA and ijB and the phase lags jxΦε ,

expressed in terms of conformal mapping coefficients, are given in a foregoing chapter.

These loads 'hiX can be expressed in terms of in-phase and out-phase components with the

harmonic oscillations:

( ) ( ) ( ) ( ) tQBPAtPBQA

g

x

aX

jijjijjijjij

ja

ja

ijhi

⋅⋅⋅−⋅+⋅⋅⋅+⋅

⋅

⋅⋅

⋅⋅

=

ωω

πζ

ωρ

sincos 0000

2

2'

with:


230

j

j

xwlja

jaj

xwlja

jaj

wl

wl

yg

xQ

yg

xP

a

ya

a

ya

a

Φ

Φ

⋅⋅⋅⋅+=

⋅⋅⋅⋅−=

=⋅=

=

==

εωπζ

εωπζ

cos

sin

2

4

2

4

2

2

0

2

0

44

42

33

24

22

Equation 6.3–3

The phase lag jxΦε between he velocity potentials and the forced motion is incorporated in the

coefficients jP0 and jQ0 and can be obtained by using:

+−

=Φj

jx Q

Pj

0

0arctanε

Equation 6.3–4

This equation will be used further on for obtaining wave load phases.Generally, these hydrodynamic loads are expressed in terms of potential mass and dampingcoefficients:

( ) ( )txNtxM

xNxMX

jaijjaij

jijjijhi

⋅⋅⋅⋅+⋅⋅⋅⋅=

⋅−⋅−=

ωωωω sincos2

' &&&

with:

444

342

233

324

222

20

20

20

20

2

2

2

2

wl

wl

wl

wl

wl

jj

ojijojijijij

jj

ojijojijijij

yb

yb

yb

yb

yb

QP

PBQAbN

QP

PBQAbM

⋅=

⋅=

⋅=

=

⋅=

⋅+

⋅−⋅⋅⋅=

+

⋅+⋅⋅⋅=

ωρ

ρ

Equation 6.3–5

Note that the phase lag information jxΦε is vanished here.

Tasai [1965] has used the following potential damping coupling coefficients in his formulationof the hydrodynamic loads for roll:

'

'44'

42wl

NN = and ''

22'

24 wlNN ⋅=


231

Equation 6.3–6

in which 'wl is the lever of the rolling moment.

Because '24

'42 NN = , one may write for the roll damping coefficient:

( ) ( )'

22

2'42

'22

2'24'

44 NN

NN

N ==

Equation 6.3–7

This relation - which has been confirmed by numerical calculations with SEAWAY - will beused further on for obtaining the wave loads for roll from those for sway.

6.3.2 Energy Considerations

The wave velocity, wavec , and the group velocity, groupc , of regular waves are defined by:

kcwave

ω= and [ ]hk

hkcc wave

group ⋅⋅⋅⋅

==2sinh

22

Equation 6.3–8

Consider a cross section which is harmonic oscillating with a frequency Tπω ⋅= 2 and an

amplitude jax in the direction j in previously still water by an oscillatory force 'hjX in the

same direction j :

( )( )

( ) ( )tXtX

tXX

txx

hjhjahjhja

hjhjahj

jaj

⋅⋅⋅−⋅⋅⋅=

+⋅⋅=

⋅⋅=

ωεωε

εω

ω

sinsincoscos

cos

cos

''''

''' for: 4,3,2=j

Equation 6.3–9

The energy required for this oscillation should be equal to the energy radiated by the dampingwaves:

groupa

T

jjjj

T

jhj

cg

dtxxNT

dtxXT

⋅⋅⋅⋅⋅=

⋅⋅⋅⋅=⋅⋅⋅ ∫∫2

0

'

0

'

21

2

11

ζρ

&&&

or:

groupa

jajjhjjahja

cg

xNxX

⋅⋅⋅=

⋅⋅⋅=⋅⋅⋅⋅

2

22'''

21

sin21

ζρ

ωεω

Equation 6.3–10

From the first part of Equation 6.3–10 follows:

a

jajj

a

hjhja xN

X

ζω

ζε

⋅⋅=⋅ '

'' sin

Equation 6.3–11


232

From the second part of Equation 6.3–10 follows the amplitude ratio of the oscillatorymotions and the radiated waves:

'

21

jj

group

a

ja

N

cgx ⋅⋅⋅⋅=

ρωζ

Equation 6.3–12

Combining these last two equations provides for the out-phase part - so the damping part - ofthe oscillatory force:

'''

2sin

jjgroupa

hjhja NcgX

⋅⋅⋅⋅=⋅

ρζ

ε for: 4,3,2=j

In here, '' sin hjhjaX ε⋅ is the in-phase with the velocity part of the exciting force or moment.

6.3.3 Wave Loads

Consider now the opposite case: the cross section is restrained and is subject to regularincoming beam waves with amplitude aζ . Let wjx represent the equivalent (or average)oscillation of the water particles with respect to the restrained cross section. The resultingwave force, '

wjX , is caused by these motions, which will be in phase with its velocity(damping waves). Then the energy consumed by this oscillation is equal to the energysupplied by the incoming waves.

( )( )';

'

cos

cos

wjwjawj

wjwjawj

tXX

txx

εω

εω

+⋅⋅=

+⋅⋅= for: 4,3,2=j

Equation 6.3–13

in which 'wjε is the phase lag with respect to the wave surface elevation at the center of the

cross section.This leads for the amplitude of the exciting wave force to:

''

2 jjgroupa

wja NcgX

⋅⋅⋅⋅= ρζ

for: 4,3,2=j

Equation 6.3–14

which is in principle the same equation as the previous one for the out-phase part of theoscillatory force in still water.However, for the phase lag of the wave force, '

wjε , an approximation has to be found.


233

Figure 6.3–1 Vector diagrams of wave components for sway and heave

6.3.3.1 Heave Mode

The vertical wave force on a restrained cross section in waves is:( )

( ) ( )tXtX

tXX

wawwaw

waww

⋅⋅⋅−⋅⋅⋅=

+⋅⋅=

ωεωε

εω

sinsincoscos

cos'

3'

3'

3'

3

'3

'3

'3

Equation 6.3–15

of which the amplitude is equal to:'

33'

3 2 NcgX groupaaw ⋅⋅⋅⋅⋅= ρζ

Equation 6.3–16

For the phase lag of this wave force, '3wε , an approximation has to be found.

The phase lag of a radiated wave, '3wRε , at the intersection of the ship's hull with the waterline,

wlb yy = , is wlwR yk ⋅='3ε . The phase lag of the wave force, '

3wε , has been approximated bythis phase:

wlwRw yk ⋅== '3

'3 εε

Equation 6.3–17

Then, the in-phase and out-phase parts of the wave loads are:


234

'3

'33

'3

'3

'32

'3

'33

'3

'3

'31

'3

sin2

sin

cos2

cos

wgroupa

waww

wgroupa

wawwFK

Ncg

XX

Ncg

XXX

ερζ

ε

ερζ

ε

⋅

⋅⋅⋅⋅⋅−=

⋅−=

⋅

⋅⋅⋅⋅⋅+=

⋅+=+

Equation 6.3–18

from which the diffraction terms, '31wX and '

32wX follow.These diffraction terms can also be written as:

'3

'33

'32

'3

'33

'31

vNX

aMX

w

w

⋅=

⋅=

Equation 6.3–19

in which '3a and '

3v are the equivalent amplitudes of the acceleration and the velocity of thewater particles around the cross section.Herewith, the equivalent acceleration and velocity amplitudes of the water particles are:

'33

'32'

3

'33

'31'

3

NX

v

MX

a

w

w

=

=

Equation 6.3–20

6.3.3.2 Sway Mode

The horizontal wave force on a restrained cross section in beam waves is:( )

( ) ( )tXtX

tXX

wawwaw

waww

⋅⋅⋅−⋅⋅⋅=

+⋅⋅=

ωεωε

εω

sinsincoscos

cos'

2'

2'

2'

2

'2

'2

'2

Equation 6.3–21

of which the amplitude is equal to:'

22'

2 2 NcgX groupaaw ⋅⋅⋅⋅⋅= ρζ

Equation 6.3–22

For the phase lag of this wave force, '2wε , an approximation has to be found.

The phase lag of an incoming undisturbed wave, '2wIε , at the intersection of the ship's hull

with the waterline, wlb yy = , is:

πεεµ

µε

+=<

⋅⋅−='

2'

2

'2

: then0sin if

sin

wIwI

wlwI yk

Equation 6.3–23


235

In very short waves - so at high wave frequencies ∞→ω - the ship's hull behaves like a

vertical wall and all waves will be diffracted. Then, the phase lag of the wave force, '2wε , is

equal to:( ) '

2'

2 wIw εωε −=∞→

Equation 6.3–24

The acceleration and velocity amplitudes of the water particles in the undisturbed surface ofthe incoming waves are:

( )( )

ωµ

ω

µ

sin

sin'

2surface water still

'2

surface water still'

2

⋅⋅=

−=

⋅⋅−=

gkav

gka

Equation 6.3–25

In very long waves - so at low wave frequencies 0→ω - the wave force is dominated by theFroude-Krylov force and the amplitudes of the water particle motions do not change verymuch over the draft of the section. Apparently, the phase lag of the wave force, '

2wε , can beapproximated by:

( ) ( )

⋅⋅

⋅

⋅⋅−⋅+−=→

ωµ

µωεsin

sinarctan0

'22

'22

'2'

2 gkN

gkMXFKw

Equation 6.3–26

When plotted against ω , the two curves ( )0'2 →ωεw and ( )∞→ωε '

2w will intersect each

other. The phase lag of the wave force, '2wε , can now be approximated by the lowest of these

two values:( )

( ) ( ) ( )0 : then0 if '2

'2

'2

'2

'2

'2

→=∞→>→

∞→=

ωεεωεωε

ωεε

wwww

ww

Equation 6.3–27

Because ( )∞→ωε '2w goes to zero in the low frequency region and ( )0'

2 →ωεw can havevalues between 0 and π⋅2 , one simple precaution has to be taken:

( ) ( ) ( ) πωεωεπωε ⋅−→=∞→>→ 20 : then20 if '2

'2

'2 www

Equation 6.3–28

Now the in-phase and out-phase terms of the wave force in beam waves are:

'2

'22

'2

'2

'22

'2

'22

'2

'2

'21

'2

cos2

sin

sin2

sin

wgroupa

waww

wgroupa

wawwFK

Ncg

XX

Ncg

XXX

ερζ

ε

ερζ

ε

⋅

⋅⋅⋅⋅⋅+=

⋅+=

⋅

⋅⋅⋅⋅⋅−=

⋅−=+

Equation 6.3–29


236


22wX follow.These terms can also be written as:

'2

'22

'22

'2

'22

'21

vNX

aMX

w

w

⋅=

⋅=

Equation 6.3–30

in which '2a and '

2v are the equivalent amplitudes of the acceleration and the velocity of thewater particles around the cross section.Then - when using an approximation for the influence of the wave direction - the equivalentacceleration and velocity amplitudes of the water particles are:

µ

µ

sin

sin

'22

'22'

2

'22

'21'

2

⋅=

⋅=

NX

v

MX

a

w

w

Equation 6.3–31

6.3.3.3 Roll Mode

The fluid is free of rotation; so the wave moment for roll consists of sway contributions only.However, the equivalent amplitudes of the acceleration and the velocity of the water particleswill differ from those of sway.From a study on potential coefficients, the following relation between sway and roll dampingcoefficients has been found:

( ) ( )'

22

2'42

'22

2'24'

44 NN

NN

N ==

The horizontal wave moment on a restrained cross section in beam waves is:( )'4

'4

'4 cos waww tXX εω +⋅⋅=

Equation 6.3–32

of which the amplitude is equal to:

( )

'22

'24'

2

'22

'24'

22

'22

2'24

'44

'4

2

2

2

N

NX

N

NNcg

NN

cg

NcgX

aw

groupa

groupa

groupaaw

⋅=

⋅⋅⋅⋅⋅⋅=

⋅⋅⋅⋅⋅=

⋅⋅⋅⋅⋅=

ρζ

ρζ

ρζ

Equation 6.3–33

The in-phase and out-phase parts of the wave moment in beam waves are:


237

( )

'22

'24'

22'

42

'22

'24'

21'

2'

41'

4

N

NXX

N

NXXXX

ww

wFKwFK

⋅=

⋅+=+

Equation 6.3–34


42wX follow.These terms can also be written as:

'24

'24

'42

'24

'24

'41

vNX

aMX

w

w

⋅=

⋅=

Equation 6.3–35

in which '24a and '

24v are the equivalent amplitudes of the acceleration and the velocity of thewater particles around the cross section.Then - when using an approximation for the influence of the wave direction - the equivalentacceleration and velocity amplitudes of the water particles are:

µ

µ

sin

sin

'24

'42'

24

'24

'41'

24

⋅=

⋅=

NX

v

MX

a

w

w

Equation 6.3–36

6.3.3.4 Surge Mode

The equivalent acceleration and velocity amplitudes of the water particles around the crosssection for surge have been found from:

ω

µ'

1'1

'2'

1tan

av

aa

−=

=

Equation 6.3–37


238

6.4 Numerical Comparisons

Figure 6.4–1 and Figure 6.4–2 give a comparison between these sway, heave and roll waveloads on a crude oil carrier in oblique waves - obtained by the classic approach and the simplediffraction approach, respectively - with the 3-D zero speed ship motions program DELFRACof Pinkster; see Dimitrieva [1017].

Figure 6.4–1: Comparison of classic wave loads with DELFRAC data

Figure 6.4–2: Comparison of simple diffraction wave loads with DELFRAC data


239

7 Transfer Functions of Motions

After dividing the left and right hand terms by the wave amplitude aζ , two sets of six coupledequations of motion are available.

The 6 variables in the coupled equations for the vertical plane motions are:

θζθζ

ζζ

ζζ

εζθε

ζθ

εζ

εζ

εζ

εζ

sinandcos:Pitch

sinz

andcosz

:Heave

sinx

andcosx

:Surge

a

a

a

a

a

a

a

a

a

a

a

a

⋅⋅

⋅⋅

⋅⋅

zz

xx

The 6 variables in the coupled equations for the horizontal plane motions are:

ψζψζ

φζφζ

ζζ

εζψε

ζψ

εζφε

ζφ

εζ

εζ

sinandcos:Yaw

sinandcos:Roll

siny

andcosy

:Sway

a

a

a

a

a

a

a

a

a

a

a

a

⋅⋅

⋅⋅

⋅⋅ yy

These sets of motions have to be solved by a numerical method. A method that providescontinuous good results, given by de Zwaan [1977], has been used in the strip theory programSEAWAY.


240

7.1 Centre of Gravity Motions

From the solutions of these in and out of phase terms follow the transfer functions of themotions (or Reponse Amplitude Operators, RAO’s), which is the motion amplitude to waveamplitude ratio:

a

axζ a

ayζ a

azζ a

a

ζφ

a

a

ζθ

a

a

ζψ

The associated phase shifts of these motions relative to the wave elevation are:

ζεx ζεy ζεz φζε θζε ψζε

The transfer functions of the translations are non-dimensional. The transfer functions of therotations can be made non-dimensional by dividing the amplitude of the rotations by theamplitude of the wave slope ak ζ⋅ in lieu of the wave amplitude aζ :

a

axζ a

ayζ a

azζ a

a

k ζφ⋅ a

a

k ζθ⋅ a

a

k ζψ⋅

Some examples of calculated transfer functions of a crude oil carrier and a containership aregiven in Figure 7.1–1, Figure 7.1–2 and Figure 7.1–3.

-720

-630

-540

-450

-360

-270

-180

-90

0

0 0.25 0.50 0.75 1.00

µ = 900

µ = 1800

Wave Frequency (rad/s )

Pha

se ε

z ζ (d

eg)

0

0 .5

1 .0

1 .5

0 0 .25 0 .5 0 0 .7 5 1 .0 0

Cru de O il Carr ierV = 0 k nHe a ve

µ = 900

µ = 1 8 00

W ave Fre qu e ncy (ra d/ s)

RA

O H

eav

e (

-)

0

0 .5

1 .0

1 .5

0 0 .2 5 0. 50 0 .7 5 1. 00

Cru de O il Ca rr ie r

V = 0 knP it ch

µ = 900

µ = 18 00

W a ve Freq u en cy (rad /s )

RA

O P

itch

(-)

-720

-630

-540

-450

-360

-270

-180

-90

0

0 0.25 0.50 0.75 1.00

µ = 900

µ = 1800

Wave F requency (rad/s )

Phas

e ε θ

ζ

Figure 7.1–1: Heave and Pitch Motions of a Crude Oil Carrier, V = 0 kn


241

0

0 .5

1 .0

1 .5

0 0 .25 0 .5 0 0 .7 5 1 .0 0

Cru de O il Carr ierV = 1 6 knHe a ve

µ = 900

µ = 18 00

W ave Fre qu e ncy (ra d/ s)

RA

O H

eav

e (

-)

-720

-630

-540

-450

-360

-270

-180

-90

0

0 0.25 0.50 0.75 1.00

µ = 900

µ = 1800

Wave Frequency (rad/s )

Pha

se ε

z ζ (d

eg)

0

0 .5

1 .0

1 .5

0 0 .2 5 0. 50 0 .7 5 1. 00

Cru de O il Ca rr ie r

V = 1 6 knP it ch

µ = 900

µ = 1 800

W a ve Freq u en cy (rad /s )

RA

O P

itch

(-)

-720

-630

-540

-450

-360

-270

-180

-90

0

0 0.25 0.50 0.75 1.00

µ = 900

µ = 1800

Wave F requency (rad/s )

Phas

e ε θ

ζ

Figure 7.1–2: Heave and Pitch Motions of a Crude Oil Carrier, V = 16 kn

0

0.25

0.50

0.75

1.00

1.25

1.50

0 0.2 0.4 0.6 0.8 1.0

RAO of pitch Head waves

ContainershipL

pp = 175 metre

V = 0 knots

V = 10 knots

V = 20 knots

wave frequency (rad/s)

Non

-dim

. RA

O o

f pitc

h (-

)

0

5

10

15

0 0.2 0.4 0.6 0.8 1.0

RAO of roll Beam waves

ContainershipL

pp = 175 metre

V = 20 knots

V = 10 knots

V = 0 knots

wave frequency (rad/s)

Non

-dim

. RAO

of r

oll (

-)

Figure 7.1–3: Roll and Pitch Motions of a Containership

Notice the different speed effects on the motions in these figures.

For motions with a spring term in the equation of motion, three frequency regions can bedistinguished:• the low frequency region ( ( )amc +<<2ω ), with motions dominated by the restoring

spring term,


242

• the natural frequency region ( ( )amc +≈2ω ), with motions dominated by the dampingterm and

• the high frequency region ( ac>>2ω ), with motions dominated by the mass term.

An example for heave motions is given in Figure 7.1–4.

Figure 7.1–4: Frequency Regions and Motional Behaviour


243

7.2 Local Absolute Displacements

Consider a point ( )bbb zyxP ,, on the ship in the ( )bbb zyxG ,, ship-bound axes system. The

harmonic displacements in the ship-bound bx , by and bz directions - or in the earth bound x ,

y and z directions - in any point ( )bbb zyxP ,, on the ship can be obtained from the six centreof gravity motions as presented below.

The harmonic longitudinal displacement is given by:

( )ζεωθψ

PxePa

bbP

tx

zyxx

+⋅⋅=

⋅+⋅−=

cos

The harmonic lateral displacement is given by:

( )ζεωφψ

PyePa

bbP

ty

zxyy

+⋅⋅=

⋅−⋅+=

cos

The harmonic vertical displacement is given by:

( )ζεωφθ

PzePa

bbP

tz

yxzz

+⋅⋅=

⋅−⋅−=

cos

With the six motions of the centre of gravity, the harmonic motions of any point ( )bbb zyxP ,,

on the ship in the ship-bound bx , by and bz directions - or in the earth bound system in x , yand z directions - can be calculated by using the previous equations.


244

7.3 Local Absolute Velocities

The harmonic velocities in the ship-bound bx , by and bz directions - or in the earth bound x ,

y and z directions - in any point ( )bbb zyxP ,, on the ship can be obtained by taking thederivative of the three harmonic displacements.

The harmonic longitudinal velocity is given by:

( )( )

( )ζ

ζ

ζ

εω

πεωω

εωωθψ

P

P

P

xePa

xePae

xePae

bbP

tx

tx

tx

zyxx

&&

&&&&

+⋅⋅=

−+⋅⋅⋅=

+⋅⋅⋅−=

⋅+⋅−=

cos

2cos

sin

The harmonic lateral velocity is given by:

( )( )

( )ζ

ζ

ζ

εω

πεωω

εωωφψ

P

P

P

yePa

yePae

yePae

bbP

ty

ty

ty

zxyy

&&

&&&&

+⋅⋅=

−+⋅⋅⋅=

+⋅⋅⋅−=

⋅−⋅+=

cos

2cos

sin

The harmonic vertical velocity is given by:

( )( )

( )ζ

ζ

ζ

εω

πεωω

εωωφθ

P

P

P

zePa

zePae

zePae

bbP

tz

tz

tz

yxzz

&&

&&&&

+⋅⋅=

−+⋅⋅⋅=

+⋅⋅⋅−=

⋅−⋅−=

cos

2cos

sin


245

7.4 Local Absolute Accelerations

In the earth-bound axes system, the harmonic accelerations on the ship are obtained by takingthe second derivative of the displacements. In the ship-bound axes system, a component of theacceleration of gravity has to be added to the accelerations in the horizontal plane direction.

7.4.1 Accelerations in the Earth-Bound Axes System

In the earth-bound axes system, ( )zyxO ,, , the harmonic accelerations in a point ( )bbb zyxP ,,on the ship in the x , y and z directions can be obtained by taking the second derivative ofthe three harmonic displacements.

Thus:

• Longitudinal acceleration:

( )( )

( )ζ

ζ

ζ

εω

πεωω

εωω

θψ

P

P

P

xePa

xePae

xePae

bbP

tx

tx

tx

zyxx

&&&&

&&&&&&&&

+⋅⋅=

−+⋅⋅⋅=

+⋅⋅⋅−=

⋅+⋅−=

cos

cos

cos2

2

• Lateral acceleration:

( )( )

( )ζ

ζ

ζ

εω

πεωω

εωω

φψ

P

P

P

yePa

yePae

yePae

bbP

ty

ty

ty

zxyy

&&&&

&&&&&&&&

+⋅⋅=

−+⋅⋅⋅=

+⋅⋅⋅−=

⋅−⋅+=

cos

cos

cos2

• Vertical acceleration:

( )( )

( )ζ

ζ

ζ

εω

πεωω

εωω

φθ

P

P

P

zePa

zePae

zePae

bbP

tz

tz

tz

yxzz

&&&&

&&&&&&&&

+⋅⋅=

−+⋅⋅⋅=

+⋅⋅⋅−=

⋅−⋅−=

cos

cos

cos2

2

7.4.2 Accelerations in the Ship-Bound Axes System

In the ship-bound axes system, ( )bbb zyxG ,, , a component of the acceleration of gravity ghas to be added to the accelerations in the longitudinal and lateral direction in the earth-boundaxes system. The vertical acceleration does not change.These accelerations are the accelerations that will be ''felt'' by for instance the cargo or sea-fastenings on the ship.

Thus:


246

• Longitudinal acceleration:

( ) ( )( )ζ

θζζ

εω

εωθεωω

θθψ

P

P

xePa

eaxePae

bbP

tx

tgtx

gzyxx

&&&&

&&&&&&&&

+⋅⋅=

+⋅⋅⋅−+⋅⋅⋅−=

⋅−⋅+⋅−=

cos

coscos2

• Lateral acceleration:

( ) ( )( )ζ

φζζ

εω

εωφεωω

φφψ

P

P

yePa

eayePae

bbP

ty

tgty

gzxyy

&&&&

&&&&&&&&

+⋅⋅=

+⋅⋅⋅++⋅⋅⋅−=

⋅+⋅−⋅+=

cos

coscos2

• Vertical acceleration:

( )( )ζ

ζ

εω

εωω

φθ

P

P

zePa

zePae

bbP

tz

tz

yxzz

&&&&

&&&&&&&&

+⋅⋅=

+⋅⋅⋅−=

⋅−⋅−=

cos

cos2


247

7.5 Local Vertical Relative Displacements

The harmonic vertical relative displacement with respect to the wave surface of a point( )bbb zyxP ,, connected to the ship can be obtained too:

( )ζεωφθζ

PsePa

bbPP

ts

yxzs

+⋅⋅=

⋅+⋅+−=

cos

with:( )µµωζζ sincoscos ⋅⋅−⋅⋅−⋅⋅= bbeaP ykxkt

It may be noted that the sign of the relative motion is chosen here in such a way that a positiverelative displacement implies a decrease of the freeboard.An oscillating ship will produce waves and these phenomena will change the relative motion.A dynamical swell up should be taken into account, which is not included in the previousformulation.Notice the different behaviours of the absolute and relative vertical motions, as given inFigure 7.5–1.

0

1

2

3

4

5

0 0.5 1 .0 1 .5

R A O te nd sto 0 .0


C o nta ine rsh ipH e ad w ave s

V = 20 kno ts

V = 10 kno ts

V = 0 kn ots

w a ve fre qu en cy (ra d/s )

RA

O o

f ve

rtic

al a

bso

lute

bo

w m

otio

ns

(m/m

)

0

1

2

3

4

5

0 0 .5 1.0 1 .5


R AO ten d sto 0.0

C on tai ne rsh ip

H ea d w ave s

V = 0 kn ots

V = 1 0 kn ots

V = 2 0 kn o ts

w a ve fre q ue ncy (r ad /s )

RA

O o

f ve

rtic

al r

ela

tive

bo

w m

otio

ns

(m/m

)

Figure 7.5–1: Absolute and Relative Vertical Motions at the Bow


248

7.6 Local Vertical Relative Velocities

The harmonic vertical relative velocity with respect to the wave surface of a certain point( )bbb zyxP ,, , connected to the ship, can be obtained by:

( )ζεωφθθζ

φθζ

PsePa

bbP

bbPP

ts

yVxz

yxzDtD

s

&&

&&&&

&

+⋅⋅=

⋅+⋅−⋅+−=

⋅+⋅+−=

cos

in which for the vertical velocity of the water surface itself:( )µµωζωζ sincossin ⋅⋅−⋅⋅−⋅⋅⋅−= bbeaP ykxkt&


249

8 Anti-Rolling Devices

Since the disappearance of sails on ocean-going ships, with their stabilising wind effect on therolling motions, naval architects have been concerned in reducing the rolling of ships amongwaves. With bilge keels they performed a first successful attack on the problem of rolling, butin several cases these bilge keels did not prove to be sufficient. Since 1880, numerous othermore or less successful ideas have been tested and used.

Four types of anti-rolling devices and its contribution to the equations of motion are describedhere:• bilge keels• passive free-surface tanks• active fin stabilisers• active rudder stabilisers.

The active fin and rudder stabilisers are not built into the program SEAWAY yet.


250

8.1 Bilge Keels

Bilge keels can deliver an important contribution to an increase the damping of the rollingmotions of ships. A reliable method to determine this contribution is given by Ikeda, Himenoand Tanaka [1978], as described before.Ikeda divides the two-dimensional quadratic bilge keel roll damping into a component due tothe normal force on the bilge keels and a component due to the pressure on the hull surface,created by the bilge keels.The normal force component of the bilge keel damping has been deduced from experimentalresults of oscillating flat plates. The drag coefficient DC depends on the period parameter orthe Keulegan-Carpenter number. Ikeda measured the quadratic two-dimensional drag bycarrying out free rolling experiments with an ellipsoid with and without bilge keels.Assuming a pressure distribution on the hull caused by the bilge keels, a quadratic two-dimensional roll damping can be defined. Ikeda carried out experiments to measure thepressure on the hull surface created by bilge keels. He found that the coefficient +

pC of the

pressure on the front face of the bilge keel does not depend on the period parameter, while thecoefficient −

pC of the pressure on the back face of the bilge keel and the length of thenegative pressure region depend on the period parameter. Ikeda defines an equivalent lengthof a constant negative pressure region 0S over the height of the bilge keels and a two-dimensional roll-damping component can be found.The total bilge keel damping has been obtained by integrating these two two-dimensional roll-damping components over the length of the bilge keels.Experiments of Ikeda showed that the effect of forward speed on the roll damping due to thebilge keels could be ignored.The equivalent linear total bilge keel damping has been obtained by linearising the result, ashas been shown in a separate Chapter.


251

8.2 Passive Free-Surface Tanks

The roll damping, caused by a passive free-surface tank, is essentially based on the existenceof a hydraulic jump or bore in the tank. Verhagen and van Wijngaarden [1965] give atheoretical approach to determine the counteracting moments by free-surface anti-rollingtanks. Van den Bosch and Vughts [1966] give extended quantitative information on thesemoments.

8.2.1 Theoretical Approach

When a tank that contains a fluid with a free surface is forced to carry out roll oscillations,resonance frequencies can be obtained with high wave amplitudes at lower water depths.Under these circumstances a hydraulic jump or bore is formed, which travels periodicallyback and forth between the walls of the tank. This hydraulic jump can be a strongly non-linearphenomenon. A theory, based on gas-dynamics for the shock wave in a gas flow under similarresonance circumstances, as given by Verhagen and van Wijngaarden [1965], has beenadapted and used to describe the motions of the fluid. For low and high frequencies and thefrequencies near to the natural frequency, different approaches have been used.Observe a rectangular tank with a length l and a breadth b , which has been filled until awater level h with a fluid with a mass density ρ . The distance of the tank bottom above thecentre of gravity of the vessel is s . Figure 8.2–1 shows a 2-D sketch of this tank with the axissystem and notations.

Figure 8.2–1: Axes System and Notations of an Oscillating Tank

The natural frequency of the surface waves in a harmonic rolling tank appears as the wavelength λ in the tank equals twice the breadth b , so: b⋅= 20λ .With the wave number and the dispersion relation:

λπ⋅

=2

k and [ ]hkgk ⋅⋅⋅= tanhω

it follows for the natural frequency of surface waves in the tank:


252

⋅

⋅⋅

=b

hb

g ππω tanh0

Verhagen and van Wijngaarden [1965] have investigated the shallow water wave loads in arolling rectangular container, with the centre of rotation at the bottom of the container. Theirexpressions for the internal wave loads are rewritten and modified to be useful for anyarbitrary vertical position of the centre of rotation by Journée [1997]. For low and highfrequencies and the frequencies close to the natural frequency, different approaches have beenused. A calculation routine has been made to connect these regions.

8.2.1.1 Low and High Frequencies

The harmonic roll motion of the tank is defined by:( )ta ⋅⋅= ωφφ sin

In the axis-system of Figure 8.2–1 and after linearisation, the vertical displacement of the tankbottom is described by:

φ⋅+= ysz

and the surface elevation of the fluid is described by:ζ++= hsz

Relative to the bottom of the tank, the linearised surface elevation of the fluid is described by:φζξ ⋅−+= yh

Using the shallow water theory, the continuity and momentum equations are:

0

0

=⋅+∂∂

⋅+∂∂

⋅+∂∂

=∂∂

⋅+∂∂

⋅+∂∂

φξ

ξξξ

gy

gyv

vtv

yv

yv

t

In these formulations, $v$ denotes the velocity of the fluid in the y -direction and the verticalpressure distribution is assumed to be hydrostatic. Therefore, the acceleration in the z -direction, introduced by the excitation, must be small with respect to the acceleration ofgravity g , so:

gba <<⋅⋅ 2ωφ

The boundary conditions for v have been determined by the velocity produced in thehorizontal direction by the excitation. Between the surface of the fluid and the bottom of thetank, the velocity of the fluid v varies between sv and [ ]hkvs ⋅cosh with a mean velocity:

( )hkvs ⋅ . However, in very shallow water v does not vary between the bottom and thesurface. When taking the value at the surface, it is required that:

( ) φ&⋅+−= hsv at: 2b

y ±=


253

For small values of aφ , the continuity equation and the momentum equation can be given in alinearised form:

0

0

=⋅+∂∂

⋅+∂∂

=∂∂

⋅+∂∂

φξ

ξξ

gy

gtv

yh

t

The solution of the surface elevation ξ in these equations, satisfying the boundary values forv , is:

( )

⋅

⋅⋅⋅

⋅

⋅⋅⋅⋅

⋅++⋅⋅

−= φωωπ

ωωπωπ

ωωξ

0

0

2

0

sin

2cos

1

byg

hsb

h

Now, the roll moment follows from the quasi-static moment of the mass of the frozen liquidhbl ⋅⋅⋅ρ and an integration of ξ over the breadth of the tank:

∫+

−

⋅⋅⋅⋅⋅+⋅

+⋅⋅⋅⋅⋅=

2

22

b

b

dyylgh

shblgM ξρφρφ

This delivers the roll moment amplitude for low and high frequencies at small water depths:

( )a

aa

ghs

blg

hshblgM

φωπ

ωωωπ

ωπωωρ

φρφ

⋅

⋅−

⋅⋅

⋅

⋅⋅⋅

⋅+

+⋅⋅⋅⋅

+⋅

+⋅⋅⋅⋅⋅=

2

0

0

3

02

3

2tanh21

2

For very low frequencies, so for the limit value 0→ω , this will result into the static moment:

a

bhshblgM φρφ ⋅

+

+⋅⋅⋅⋅⋅=

122

3

The phase lags between the roll moments and the roll motions have not been obtained here.However, they can be set to zero for low frequencies and to π− for high frequencies:

πε

ε

φ

φ

φ

φ

−=

=

M

M 0 for:

0

0

ωωωω

>><<

8.2.1.2 Natural Frequency Region

For frequencies near to the natural frequency 0ω , the expression for the surface elevation ofthe fluid ξ goes to infinity. Experiments showed the appearance of a hydraulic jump or a boreat these frequencies. Obviously, then the linearised equations are not valid anymore.Verhagen and van Wijngaarden [1965] solved the problem by using the approach in gasdynamics, when a column of gas has been oscillated at small amplitude, e.g. by a piston. At


254

frequencies near to the natural frequency at small water depths, they found a roll momentamplitude, defined by:

( )

⋅⋅−⋅⋅−⋅

⋅⋅⋅⋅

⋅⋅⋅⋅=

a

aa g

bb

hblgMφωωπφ

πρφ 32

13

2412

20

243

The phase shifs between the roll moment and the roll motion at small water depths are givenby:

απε

απε

φ

φ

φ

φ

−−=

+−=

2

2

M

M for:

0

0

ωωωω

>><<

with:

( )

( )( )

−⋅⋅⋅−⋅⋅−⋅⋅

−

⋅⋅−⋅⋅

⋅=

20

2

20

2

20

2

396arcsin

24arcsin2

ωωπφωωπ

φωωπα

bgb

gb

a

a

Because that the arguments of the square roots in the expression for φφεM have to be positive,

the limits for the frequency ω are at least:

2020

2424π

φωωπ

φω⋅

⋅⋅+<<

⋅⋅⋅

−b

gb

g aa

8.2.1.3 Comparison with Experimental Data

An example of the results of this theory with experimental data of an oscillating free-surfacetank by Verhagen and van Wijngaarden [1965] is given in Figure 8.2–2.

Figure 8.2–2: Comparison between Theoretical and Experimental Data


255

The roll moments have been calculated here for low and high frequencies and for frequenciesnear to the natural frequency of the tank. A calculation routine connects these three regions.

8.2.2 Experimental Approach

Van den Bosch and Vugts [1966] have described the physical behaviour of passive free-surface tanks, used as an anti-rolling device. Extended quantitative information on thecounteracting moments, caused by the water transfer in the tank, has been provided.With their symbols, the roll motions and the exciting moments of an oscillating rectangularfree-surface tank, are defined by:

( )( )ϕεω

ωϕϕ

ttat

a

tKK

t

+⋅⋅=⋅⋅=

cos

cos

and the dimensions of the rectangular free-surface tank are given by:l length of the tankb breadth of the tanks distance of tank bottom above rotation pointh water depth in the tank at rest

*ρ mass density of the fluid in the tank\endtabular

A non-dimensional frequency range is defined by:

60.100.0 <⋅<gbω

In this frequency range, van den Bosch and Vugts have presented extended experimental dataof:

3* blgKta

a ⋅⋅⋅=

ρµ and ϕεt

for:=aϕ 0.0333, 0.0667 and 0.1000 radians=bs -0.40, -0.20, 0.00 and +0.20=bh 0.02, 0.04, 0.06, 0.08 and 0.10

An example of a part of these experimental data has been shown for 40.0−=bs and1000.0=aϕ radians in Figure 8.2–3, taken from the report of van den Bosch and Vugts

[1966].


256

Figure 8.2–3: Experimental Data on Anti-Rolling Free-Surface Tanks

When using these experimental data, the external roll moment due to an, with a frequency ω ,oscillating free surface tank can be written as:

ϕϕϕ ϕϕϕ ⋅+⋅+⋅= 444 cbaKt &&&

with:

ϕϕ

ϕ

ϕ

ϕ

εϕ

ω

εϕ

ta

ta

ta

ta

Kc

K

b

a

cos

sin

0

4

4

4

⋅=

⋅=

=

It is obvious that for an anti-rolling free-surface tank, built into a ship, it holds:

aa ϕφ = and ωω =e

So it can be written:( )

( )ϕφξ

φζ

εεω

εωφφ

ttat

a

tKK

t

++⋅⋅=

+⋅⋅=

cos

cos

Then, an additional moment has to be added to the right-hand side of the equations of motionfor roll:

φφφ ⋅+⋅+⋅= tank44tank44tank44tank4 cbaX &&&

with:


257

ϕ

ϕ

εφ

ω

εφ

ta

ta

ta

ta

Kc

K

b

a

cos

sin

0

tank44

tank44

tank44

⋅=

⋅=

=

This holds that the anti-rolling coefficients tank44a , tank44b and tank44c have to be subtracted

from the coefficients 44a , 44b and 44c in the left-hand side of the equations of motion for roll.

8.2.3 Effect of Free-Surface Tanks

Figure 8.2–4 shows the significant reduction of the roll transfer functions and the significantroll amplitude of a trawler, being obtained by a free-surface tank.

0

1 0

2 0

3 0

4 0

0 0.5 1.0 1 .5 2 .0 2.5

T ra w ler L = 2 3.9 0 m e tre

W ith tan k

W itho ut ta nk

c ircu lar w ave fre qu en cy (1 /s)

Tra

nsfe

r fu

nctio

n ro

ll (d

eg/m

)

0

5

1 0

1 5

2 0

2 5

3 0

0 1 2 3 4 5 6

T r aw l er L = 2 3.9 0 m e tre

W ith tan k

W i th ou t ta nk

S ig nif ica nt w a ve he ig ht (m )

Sig

nifi

cant

ro

ll a

mp

litu

de (

deg

)

Figure 8.2–4: Effect of a Free-Surface Tank on Roll Motions in Beam Waves


258

8.3 Active Fin Stabilisers

To determine the effect of active fin stabilisers on ship motions, use has been made here ofreports published by Schmitke [1978] and Lloyd [1989].

The oscillatory angle of the portside fin is given by:( )βφεωββ +⋅⋅= ta cos

The exciting forces and moments, caused by an oscillating fin pair are given by:

βββ

βββ

βββ

βββ

βββ

βββ

⋅+⋅+⋅=

⋅+⋅+⋅=

⋅+⋅+⋅=

666fin6

444fin4

222fin2

cbaX

cbaX

cbaX

&&&

&&&

&&&

with:

( )( )( )

ββ

ββ

ββ

ββ

ββ

ββ

ββ

ββ

ββ

γ

γ

γ

γγ

γγ

γγ

γ

γ

γ

cxc

bxb

axa

czyb

bzyb

azya

cc

bb

aa

b

b

b

bb

bb

bb

⋅⋅⋅−=

⋅⋅⋅−=

⋅⋅⋅−=

⋅⋅+⋅⋅+=

⋅⋅+⋅⋅+=

⋅⋅+⋅⋅+=

⋅⋅−=

⋅⋅−=

⋅⋅−=

sin2

sin2

sin2

sincos2

sincos2

sincos2

sin2

sin2

sin2

fin6

fin6

fin6

finfin4

finfin4

finfin4

2

2

2

and:

( )

( )kCC

AVc

kCCc

AVb

csa

fin

Lfin

fin

Lfinfin

finfin

⋅

∂∂

⋅⋅⋅⋅=

⋅

∂∂

+⋅⋅⋅⋅⋅=

⋅

⋅⋅⋅=

αρ

απρ

πρ

β

β

β

2

3

21

221

221

In here:γ angle of port fin

fin

LC

∂∂

αlift curve slope of fin

( )kC circulation delay function

Vc

k re

⋅⋅

=2

ωreduced frequency

finA projected fin area

fins span of fin

finc mean chord of fin


259

finbx bx -co-ordinate of the centroid of fin forces

finby by -co-ordinate of the centroid of fin forces

finbz bz -co-ordinate of the centroid of fin forces

The nominal lift curve slope of a fin profile in a uniform flow is approximated by:( )( )

0.4cos

cos80.1

80.1

4

2

+Λ

⋅Λ+

⋅⋅=

∂∂

E

EL

AR

ARC πα

with:Λ sweep angle of fin profile( )EAR effective aspect ratio of fin profile

Of normal fins, the sweep angle of the fin profile is zero, so 0=Λ or 1cos =Λ .The fin acts in the boundary layer of the ship, which will reduce the lift. This effect istranslated into a reduced lift curve slope of the fin.

The velocity distribution in the hull boundary layer is estimated by the following twoequations:

( ) τδδ

δBL

VV ⋅= with: BLδδ <

2.0377.0 −⋅⋅= xfinBL Rxδ with: ν

xVRx

⋅=

in which:( )δV flow velocity inside boundary layer

V forward ship speedδ normal distance from hull

BLδ thickness of boundary layer

finx distance aft of forward perpendicular of fin

xR local Reynolds numberν kinematic density of fluid

The kinematic viscosity of seawater can be found from the water temperature T in degreescentigrade by:

26

000221.00336.00.178.1

10TT ⋅+⋅+

=⋅ν m2/s

It is assumed here that the total lift of the fin can be found from:

( ) ( ) finLfin

s

L AVCdcVCfin

⋅⋅⋅⋅=⋅⋅⋅⋅⋅ ∫ 2

0

2

21

21 ρδδδρ

where ( )δc is the chord at span-wise location δ .For rectangular fins, this is simply an assumption of a uniform loading.Because:

( ) ( )fin

tfinrfinrfin scccc δδ ⋅−−=


260

in which:

rfinc root chord of fin

tfinc tip chord of fin

2tfinrfin

fin

ccc

+= mean chord of fin

the correction to the lift curve slope is:

⋅−⋅

−−

⋅⋅

−⋅=fin

BLtfinrfinBL

fin

rfinBL s

cc

sc

cE

fin8

129

21

2δδ

Then the corrected lift curve slope of the fin is:( )

( ) 0.480.1

80.12 ++

⋅⋅⋅=

∂∂

finE

finEBL

fin

L

AR

ARE

C πα

Generally a fin is mounted close to the hull, so the effective aspect ratio is about twice thegeometric aspect ratio:

( ) ( )fin

finfinfinE c

sARAR ⋅=⋅= 22


261

8.4 Active Rudder Stabilisers

To determine the effect of rudder stabilisers on ship motions, use has been made of reportspublished by Lloyd [1989] and Schmitke [1978].

The oscillatory rudder angle is given by:( )δφεωδδ +⋅⋅= tea cos

with δ is positive in a counter-clockwise rotation of the rudder.So, a positive δ results in a positive side force, a positive roll moment and a negative yawmoment.The exciting forces and moments, caused by this oscillating rudder are given by:

δδδ

δδδ

δδδ

δδδ

δδδ

δδδ

⋅+⋅+⋅=

⋅+⋅+⋅=

⋅+⋅+⋅=

666rud6

444rud4

222rud2

cbaX

cbaX

cbaX

&&&

&&&

&&&

with:

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ

δδ

cxc

bxb

axa

czb

bzb

aza

cc

bb

aa

b

b

b

b

b

b

⋅+=⋅+=

⋅+=⋅−=⋅−=

⋅−=+=

+=+=

rud6

rud6

rud6

rud4

rud4

rud4

2

2

2

and:

( )

( )kCC

AVc

kCCc

AVb

csa

rud

Lrud

rud

Lrudrud

rudrud

⋅

∂∂

⋅⋅⋅⋅=

⋅

∂∂

+⋅⋅⋅⋅⋅=

⋅

⋅⋅⋅=

αρ

απρ

πρ

δ

δ

δ

2

3

21

221

221

In here:VVrud ⋅≈ 125.1 equivalent flow velocity at rudder

rud

LC

∂∂

αlift curve slope of rudder

Vc

k rude

⋅⋅

=2

ωcirculation delay function

rudA projected area of rudder

ruds span of rudder

rudc mean chord of rudder


262

rudbx bx -co-ordinate of centroid of rudder forces

rudbz bz -co-ordinate of centroid of rudder forces

The lift curve slope of the rudder is approximated by:( )

( ) 0.480.1

80.12 ++

⋅⋅=

∂∂

rudE

rudE

rud

L

AR

ARC πα

Generally a rudder is not mounted close to the hull, so the effective aspect ratio is equal to thegeometric aspect ratio:

( ) ( )rud

rudrudrudE c

sARAR ==


263

9 External Linear Springs

Suppose a linear spring connected to point P on the ship, see Figure 8.4–1.

Figure 8.4–1: Co-ordinate System of Springs

The harmonic longitudinal, lateral and vertical displacements of a certain point P on the shipare given by:

( )( )( ) φθ

φψ

θψ

⋅+⋅−=

⋅−⋅+=

⋅+⋅−=

pp

pp

pp

yxzPz

zxyPy

zyxPx

The linear spring coefficients in the three directions in a certain point P are defined by( )pzpypx CCC ,, . The units of these coefficients are N/m or kN/m.

9.1 External Loads

The external forces and moments, caused by these linear springs, acting on the ship are givenby:

( )( )( )

pspss

pspss

pspss

pppzs

pppys

pppxs

xXyXX

xXzXX

yXzXX

yxzCX

zxyCX

zyxCX

⋅+⋅−=

⋅−⋅+=

⋅+⋅−=

⋅+⋅−⋅−=

⋅−⋅+⋅−=

⋅+⋅−⋅−=

216

315

324

3

2

1

φθ

φψ

θψ


264

9.2 Additional Coefficients

After a change of sign, this results into the following coefficients ijc∆ , which have to be

added to the restoring spring coefficients ijc of the hydromechanical loads in the left-handside of the equations of motions:

• Surge:

ppx

ppx

px

yCc

zCc

c

c

c

Cc

⋅−=∆

⋅+=∆=∆

=∆=∆

+=∆

16

15

14

13

12

11

0

0

0

• Sway:

ppy

ppy

py

xCc

c

zCc

c

Cc

c

⋅+=∆=∆

⋅−=∆

=∆

+=∆=∆

26

25

24

23

22

21

0

0

0

• Heave:

0

0

0

36

35

34

33

32

31

=∆

⋅−=∆

⋅+=∆

+=∆

=∆=∆

c

xCc

yCc

Cc

c

c

ppz

ppz

pz

• Roll:

pppy

pppz

ppzppy

ppz

ppy

zxCc

yxCc

yCzCc

yCc

zCc

c

⋅⋅−=∆

⋅⋅−=∆

⋅+⋅+=∆

⋅+=∆

⋅−=∆

=∆

46

45

2244

43

42

41 0


265

• Pitch:

pppx

ppzppx

pppz

ppz

ppx

zyCc

xCzCc

yxCc

xCc

c

zCc

⋅⋅−=∆

⋅+⋅+=∆

⋅⋅−=∆

⋅−=∆=∆

⋅+=∆

56

2255

54

53

52

51

0

• Yaw:

2266

65

64

63

62

61

0

ppyppx

pppx

pppy

ppy

ppx

xCyCc

zyCc

zxCc

c

xCc

yCc

⋅+⋅+=∆

⋅⋅−=∆

⋅⋅−=∆=∆

⋅+=∆

⋅−=∆

It is obvious that in case of several springs, a linear superposition of the coefficients can beused.

When using linear springs, generally 12 sets of coupled equations with the in and out of phaseterms of the motions have to be solved. Because of these springs, the surge, heave and pitchmotions will be coupled then with the sway, roll and yaw motions.


266

9.3 Linearised Mooring Coefficients

Figure Figure 9.3–1 shows an example of results of static catenary line calculations, see forinstance Korkut and Hebert [1970], for an anchored platform.

Figure 9.3–1: Horizontal Forces on a Floating Structure as Function of Surge Displacements

Figure 9.3–1-a shows the platform anchored by two anchor lines of chain at 100 m waterdepth. Figure 9.3–1-b shows the horizontal forces at the suspension points of both anchorlines as a function of the horizontal displacement of the platform. Finally, Figure 9.3–1-cshows the relation between the total horizontal force on the platform and its horizontaldisplacement.This figure shows clearly the non-linear relation between the horizontal force on the platformand its horizontal displacement.

A linearised spring coefficient, to be used in frequency domain computations, can be obtainedfrom Figure 9.3–1-c by determining an average restoring spring coefficient, pxC , in the surgedisplacement region:

=ntDisplaceme

Force TotalMEANCpx


267

10 Added Resistance due to Waves

A ship moving forward in a wave field will generate ''two sets of waves'': waves associatedwith forward speed through still water and waves associated with its vertical relative motionresponse to waves. Since both wave patterns dissipate energy, it is logical to conclude that aship moving through still water will dissipate less energy than one moving through waves.The extra wave-induced loss of energy can be treated as an added propulsion resistance.

Figure 9.3–1 shows the resistance in regular waves as a function of the time: a constant partdue the calm water resistance and an oscillating part due to the motions of the ship, relative tothe incoming regular waves. The time-averaged part of the increase of resistance is called: theadded resistance due to waves, awR .

0

50 0

1 00 0

1 50 0

2 00 0

2 50 0

0 10 20 3 0

Re sistan ce

S til l water re sista n ce RSW

+M e an a dd ed re sista nce R

AW

Stil l water re sista n ce RSW

T im e (s)

Re

sist

ance

(kN

)

Figure 9.3–1: Increase of Resistance in Regular Waves

Two theoretical methods have been used for the estimation of the time-averaged addedresistance of a ship due to the waves and the resulting ship motions:

• a radiated wave energy method, as introduced by Gerritsma and Beukelman [1972],suitable for head to beam waves.

• an integrated pressure method, as introduced by Boese [1970], suitable for all wavedirections.

Because of the added resistance of a ship due to the waves is proportional to the relativemotions squared, its inaccuracy will be gained strongly by inaccuracies in the predictedmotions.The transfer function of the mean added resistance is presented as:


268

2"

a

awaw

RR

ζ=

In a non-dimensional way the transfer function of the mean added resistance is presented as:

LBgR

Ra

awaw 22

"

⋅⋅⋅=

ζρin which:

L length between perpendicularsB maximum breadth of the waterline

Both methods will be described here.


269

10.1 Radiated Energy Method

The wave energy - radiated during one period of oscillation of a ship in regular waves - hasbeen defined by Gerritsma and Beukelman [1972] as:

∫ ∫ ⋅⋅⋅=eT

Lbz dtdxVbP

0

2*'33

in which:'

33b hydrodynamic damping coefficient of the vertical motion of the cross section*

zV vertical average velocity of the water particles, relative to the cross sections

eT period of vertical oscillation of the cross section

The speed dependent hydrodynamic damping coefficient for the vertical motion of a crosssection is defined here as showed before:

bdxdM

VNb'

33'33

'33 ⋅−=

The harmonic vertical relative velocity of a point on the ship with respect to the waterparticles is defined by:

( )

( )φθθζ

φθζ

&&&&

&

⋅+⋅+⋅−−=

⋅+⋅−−=

bbw

bbwz

yVxz

yxzDtD

V

'3

'3

For a cross section of the ship, an equivalent harmonic vertical relative velocity has to befound, defined here by:

( )( )

ζεω

θθζ

*cos*

*3

*

zVeza

bwz

tV

VxzV

+⋅⋅=

⋅+⋅−−= &&&

With this the radiated energy during one period of oscillation is given by:

∫ ⋅⋅

⋅−⋅=

L

bzabe

dxVdx

dMVNP2*

'33'

33ωπ

To maintain a constant forward ship speed, this energy should be delivered by the ship'spropulsion plant. A mean added resistance awR has to be gained.The energy delivered to the surrounding water is given by:

µπ

µ

cos2

cos

⋅−⋅

⋅=

⋅

−⋅=

kR

Tc

VRP

aw

eaw

From this the transfer function of the mean added resistance according to Gerritsma andBeukelman can be found:


270

∫ ⋅

⋅

⋅−⋅

⋅⋅−

=L

ba

za

bea

aw dxV

dxdM

VNkR

2*'33'

332 2cos

ζωµ

ζ

Equation 10.1–1

This method gives good results in head to beam waves. However, in following waves thismethod could fail.When the wave speed in following waves approaches the ship speed the frequency ofencounter in the denominator tends to zero, 0→eω . At these low frequencies, the potentialsectional mass is very high and the potential sectional damping is almost zero. The dampingmultiplied with the relative velocity squared in the nominator does not tend to zero, as fast asthe frequency of encounter. This is caused by the presence of a natural frequency for heaveand pitch at this low eω , so a high motion peak can be expected. This results into an extremepositive or negative added resistance.


271

10.2 Integrated Pressure Method

Boese [1970] calculates the added resistance by integrating the longitudinal components ofthe oscillating pressures on the wetted surface of the hull. A second small contribution of thelongitudinal component of the vertical hydrodynamic and wave forces has been added.

The wave elevation is given by:( )µµωζζ sincoscos ⋅⋅−⋅⋅−⋅⋅= bbea ykxkt

The pressure in the undisturbed waves is given by:( )[ ][ ]( )[ ][ ] ( )µµωζρρ

ζρρ

sincoscoscosh

cosh

coshcosh

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅⋅+⋅⋅−=

⋅⋅+⋅

⋅⋅+⋅⋅−=

bbeab

b

ykxkthk

zhkgzg

hkzhk

gzgp

The horizontal force on an oscillating cross section is given by:

( )

( )[ ] ( )

−+⋅

⋅+

+−+−⋅⋅=

⋅= ∫+−

xsxs

zDbb

zDhk

zDg

dzptxfxs

ζζζρ

ζ

tanh2

,

22

with: θ⋅−= bx xzz .

As the mean added resistance during one period will be calculated, the constant term and thefirst harmonic term can be ignored. So:

( ) ( )[ ]

⋅

−⋅++−⋅⋅=hkzzgtxf xx

b tanh2,

22* ζζζρ

The vertical relative motion is defined by xzs −=ζ , so:

( ) [ ]

⋅

⋅++−⋅⋅=hk

szgtxf xb tanh2,

22* ζζρ

The average horizontal force on a cross section follows from:

( ) ( )

( )[ ]

⋅⋅

−⋅⋅−⋅⋅++−⋅

⋅⋅=

⋅= ∫

hk

xkszg

dttxfxf

a

sba

a

xaa

T

bb

e

tanh

coscos21

2

,

2

22

0

**

ζεµ

ζζρ ζ

The added resistance due to this force is:


272

( )

( )[ ] b

b

w

L a

sba

a

xaa

b

L b

wbaw

dxdxdy

hk

xkszg

dxdxdy

xfR

⋅

−⋅

⋅⋅

−⋅⋅−⋅⋅−−−⋅

⋅⋅=

⋅

−⋅⋅=

∫

∫

tanh

coscos21

2

2

2

22

*1

ζεµ

ζζρ ζ

where wy is the still water line.

For deep water, this part of the mean added resistance reduces to:

∫ ⋅⋅⋅⋅−

=L

bb

waaw dx

dxdy

sg

R 21 2

ρ (as given by Boese for deep water)

The integrated vertical hydromechanical and wave forces in the ship-bounded system vary notonly in time but also in direction with the pitch angle.From this follows a second contribution to the mean added resistance:

( ) ( ) ( )

( ) ( ) dtttzT

dtttZtZT

R

e

e

T

e

T

whe

aw

⋅⋅⋅∇⋅⋅−

=

⋅⋅+⋅−

=

∫

∫

0

02

1

1

θρ

θ

&&

For this second contribution can be written:

( )θζζ εεθωρ −⋅⋅⋅⋅∇⋅⋅= zaaeaw zR cos21 2

2

So the transfer function of the total mean added resistance according to Boese is given by:( )

[ ]

( )θζζ

ζ

εεθωρ

ζεµ

ζρ

ζ

−⋅⋅⋅⋅∇⋅⋅+

⋅

−⋅

⋅⋅−⋅⋅−⋅⋅

−−−⋅⋅⋅= ∫

zaae

bb

w

L a

sba

a

xa

a

aw

z

dxdxdy

hk

xkszg

R

cos21

tanh

coscos21

21

2

2

2

2

Equation 10.2–1


273

10.3 Comparison of Results

Figure 10.3–1 shows an example of a comparison between computed and experimental data.

Figure 10.3–1: Added Resistance of the S-175 Containership Design


274


275

11 Bending and Torsion Moments

The axes system (of which the hydrodynamic sign convention differs from that commonlyused in structural engineering) and the internal load definitions are given in Figure 10.3–1.

Figure 10.3–1: Axis System and Internal Load Definitions

To obtain the vertical and lateral shear forces and bending moments and the torsion momentsthe following information over a length mL on the solid mass distribution of the shipincluding its cargo is required:

( )bxm' distribution over the ship length of the solid mass of the ship per unitlength, see Figure 10.3–2

( )bm xz ' distribution over the ship length of the vertical bz -values of the centreof gravity of the solid mass of the ship per unit length

( )bxx xk ' distribution over the ship length of the radius of inertia of the solidmass of the ship per unit length, about a horizontal longitudinal axisthrough the centre of gravity

Figure 10.3–2: Distribution of Solid Mass


276

The input values for the calculation of shear forces and bending and torsion moments areoften more or less inaccurate. Mostly small adaptations are necessary, for instance to avoid aremaining calculated bending moment at the forward end of the ship.The total mass of the ship is found by an integration of the mass per unit length:

( )∫ ⋅=mL

bb dxxmm '

It is obvious that this integrated mass should be equal to the mass of displacement, calculatedfrom the underwater hull form:

∇⋅= ρm

Both terms will be calculated from independently derived data, so small deviations arepossible. A proportional correction of the masses per unit length ( )bxm' can be used, seeFigure 10.3–3.Then ( )bxm' will be replaced by:

( )m

xm b

∇⋅⋅ρ'

Figure 10.3–3: Mass Correction for Buoyancy

The longitudinal position of the centre of gravity is found from the distribution of the massper unit length:

( )∫ ⋅⋅⋅=mL

bbbG dxxxmm

x '1

An equal longitudinal position of the ship's centre of buoyancy Bx is required, so:

BG xx =

Again, because of independently derived data, a small deviation is possible.Then, for instance, ( )bxm' can be replaced by ( ) ( )bb xcxm +' , with:

( ) ( )01 −−⋅−= Abb xxcxc for: 40 mAb Lxx <−<( ) ( )21 mAbb Lxxcxc −−⋅+= for: 434 mAbm LxxL ⋅<−<( ) ( )mAbb Lxxcxc −−⋅−= 1 for: mAbm LxxL <−<⋅ 43


277

with:( )31

32

m

GB

Lxx

c−⋅∇⋅⋅

=ρ

In here:

Ax bx -co-ordinate of hindmost part of mass distribution

mL total length of mass distribution

Figure 10.3–4: Mass Correction for Cent re of Buoyancy

For relative slender bodies, the longitudinal radius of inertia of the mass can be found fromthe distribution of the mass per unit length:

( )∫ ⋅⋅⋅=mL

bbbyy dxxxmm

k 2'2 1

It can be desirable to change the mass distribution in such a way that a certain requiredlongitudinal radius of inertia ( )newk yy or ( )newkzz will be achieved, without changing thetotal mass or the position of its centre of gravity.Then, for instance, ( )bxm' can be replaced by ( ) ( )bb xcxm +' , see Figure 10.3–5, with:

( ) ( )02 −−⋅+= Abb xxcxc for: 80 mAb Lxx <−<( ) ( )822 mAbb Lxxcxc ⋅−−⋅−= for: 838 mAbm LxxL ⋅<−<( ) ( )842 mAbb Lxxcxc ⋅−−⋅+= for: 8483 mAbm LxxL ⋅<−<⋅( ) ( )842 mAbb Lxxcxc ⋅−−⋅−= for: 8584 mAbm LxxL ⋅<−<⋅( ) ( )862 mAbb Lxxcxc ⋅−−⋅+= for: 8785 mAbm LxxL ⋅<−<⋅( ) ( )mAbb Lxxcxc −−⋅−= 2 for: mAbm LxxL <−<⋅ 87

with:( ) ( )

3

22

2 9

3204

m

yyyy

L

oldknewkc

⋅

−⋅∇⋅⋅=

ρ

In here:

Ax bx -co-ordinate of hindmost part of mass distribution

mL total length of mass distribution


278

Figure 10.3–5: Mass Correction for Radius of Inertia

The position in height of the centre of gravity is found from the distribution of the heights ofthe centre of gravity of the masses per unit length:

( ) ( )∫ ⋅⋅⋅=mL

bbmbG dxxzxmm

z ''1

It is obvious that this value should be zero. If not so, this value has to be subtracted from( )bm xz ' .

So, ( )bm xz ' will be replaced by ( ) Gbm zxz −' .

The transverse radius of inertia xxk is found from the distribution of the radii of inertia of themasses per unit length:

( )∫ ⋅⋅⋅=mL

bxxbyy dxkxmm

k2''2 1

If this value of xxk differs from a required value ( )newkxx of the radius of inertia, aproportional correction of the longitudinal distribution of the radii of inertia can be used:

( ) ( ) ( )( )oldknewk

oldxknewxkxx

xxbxxbxx '

''' ,, ⋅=

Consider a section of the ship with a length bdx to calculate the shear forces and the bendingand the torsion moments.


279

Figure 10.3–6: Loads on a Cross Section

When a load ( )bxq loads the disk, this implies for the disk:

( ) ( )bbb xdQdxxq −=⋅ so: ( ) ( )b

b

b xqdx

xdQ−=

( ) ( )bbb xdMdxxQ +=⋅ so: ( ) ( )bb

b xQdx

xdM+=

in which:( )bxQ shear force

( )bxM bending moment

The shear force and the bending moment in a cross section 1x follows from an integration ofthe loads from the hindmost part of the ship 0x to this cross section 1x :

( ) ( )

( ) ( )

( )b

x

x

x

xb

b

b

x

xbb

x

x

bb

b

dxdxdx

xdQ

dxxQxM

dxdx

xdQxQ

b

⋅

⋅+=

⋅−=

⋅−=

∫ ∫

∫

∫

1

0 0

1

0

1

0

1

1

So, the shear force ( )1xQ and the bending moment ( )1xM in a cross section can be expressedin the load ( )bxq by the following integrals:


280

( ) ( )

( ) ( ) ( )

( ) ( )∫∫

∫

∫

⋅⋅−⋅⋅+=

⋅−⋅+=

⋅−=

1

0

1

0

1

0

1

0

1

11

1

x

x

bb

x

x

bbb

x

xbbb

x

x

bb

dxxqxdxxxq

dxxxxqxM

dxxqxQ

For the torsion moment an approach similar to the approach for the shear force can be used.The load ( )bxq consists of solid mass and hydromechanical terms. The ordinates of theseterms will differ generally, so numerical integrations of these two terms have to be carried outseparately.


281

11.1 Still Water Loads

Consider the forces acting on a section of the ship with a length bdx .

Figure 11.1–1: Still Water Loads on a Cross Section

According to Newton's second law of dynamics, the vertical forces on the unfastened disk of aship in still water are given by:

( ) ( ) ( ) bbswb dxxqgdxm ⋅=−⋅⋅ 3'

with:gmgAq ssw ⋅−⋅⋅= '

3 ρ

So, the vertical shear force ( )13 xQ sw and the bending moment ( )15 xQ sw in still water in a cross

section can be obtained from the vertical load ( )13 xq sw by the following integrals:

( ) ( )

( ) ( ) ( )∫∫

∫

⋅⋅−⋅⋅+=

⋅−=

1

0

1

0

1

0

31315

313

x

x

bbsw

x

x

bbbswsw

x

xbbswsw

dxxqxdxxxqxQ

dxxqxQ

For obtaining the dynamic parts of the vertical shear forces and the vertical bending momentsin regular waves, reference is given to Fukuda [1962]. For the lateral mode and the roll modea similar procedure can be followed. This will be showed in the following Sections.


282

11.2 Dynamical Lateral Loads


Figure 11.2–1: Lateral Loads on a Cross Section in Waves

According to Newton's second law of dynamics, the harmonic lateral dynamic load per unitlength on the unfastened disk is given by:

( ) ( ) ( )( ) ( )φφψφρ ⋅+⋅−⋅+⋅−⋅⋅⋅+

++=

gzxyxmAg

xXxXxq

mbbs

bwbhb

&&&&&& ''

'2

'22

The sectional hydromechanical loads for sway are given by:

ψψψφφφ

⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−=

'26

'26

'26

'24

'24

'24

'22

'22

'22

'2

cba

cba

ycybyaXh

&&&

&&&&&&

with:

0

2

0

0

'26

'22

2

2'

22

'22'

22'

26

'22

2'

222

'22'

222'

22'

26

'24

'22'

22

'24'

24'

24

'22

2

'24

2'

22'

24'

24

'22

'22'

22'

22

'22

2'

22'

22

=

⋅+⋅⋅−⋅

⋅−+=

⋅⋅+⋅+

⋅−⋅+⋅+=

=

⋅−⋅+⋅−+=

⋅⋅+⋅+⋅++=

=

⋅−+=

⋅++=

c

dxdNV

MVxdx

dMVNb

xdx

dNVN

Vdx

dMVN

VxMa

c

dxdM

VNOGdx

dMVNb

dxdN

OGV

dxdNV

MOGMa

c

dxdM

VNb

dxdNV

Ma

beb

b

bbeebe

b

bb

bebe

b

be

ω

ωωω

ωω

ω


283

The sectional wave loads for sway are given by:

'2

*2

'22'

22

*2

'22*

2'

22'

2

FK

wbe

wbe

ww

X

dxdM

VN

dxdNV

MX

+

⋅

⋅−⋅+

⋅⋅⋅

+⋅+=

ζωω

ζωω

ζ

&

&&&&

The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.Then the harmonic lateral shear forces ( )12 xQ and the bending moments ( )16 xQ in waves in

cross section 1x can be obtained from the horizontal load ( )bxq2 by the following integrals:( ) ( )

( )

( ) ( )

( ) ( )∫∫

∫

⋅⋅−⋅⋅+

+⋅⋅=

⋅−

+⋅⋅=

1

0

1

0

6

1

0

2

212

616

2

212

cos

cos

x

x

bb

x

x

bbb

Qea

x

x

bb

Qea

dxxqxdxxxq

tQxQ

dxxq

tQxQ

ζ

ζ

εω

εω


284

11.3 Dynamical Vertical Loads


Figure 11.3–1: Vertical Loads on a Cross Section in Waves

According to Newton's second law of dynamics, the harmonic longitudinal and verticaldynamic loads per unit length on the unfastened disk are given by:

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )θ

θ&&&&

&&&&

⋅−⋅−++=

⋅−⋅−++=

bbbwbhb

bbwbhb

xzxmxXxXxq

bGxxmxXxXxq''

3'

33

''1

'11

The sectional hydromechanical loads for surge are given by:

θθθ ⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−=

'15

'15

'15

'13

'13

'13

'11

'11

'11

'1

cba

zczbza

xcxbxaXh

&&&&&&

&&&

with:

0

0

0

0

0

'15

11

'11'

11'

15

'11

2'

11'

15

'13

'13

'13

'11

11

'11'

11'

11

'11

2'

11'

11

=

⋅−⋅

⋅−−=

⋅⋅+⋅−=

=

=

=

=

+−+=

⋅++=

c

bGbbGdx

dMVNb

bGdx

dNVbGMa

c

b

a

c

bdx

dMNb

dxdNV

Ma

Vb

be

Vb

be

ω

ω


285

The sectional hydromechanical loads for heave are given by:

θθθ ⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−=

'35

'35

'35

'33

'33

'33

'31

'31

'31

'3

cba

zczbza

xcxbxaXh

&&&&&&

&&&

with:

bw

beLb

Lbb

b

bbeebe

b

w

b

be

xygc

dxdNV

dxMVdxxdx

dMVNb

xdx

dNVN

Vdx

dMN

VxMa

ygc

dxdM

VNb

dxdNV

Ma

c

b

a

⋅⋅⋅⋅−=

⋅+⋅⋅⋅+⋅⋅

⋅−−=

⋅⋅−

+⋅−

+

−⋅−⋅−=

⋅⋅⋅+=

⋅−+=

⋅++=

=

=

=

∫∫

ρ

ω

ωωω

ρ

ω

2

2

2

0

0

0

'35

'33

2

2'

33

'33'

33'

35

'33

2'

332

'33'

332'

33'

35

'33

'33'

33'

33

'33

2'

33'

33

'31

'31

'31

The sectional wave loads for surge and heave are given by:

'1

*1

'11'

11

*1

'11*

1'

11'

1

FK

wbe

wbe

ww

X

dxdM

VN

dxdNV

MX

+

⋅

⋅−⋅+

⋅⋅⋅

+⋅+=

ζωω

ζωω

ζ

&

&&&&

'3

*3

'33'

33

*3

'33*

3'

33'

3

FK

wbe

wbe

ww

X

dxdM

VN

dxdNV

MX

+

⋅

⋅−⋅+

⋅⋅⋅

+⋅+=

ζωω

ζωω

ζ

&

&&&&



286

Then the harmonic vertical shear forces ( )13 xQ and the bending moments ( )15 xQ in waves in

cross section 1x can be obtained from the longitudinal and vertical load ( )bxq1 and ( )bxq3 bythe following integrals:

Figure 11.3–2 shows a comparison between measured and calculated distributions of thevertical wave bending moment amplitudes over the length of the ship.

Figure 11.3–2: Distribution of Vertical Bending Moment Amplitudes


287

11.4 Dynamical Torsion Loads


Figure 11.4–1: Torsion Loads on a Cross Section in Waves

According to Newton's second law of dynamics, the harmonic torsion dynamic load per unitlength on the unfastened disk about a longitudinal axis at a distance 1z above the ship's centreof gravity is given by:

( ) ( ) ( )( ) ( )

( )b

bmxxb

bwbhb

xqz

gxyzkxm

xXxXzxq

21

'2''

'4

'414 ,

⋅+⋅+⋅+−⋅⋅−

++=

φψφ &&&&&&

The sectional hydromechanical load for roll is given by:

ψψψφφφ

⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−=

'46

'46

'46

'44

'44

'44

'42

'42

'42

'4

cba

cba

ycybyaXh

&&&

&&&&&&

with:


288

'26

'46

'42

2

2'

26'

42

'42'

42'

46

'42

2

'422

'26

'42'

422

'42

'46

3'

44

'24

'44

'42'

42

'44'

44'

44

'42

2

'44

2

'24

'42

'44

'44

'24

'22

'42'

42'

42

'24

2

'22

'42

'42

2

232

0

cOGc

dxdNV

bOGMVxdx

dMVNb

xdx

dNVN

VaOG

dxdM

VNV

xMa

bGAy

gc

bOGbdx

dMVNOG

dxdM

VNb

dxdN

OGV

dxdNV

aOGMOGMa

c

bOGdx

dMVNb

dxdNV

aOGMa

beb

b

bbeebe

b

sw

Vbb

bebe

b

be

⋅+=

⋅+⋅+⋅⋅−⋅

⋅−+=

⋅⋅+⋅+⋅+

⋅−⋅+⋅+=

⋅+⋅⋅⋅+=

⋅++

⋅−⋅+⋅−+=

⋅⋅+⋅+⋅+⋅++=

=

⋅+⋅−+=

⋅+⋅++=

ω

ωωω

ρ

ωω

ω

In here, bG is the vertical distance of the centre of gravity of the ship G above the centroidb of the local submerged sectional area.

The sectional wave load for roll is given by:

'2

'4

*2

'42'

42

*2

'42*

2'

42'

4

wFK

wbe

wbe

ww

XOGX

dxdM

VN

dxdNV

MX

⋅++

⋅

⋅−⋅+

⋅⋅⋅

+⋅+=

ζωω

ζωω

ζ

&

&&&&


Then the harmonic torsion moments ( )114 , zxQ in waves in cross section 1x at a distance 1zabove the centre of gravity can be obtained from the torsion load ( )14 , zxq b by the followingintegral:

( ) ( )

( ) b

x

xb

Qea

dxzxq

tQzxQ

⋅−=

+⋅⋅=

∫1

0

4

14

4114

,

cos, ζεω


289

12 Statistics in Irregular Waves

To compare the calculated behaviour of different ship designs or to get an impression of thebehaviour of a specific ship design in a seaway, standard representations of the wave energydistributions are necessary.

Three well-known types of normalised wave energy spectra are described here:• the Neumann wave spectrum, a somewhat wide wave spectrum, which is sometimes used

for open sea areas• the Bretschneider wave spectrum, an average wave spectrum, frequently used in open sea

areas• the Mean JONSWAP wave spectrum, a narrow wave spectrum, frequently used in North

Sea areas.

The mathematical formulations of these normalised uni-directional wave energy spectra arebased on two parameters:• the significant wave height 3/1H

• the average wave period 1T , based on the centroid of the spectral area curve.To obtain the average zero-crossing period 2T or the spectral peak period pT , a fixed relation

with 1T can be used not-truncated spectra.

From these wave energy spectra and the transfer functions of the responses, the responseenergy spectra can be obtained.Generally the frequency ranges of the energy spectra of the waves and the responses of theship on these waves are not very wide. Then the Rayleigh distribution can be used to obtain aprobability density function of the maximum and minimum values of the waves and theresponses. With this function, the probabilities on exceeding threshold values by the shipmotions can be calculated.Bow slamming phenomena are defined by a relative bow velocity criterion and a peak bottomimpact pressure criterion.


290

12.1 Normalised Wave Energy Spectra

Three mathematical definitions with two parameters of normalized spectra of irregular uni-directional waves have been described:• the Neumann wave spectrum, a somewhat wide spectrum• the Bretschneider wave spectrum, an average spectrum• the mean JONSWAP wave spectrum, a narrow spectrumA comparison of the Neumann, the Bretschneider and the mean JONSWAP wave spectra isgiven here for a sea state with a significant wave height of 4 meters and an average waveperiod of 8 seconds.

Figure 12.1–1: Comparison of Three Spectral Formulations

12.1.1 Neumann Wave Spectrum

In some cases in literature the Neumann definition of a wave spectrum for open sea areas isused:

( )

⋅−

⋅⋅⋅

= −− 22

1

65

1

23/1 8.69

exp3832 ωωωζ TT

HS

12.1.2 Bretschneider Wave Spectrum

A very well known two-parameter wave spectrum of open seas is defined by Bretschneider as:

( )

⋅−

⋅⋅⋅

= −− 42

1

54

1

23/1 2.691

exp8.172 ωωωζ TT

HS


291

Another name of this wave spectrum is the Modified Two-Parameter Pierson-MoskowitzWave Spectrum.

This formulation is accepted by the 2nd International Ship Structures Congress in 1967 andthe 12th International Towing Tank Conference in 1969 as a standard for seakeepingcalculations and model experiments. This is reason why this spectrum is also called I.S.S.C.or I.T.T.C. Wave Spectrum.The original One-Parameter Pierson-Moskowitz Wave Spectrum for fully developed seas canbe obtained from this definition by using a fixed relation between the significant wave heightand the average wave period in this Bretschneider definition: 3/11 861.3 HT ⋅= .

12.1.3 Mean JONSWAP Wave Spectrum

In 1968 and 1969 an extensive wave measurement program, known as the Joint North SeaWave Project (JONSWAP) was carried out along a line extending over 100 miles into theNorth Sea from Sylt Island. From analysis of the measured spectra, a spectral formulation ofwind generated seas with a fetch limitation was found.The following definition of a Mean JONSWAP wave spectrum is advised by the 15th ITTC in1978 for fetch limited situations:

( ) BATT

HS γωωωζ ⋅⋅

⋅−

⋅⋅⋅

= −− 42

1

54

1

23/1 2.691

exp8.172

with:658.0=A

⋅

−−=

2

2

0.1

expσ

ωω

pB

3.3=γ (peakedness factor)

pp T

πω ⋅= 2

(circular frequency at spectral peak)

=σ a step function of ω: if pωω < then: 07.0=σif pωω > then: 09.0=σ

The JONSWAP expression is equal to the Bretschneider definition multiplied by thefrequency function BA γ⋅ .Sometimes, a third free parameter is introduced in the JONSWAP wave spectrum by varyingthe peakedness factor γ .

12.1.4 Definition of Parameters

The nth order spectral moments of the wave spectrum, defined as a function of the circularwave frequency ω , are:


292

( )∫∞

⋅⋅=0

ωωωζζ dSm nn

The breadth of a wave spectrum is defined by:

ζζ

ζε40

221mm

m

⋅−=

The significant wave height is defined by:

ζ03/1 4 mH ⋅=

The several definitions of the average wave period are:

pT peak or modal wave period, corresponding to peak of spectralcurve

ζ

ζπ1

01 2

m

mT ⋅⋅= average wave period, corresponding to centroid of spectral

curve

ζ

ζπ2

02 2

m

mT ⋅⋅= average zero-crossing wave period, corresponding to radius of

inertia of spectral curve

For not-truncated mathematically defined spectra, the theoretical relations between the periodsare tabled below:

=⋅=⋅⋅==⋅⋅=⋅=

p

p

p

TTTTTT

TTT

21

21

21

407.1296.1711.0921.0

772.0086.1 for Bretschneider wave spectra

=⋅=⋅⋅==⋅⋅=⋅=

p

p

p

TTTTTT

TTT

21

21

21

287.1199.1777.0932.0

834.0073.1 for JONSWAP wave spectra

Truncation of wave spectra during numerical calculations can cause differences between inputand calculated wave periods. Generally, the wave heights will not differ much.

In Figure 12.1–2 and Table 12.1–1 - for ''Open Ocean Areas'' and ''North Sea Areas'' - anindication is given of a possible average relation between the scale of Beaufort or the windvelocity at 19.5 meters above the sea level and the significant wave height 3/1H and theaverage wave periods 1T or 2T .Notice that these data are an indication only. A generally applicable fixed relation betweenwave heights and wave periods does not exist.


293

Figure 12.1–2: Wave Spectra Parameter Indications

Table 12.1–1: Wave Spectra Parameter Indications


294

Other open ocean definitions for the North Atlantic and the North Pacific, obtained fromBales [1983] and adopted by the 17th ITTC (1984), are given in Table 12.1–2. The modal orcentral periods in these tables correspond with the peak period pT . For not-truncated spectra,

the relations with 1T and 2T are defined before.

Table 12.1–2: Sea State Parameters


295

12.2 Response Spectra and Statistics

The energy spectrum of the responses ( )tr of a sailing ship in the irregular waves followsfrom the transfer function of the response and the wave energy spectrum by:

( ) ( )ωζ

ω ζSr

Sa

ar ⋅

=

2

or ( ) ( )ea

aer S

rS ω

ζω ζ⋅

=

2

This has been visualized for a heave motion in Figure 12.2–1and Figure 12.2–2.

Figure 12.2–1: Principle of Transfer of Waves into Responses


296

0

0 .5

1 .0

1 .5

2 .0

0 0.5 1.0 1.5 2.0

C on tain ersh ipL = 1 75 m etre H ea d w ave sV = 2 0 kn ots

T ran s ferfu nc t io nhe ave

0

1

2

3

4

5

0 0.5 1.0 1.5 2.0

W avesp ec trum

H1/3

= 5.00 m

T2 = 8 .0 0 s

0

1

2

3

4

5

0 0.5 1.0 1 .5 2 .0

W avespe c tru m

H1/3

= 5 .0 0 m

T2 = 8.00 s

Sp

ectr

al d

ens

ity w

ave

(m2 s)

0

0.5

1.0

1.5

2.0

0 0.5 1.0 1 .5 2 .0

C onta ine rshi pL = 17 5 m e treH ead w ave sV = 20 kno ts

T ra ns ferfun c ti onh eave

Tra

nsfe

r fu

nctio

n h

eave

(m

/m)

0

2

4

6

8

0 0.5 1.0 1 .5 2 .0

H e avesp ec tru m

za

1 /3

= 1.92 m

Tz

2

= 7.74 s

w a ve fre qu en cy (ra d/s )

Sp

ectr

al d

ens

ity h

eav

e (

m2s)

0

2

4

6

8

0 0.5 1 .0 1.5 2 .0

za

1/3

= 1 .92 m

Tz

2

= 7 .74 s

fr eq ue ncy of en cou nter ( rad /s )

Figure 12.2–2: Heave Spectra in the Wave and Encounter Frequency Domain

The moments of the response spectrum are given by:

( )∫∞

⋅⋅=0

en

eernr dSm ωωω with: ,...2,1,0=n

From the spectral density function of a response the significant amplitude can be calculated.The significant amplitude is defined to be the mean value of the highest one-third part of thehighest wave heights, so:

ra mr 03/1 2 ⋅=

A mean period can be found from the centroid of the spectrum by:

r

rr m

mT

1

01 2 ⋅⋅= π

Another definition, which is equivalent to the average zero-crossing period, is found from thespectral radius of inertia by:

r

rr m

mT

2

02 2 ⋅⋅= π

The probability density function of the maximum and minimum values, in case of a spectrumwith a frequency range that is not too wide, is given by the Rayleigh distribution:


297

( )

⋅−⋅=

r

a

r

aa m

rmrrf

0

2

0 2exp

This implies that the probability of exceeding a threshold value a by the response amplitude

ar becomes:

⋅−

=

⋅

⋅−

⋅=> ∫∞

r

a

a r

a

r

aa

ma

drmr

mr

arP

0

2

0

2

0

2exp

2exp

The number of times per hour that this happens follows from:

arPT

N ar

hour >⋅=2

3600

The spectral value of the waves ( )eS ωζ , based on eω , is not equal to the spectral value

( )ωζS , based on ω . Because of the requirement of an equal amount of energy in the

frequency bands eω∆ and ω∆ , it follows:

( ) ( ) ωωωω ζζ dSdS ee ⋅=⋅

From this the following relation is found:

( ) ( )ωω

ωω ζ

ζ dd

SS

ee =

The relation between the frequency of encounter and the wave frequency, of which anexample is illustrated in Figure 12.2–3, is given by:

µωω cos⋅⋅−= Vke

Figure 12.2–3: Example of Relation Between eω and ω


298

From the relation between eω and ω follows:

dkdV

dd e

ωµ

ωω cos

0.1⋅

−=

The derivative dkdω follows from the relation between ω and k :

[ ]hkgk ⋅⋅⋅= tanhω

So:

[ ] [ ][ ]hkgk

hkhgk

hkg

dkd

⋅⋅⋅⋅⋅⋅

⋅+⋅⋅

=tanh2

coshtanh 2ω

As can be seen in Figure 12.2–3, in following waves the derivative ωω dd e can approachfrom both sides, a positive or a negative side, to zero. As a result of this, around a wave speedequal to twice the forward ship speed component in the direction of the wave propagation, thetransformed spectral values will range from plus infinite to minus infinite. This implies thatnumerical problems will arise in the numerical integration routine.

This is the reason why the spectral moments have to be written in the following format:

( ) ( )

( ) ( )

( ) ( )∫∫

∫∫

∫∫

∞∞

∞∞

∞∞

⋅⋅=⋅⋅=

⋅⋅=⋅⋅=

⋅=⋅=

0

2

0

22

00

1

000

ωωωωωω

ωωωωωω

ωωωω

dSdSm

dSdSm

dSdSm

ereeerr

ereeerr

reerr

with:

( ) ( )ωζ

ω ζSr

Sa

ar ⋅

=

2

If ( )erS ω has to be known, for instance for a comparison of the calculated response spectra

with measured response spectra, these values can be obtained from this ( )ωrS and thederivative ωω dd e . So an integration of ( )erS ω over eω has to be avoided.

Because of the linearities, the calculated significant values can be presented by:

3/1

3/1

Hra versus 1T or 2T

with:

3/1H significant wave height

21,TT average wave periods


299

The mean added resistance in a seaway follows from:

( ) ωωζ ζ dSR

Ra

awAW ⋅⋅⋅= ∫

∞

022

Because of the linearities of the motions, the calculated mean added resistance values can bepresented by:

23/1H

RAW versus 1T or 2T


300

12.3 Shipping Green Water

The effective dynamic freeboard will differ from the results obtained from the geometricfreeboard at zero forward speed in still water and the calculated vertical relative motions of asailing ship in waves.When sailing in still water, sinkage, trim and the ship's wave system will effect the localgeometric freeboard. A static swell up should be taken into account.An empirical formula, based on model experiments, for the static swell up at the forwardperpendicular is given by Tasaki [1963]:

275.0 FnLL

BffE

e ⋅⋅⋅−=

with:

ef effective freeboard at the forward perpendicularf geometric freeboard at the forward perpendicularL length of the shipB breadth of the ship

EL length of entrance of the waterlineFn Froude number

An oscillating ship will produce waves and these dynamic phenomena will influence theamplitude of the relative motion. A dynamic swell up should be taken into account.Tasaki [1963] carried out forced oscillation tests with ship models in still water and obtainedan empirical formula for the dynamic swell-up at the forward perpendicular in head waves:

gLC

ss eB

a

a ⋅⋅

−=

∆ 2

345.0 ω

with the restrictions:block coefficient: 80.060.0 << BCFroude number: 29.016.0 << Fn

In this formula as is the amplitude of the relative motion at the forward perpendicular asobtained in head waves, calculated from the heave, the pitch and the wave motions.Then the actual amplitude of the relative motions becomes:

aaa sss ∆+=*

Then, shipping green water is defined by:

ea fs >* at the forward perpendicular

The spectral density of the vertical relative motion at the forward perpendicular is given by:

( ) ( )ωζ

ω ζSs

Sa

as

⋅

=

2*

*

The spectral moments are given by:

( )∫∞

⋅⋅=0

** ωωω dSm nesns

with: ,...2,1,0=n


301

When using the Rayleigh distribution the probability of shipping green water is given by:

⋅−=>

*0

2*

2exp

s

eea m

ffsP

The average zero-crossing period of the relative motion is found from the spectral radius ofinertia by:

*

*

*

2

02

2s

ss m

mT ⋅⋅= π

The number of times per hour that green water will be shipped follows from:

ea

s

hour fsPT

N >⋅= *

2 *

3600


302

12.4 Bow Slamming

Slamming is a two-node vibration of the ship caused by suddenly pushing the ship by thewaves. A complete prediction of slamming phenomena is a complex task, which is beyond thescope of any existing theory.Slamming impact pressures are affected by the local hull section shape, the relative velocitybetween ship and waves at impact, the relative angle between the keel and the water surface,the local flexibility of the ship's bottom plating and the overall flexibility of the ship'sstructure.

12.4.1 Criterium of Ochi

Ochi [1964] translated slamming phenomena into requirements for the vertical relativemotions of the ship.He defined slamming by:• an emergence of the bow of the ship at 10 percentile of the length aft of the forward

perpendiculars• an exceeding of a certain critical value at the instance of impact by the vertical relative

velocity, without forward speed effect, between the wave surface and the bow of the ship

Ochi defines the vertical relative displacement and velocity of the water particles with respectto the keel point of the ship by:

θζ

θζ&&&& ⋅+−=

⋅+−=

bx

bx

xzs

xzs

b

b

with:( )

( )µωζωζ

µωζζ

cossin

coscos

⋅⋅−⋅⋅⋅−=

⋅⋅−⋅⋅=

beaex

beax

xkt

xkt

b

b

&

So a forward speed effect ( θ⋅V -term) is not included in this vertical relative velocity. Thespectral moments of the vertical relative displacements and velocities are defined by sm0 and

sm &0 .Emergence of the bow of the ship happens when the vertical relative displacement amplitude

as at L⋅90.0 is larger than the ship's draft sD at this location.The probability of emergence of the bow follows from:

⋅−=>

s

ssa m

DDsP0

2

2exp

The second requirement states that the vertical relative velocity exceeds a threshold value.According to Ochi, 12 feet per second can be taken as a threshold value for a ship with alength of 520 feet.Scaling results into:

Lgscr ⋅⋅= 0928.0&

The probability of exceeding this threshold value is:


303

⋅−=>

s

crcra m

sssP&

&&&

0

2

2exp

Both occurrences, emergence of the bow and exceeding the threshold velocity, are statisticallyindependent. In case of slamming both occurrences have to appear at the same time.So the probability on a slam is the product of the both independent probabilities:

⋅−

+⋅

−=

>⋅>=

s

cr

s

s

crasa

ms

mD

ssPDsPslamP

&

&

&&

0

2

0

2

22exp

12.4.2 Criterium of Conolly

Conolly [1974] translated slamming phenomena into requirements for the peak impactpressure of the ship.He defined slamming by:• an emergence of the bow of the ship• an exceeding of a certain critical value by the peak impact pressure at this location.

The peak impact pressure is defined by:2

21

crp sCp &⋅⋅⋅= ρ

The coefficient pC has been taken from experimental data of slamming drop tests withwedges and cones, as given in literature.Some of these data, as for instance presented by Lloyd [1989] as a function of the deadriseangle β , are illustrated in Figure 12.4–1.

Figure 12.4–1: Peak Impact Pressure Coefficients

An equivalent deadrise angle β is defined here by the determination of an equivalent wedge.The contour of the cross section inside 10 percentile of the half breadth 2B of the ship hasbeen used to define an equivalent wedge with a half breadth: 210.0 Bb ⋅= .


304

The accessory draught t of the wedge follows from the section contour. In the fore body ofthe ship, this draught can be larger than 10 percentile of the amidships draught T . If so, thesection contour below T⋅10.0 has been used to define an equivalent wedge: Tt ⋅= 10.0 . Ifthis draught is larger than the local draught, the local draught has been used.The accessory half breadth b of the wedge follows from the section contour.

Figure 12.4–2: Definition of an Equivalent Wedge

Then the sectional area sA below local draught t has to be calculated.Now the equivalent deadrise angle β follows from:

=

baarctanβ 20 πβ ≤≤

( )b

Atba s−⋅⋅

=2

Critical peak impact pressures crp have been taken from Conolly [1974]. He gives measuredimpact pressures at a ship with a length of 112 meter over 30 per cent of the ship length fromforward. From this, a lower limit of crp has been assumed. This lower limit is presented inFigure 12.4–3.

Figure 12.4–3: Measured Impact Pressures of a 112 Meter Ship

These values have to be scaled to the actual ship size. Bow emergence and exceeding of thislimit is supposed to cause slamming.


305

This approach can be translated into local hull-shape-depending threshold values of thevertical relative velocity too:

p

crcr C

ps

⋅⋅

=ρ2

&

The vertical relative velocity, including a forward speed effect, of the water particles withrespect to the keel point of the ship is defined by:

θθζ

θζ

⋅−⋅+−=

⋅+−=

Vxz

xzDtD

s

bx

bx

b

b

&&&

&

with:( )

( )µωζωζ

µωζζ

cossin

coscos

⋅⋅−⋅⋅⋅−=

⋅⋅−⋅⋅=

beaex

beax

xkt

xkt

b

b

&

Then:

⋅−+

⋅−=

s

cr

s

s

ms

mDslamP

&

&

0

2

0

2

22exp

Notice that, because of including the forward speed effect, the spectral moment of thevelocities does not follow from the spectral density of the relative displacement as showed inthe definition of Ochi.The average period of the relative displacement is found by:

s

s

s

ss m

mmm

T&0

0

2

02 22 ⋅⋅=⋅⋅= ππ

Then the number of times per hour that a slam will occur follows from:

slamPT

Ns

hour ⋅=2

3600


306


307

13 Twin-Hull Ships

When not taking into account the interaction effects between the two individual hulls, thewave loads and motions of twin-hull ships can be calculated easily. Each individual hull has tobe symmetric with respect to its centre plane. The distance between the two centre planes ofthe single hulls ( Ty⋅2 ) should be constant. The co-ordinate system for the equations ofmotion of a twin-hull ship is given in Figure 12.4–1.

Figure 12.4–1: Co-ordinate System of Twin-Hull Ships

13.1 Hydromechanical Coefficients

The hydromechanical coefficients ija , ijb and ijc in this section are those of one individualhull, defined in the co-ordinate system of the single hull, as given and discussed before.


308

13.2 Equations of Motion

The equations of motion for six degrees of freedom of a twin-hull ship are defined by:

66

55

44

33

22

11

:Yaw

:Pitch

:Roll

Heave

:Sway

:Surge

TwThTzxTzz

TwThTyy

TwThTxzTxx

TwThT

TwThT

TwThT

XXII

XXI

XXII

XXz

XXy

XXx

=−⋅−⋅=−⋅=−⋅−⋅=−⋅∇⋅=−⋅∇⋅=−⋅∇⋅

φψθ

ψφρρρ

&&&&

&&

&&&&&&

&&

&&

in which:

T∇ volume of displacement of the twin-hull ship

TijI solid mass moment of inertia of the twin-hull ship

321 ,, ThThTh XXX hydromechanical forces in the x -, y - and z -directions

654 ,, ThThTh XXX hydromechanical moments about the x -, y - and z -axes

321 ,, TwTwTw XXX exciting wave forces in the x -, y - and z -directions

654 ,, TwTwTw XXX exciting wave moments about the x -, y - and z -axes


309

13.3 Hydromechanical Forces and Moments

The equations of motion for six degrees of freedom and the hydromechanical forces andmoments in here, are defined by:

θθθ ⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−

151515

131313

1111111

222

222

222

cba

zczbza

xcxbxaXTh

&&&&&&

&&&

ψψψφφφ

⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−

262626

242424

2222222

222

222

222

cba

cba

ycybyaXTh

&&&

&&&

&&&

θθθ ⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−

353535

333333

3131313

222

222

222

cba

zczbza

xcxbxaXTh

&&&&&&

&&&

ψψψφφφ

φφφ

⋅⋅+⋅⋅+⋅⋅+⋅⋅⋅+⋅⋅⋅+⋅⋅⋅+

⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−

464646

332

332

332

444444

4242424

222

222

222

222

cba

cybyay

cba

ycybyaX

TTT

Th

&&&

&&&

&&&

&&&

θθθ ⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−

555555

535353

5151515

222

222

222

cba

zczbza

xcxbxaXTh

&&&&&&

&&&

ψψψψψψφφφ

⋅⋅⋅+⋅⋅⋅+⋅⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−

112

112

112

666666

646464

6262626

222

222

222

222

cybyay

cba

cba

ycybyaX

TTT

Th

&&&

&&&

&&&

&&&

In here, Ty is half the distance between the centre planes.


310

13.4 Exciting Wave Forces and Moments

The first order wave potential for an arbitrary water depth h is defined in the new co-ordinatesystem by:

( )[ ][ ] ( )µµωζ

ωsincossin

coshcosh

⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅

⋅−

=Φ bbeab

w ykxkthk

zhkg

This holds that for the port side ( )ps and starboard ( )sb hulls the equivalent components ofthe orbital accelerations and velocities in the surge, sway, heave and roll directions are equalto:

( ) ( )( ) ( )

( ) ( )

( ) ( )µµωζω

µζ

µµωζω

µζ

µµωζµζ

µµωζµζ

sincoscoscos

sincoscoscos

sincossincos

sincossincos

*1

*1

*1

*1

*1

*1

*1

*1

⋅⋅+⋅⋅−⋅⋅⋅⋅⋅+

=

⋅⋅−⋅⋅−⋅⋅⋅⋅⋅+

=

⋅⋅+⋅⋅−⋅⋅⋅⋅⋅−=

⋅⋅−⋅⋅−⋅⋅⋅⋅⋅−=

Tbeaw

Tbeaw

Tbeaw

Tbeaw

ykxktgk

sb

ykxktgk

ps

ykxktgksb

ykxktgkps

&

&

&&

&&

( ) ( )( ) ( )

( ) ( )

( ) ( )µµωζω

µζ

µµωζω

µζ

µµωζµζ

µµωζµζ

sincoscossin

sincoscossin

sincossinsin

sincossinsin

*2

*2

*2

*2

*2

*2

*2

*2

⋅⋅+⋅⋅−⋅⋅⋅⋅⋅+

=

⋅⋅−⋅⋅−⋅⋅⋅⋅⋅+

=

⋅⋅+⋅⋅−⋅⋅⋅⋅⋅−=

⋅⋅−⋅⋅−⋅⋅⋅⋅⋅−=

Tbeaw

Tbeaw

Tbeaw

Tbeaw

ykxktgk

sb

ykxktgk

ps

ykxktgksb

ykxktgkps

&

&

&&

&&

( ) ( )( ) ( )

( ) ( )

( ) ( )µµωζω

ζ

µµωζω

ζ

µµωζζ

µµωζζ

sincossin

sincossin

sincoscos

sincoscos

*3

*3

*3

*3

*3

*3

*3

*3

⋅⋅+⋅⋅−⋅⋅⋅⋅+

=

⋅⋅−⋅⋅−⋅⋅⋅⋅+

=

⋅⋅+⋅⋅−⋅⋅⋅⋅−=

⋅⋅−⋅⋅−⋅⋅⋅⋅−=

Tbeaw

Tbeaw

Tbeaw

Tbeaw

ykxktgk

sb

ykxktgk

ps

ykxktgksb

ykxktgkps

&

&

&&

&&

From this follows the total wave loads for the degrees of freedom. In these loads on thefollowing pages, the ''Modified Strip Theory'' includes the outlined terms. When ignoringthese outlined terms the ''Ordinary Strip Theory'' is presented.


311

The exciting wave forces for surge are:

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ∫

∫

∫

∫

⋅++

⋅+⋅

⋅−⋅+

⋅+⋅⋅⋅

+

⋅+⋅+=

LbFKFK

bww

L be

L

bwwbe

LbwwTw

dxsbXpsX

dxsbpsdx

dMVN

dxsbpsdx

dNV

dxsbpsMX

'1

'1

*1

*1

'11'

11

*1

*1

'11

*1

*1

'111

ζζωω

ζζωω

ζζ

&&

&&&&

&&&&

The exciting wave forces for sway are:

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ∫

∫

∫

∫

⋅++

⋅+⋅

⋅−⋅+

⋅+⋅⋅⋅

+

⋅+⋅+=

LbFKFK

bww

L be

L

bwwbe

LbwwTw

dxsbXpsX

dxsbpsdx

dMVN

dxsbpsdx

dNV

dxsbpsMX

'2

'2

*2

*2

'22'

22

*2

*2

'22

*2

*2

'222

ζζωω

ζζωω

ζζ

&&

&&&&

&&&&

The exciting wave forces for heave are:

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ∫

∫

∫

∫

⋅++

⋅+⋅

⋅−⋅+

⋅+⋅⋅⋅

+

⋅+⋅+=

LbFKFK

bww

L be

L

bwwbe

LbwwTw

dxsbXpsX

dxsbpsdx

dMVN

dxsbpsdx

dNV

dxsbpsMX

'3

'3

*3

*3

'33'

33

*3

*3

'33

*3

*3

'333

ζζωω

ζζωω

ζζ

&&

&&&&

&&&&


312

The exciting wave moments for roll are:

( ) ( )

( ) ( )

( ) ( )

( ) ( )

32

'4

'4

*2

*2

'42'

42

*2

*2

'42

*2

*2

'424

TwTTw

L

bFKFK

bwwL be

L

bwwbe

L

bwwTw

XyXOG

dxsbXpsX

dxsbpsdx

dMVN

dxsbpsdx

dNV

dxsbpsMX

⋅+⋅+

⋅++

⋅+⋅

⋅−⋅+

⋅+⋅⋅⋅

+

⋅+⋅+=

∫

∫

∫

∫

ζζωω

ζζωω

ζζ

&&

&&&&

&&&&

The exciting wave moments for pitch are:

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ∫

∫

∫

∫

∫

∫

∫

∫

⋅⋅+−

⋅+⋅⋅

⋅−⋅−

⋅+⋅⋅⋅⋅

−+

⋅+⋅⋅−

⋅⋅+−

⋅+⋅⋅

⋅−⋅−

⋅+⋅⋅⋅⋅

−+

⋅+⋅⋅−=

LbbFKFK

bwwbL be

Lbwwb

be

Lbwwb

LbFKFK

bwwL be

Lbww

be

Lbwww

dxxsbXpsX

dxsbpsxdx

dMVN

dxsbpsxdx

dNV

dxsbpsxM

dxbGsbXpsX

dxsbpsbGdx

dMVN

dxsbpsbGdx

dNV

dxsbpsbGMX

'3

'3

*3

*3

'33'

33

*3

*3

'33

*3

*3

'33

'1

'1

*1

*1

'11'

11

*1

*1

'11

*1

*1

'115

ζζωω

ζζωω

ζζ

ζζωω

ζζωω

ζζ

&&

&&&&

&&&&

&&

&&&&

&&&&


313

The exciting wave moments for yaw are:

( ) ( )

( ) ( )

( ) ( )

( ) ( ) 1

'2

'2

*2

*2

'22'

22

*2

*2

'22

*2

*2

'226

TwT

LbbFKFK

bwwb

L be

L

bwwbbe

L

bwwbTw

Xy

dxxsbXpsX

dxsbpsxdx

dMVN

dxsbpsxdx

dNV

dxsbpsxMX

⋅+

⋅⋅++

⋅+⋅⋅

⋅−⋅+

⋅+⋅⋅⋅⋅

+

⋅+⋅⋅+=

∫

∫

∫

∫

ζζωω

ζζωω

ζζ

&&

&&&&

&&&&


314

13.5 Added Resistance due to Waves

The added resistance can be found easily from the definitions of the mono-hull ship by usingthe wave elevation at each individual centre line and replacing the heave motion z by:

( ) φ⋅+= Tyzpsz and ( ) φ⋅−= Tyzsbz

13.5.1 Radiated Energy Method

The transfer function of the mean added resistance of twin-hull ships according to the methodof Gerritsma and Beukelman [1972] becomes:

( ) ( )∫ ⋅

+

⋅

⋅−⋅

⋅⋅−=

L

ba

za

a

za

bea

aw dxsbVpsVdx

dMVNkR2*2*'

33'332 2

cosζζω

µζ

with:( ) ( ) ( )( ) ( ) ( )( ) ( )

( ) ( )µµωζω

ζ

µµωζω

ζ

φθθζ

φθθζ

sincossin

sincossin

*3

*3

*3

*3

*3

*

*3

*

⋅⋅+⋅⋅−⋅⋅⋅⋅+

=

⋅⋅−⋅⋅−⋅⋅⋅⋅+

=

⋅+⋅+⋅−−=

⋅−⋅+⋅−−=

Tbeaw

Tbeaw

Tbwz

Tbwz

ykxktgk

sb

ykxktgk

ps

yVxzsbsbV

yVxzpspsV

&

&

&&&&

&&&&

13.5.2 Integrated Pressure Method

The transfer function of the mean added resistance of twin-hull ships according to the methodof Boese [1970] becomes:

( )[ ]

( )[ ]

( )

( )sbzaae

pszaae

bb

w

sbL a

sba

a

xa

bb

w

psL a

sba

a

xa

a

aw

z

z

dxdxdy

hk

xkszg

dxdxdy

hk

xkszg

R

θζζ

θζζ

ζ

ζ

εεθωρ

εεθωρ

ζεµ

ζρ

ζεµ

ζρ

ζ

−⋅⋅⋅⋅∇⋅⋅+

−⋅⋅⋅⋅∇⋅⋅+

⋅

−⋅

⋅⋅

−⋅⋅−⋅⋅−−−⋅⋅⋅

⋅

−⋅

⋅⋅−⋅⋅−⋅⋅

−−−⋅⋅⋅=

∫

∫

cos21

cos21

tanh

coscos21

21

tanh

coscos21

21

2

2

2

2

2

2

2

with:( ) ( )( ) ( )( )( )( ) ( )( ) ( ) φθζ

φθζφθφθ

µµωζζµµωζζ

⋅+⋅+−=⋅−⋅+−=

⋅−⋅−=⋅+⋅−=

⋅⋅+⋅⋅−⋅⋅=

⋅⋅−⋅⋅−⋅⋅=

Tb

Tb

Tbx

Tbx

Tbea

Tbea

yxzsbsbs

yxzpspss

yxzsbz

yxzpsz

ykxktsb

ykxktps

sincoscos

sincoscos


315

13.6 Bending and Torsion Moments

According to Newton's second law of dynamics, the harmonic lateral, vertical and torsiondynamic loads per unit length on the unfastened disk of a twin-hull ship are given by:

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( )bT

bmTxxbT

bTwbThbT

bbT

bTwbThbT

mbbT

sbTwbThbT

bT

bTwbThbT

xqz

gxyzkxm

xXxXzxq

xzxm

xXxXxq

gzxyxm

AgxXxXxq

bGxxm

xXxXxq

21

'2''

'4

'414

'

'3

'33

''

'2

'22

'

'1

'11

,

2

⋅+⋅+⋅+⋅−⋅⋅−

++=

⋅−⋅−

++=

⋅+⋅−⋅+⋅−

⋅⋅⋅⋅+++=

⋅−⋅−

++=

φψφ

θ

φφψ

φρ

θ

&&&&&&

&&&&

&&&&&&

&&&&

In here:'

Tm mass per unit length of the twin-hull ship'

Txxk local sold mass radius of inertia for roll

sA sectional area of one hull

The calculation procedure of the forces and moments is similar to the procedure given beforefor mono-hull ships.


316


317

14 Numerical Recipes

Some typical numerical recipes, as used in the strip theory program SEAWAY, are describedin more detail here.

14.1 Polynomials

Discrete points can be connected by a first degree or a second degree polynomial, see Figure14.1–1-a,b.

Figure 14.1–1: First and Second Order Polynomials Through Discrete Points

14.1.1 First Degree Polynomial

A first degree - or linear - polynomial, as given in Figure 14.1–1-a, is defined by:( ) bxaxf +⋅=

with in the interval 0xxxm << the following coefficients:( ) ( )

( ) 00

0

0

xaxfb

xxxfxf

am

m

⋅−=

−−

=

and in the interval pxxx <<0 the following coefficients:

( ) ( )

( ) 00

0

0

xaxfb

xx

xfxfa

p

p

⋅−=

−−

=

Notice that only one interval is required for obtaining the coefficients in that interval.


318

14.1.2 Second Degree Polynomial

A second-degree polynomial, as given in Figure 14.1–1-b, is defined by:( ) cxbxaxf +⋅+⋅= 2

with in the interval pm xxx << the following coefficients:

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) 02

00

00

0

0

0

0

0

xbxaxfc

xxaxx

xfxfb

xx

xxxfxf

xx

xfxf

a

pp

p

mp

m

m

p

p

⋅−⋅−=

+⋅−−−

=

−−−

−−−

=

Notice that two intervals are required for obtaining these coefficients, valid in both intervals.


319

14.2 Integrations

Either the trapezoid rule or Simpson’s general rule can carry out numerical integration.SEAWAY uses Simpson's general rule as a standard. Then, the integration has to be carriedout over a number of sets of two intervals, see Figure 14.1–1-b. Numerical inaccuracies canbe expected when 00 xxxx pm −<<− or mp xxxx −<<− 00 . In those cases the trapezoid rulehas to be preferred, see Figure 14.1–1-a.SEAWAY makes the choice between the use of the trapezoid rule and Simpson's ruleautomatically, based on the following requirements:

Trapezoid rile, if: 2.00 <−−

mp

m

xxxx

or 0.50 >−−

mp

m

xxxx

Simpson’s rule, if: 0.52.0 0 <−−

<mp

m

xxxx

14.2.1 First Degree Integration

First-degree integration - carried out by the trapezoid rule, see Figure 14.1–1-a - means theuse of a linear function:

( ) bxaxf +⋅=

The integral over the interval 0xxp − becomes:

( ) ( )

p

pp

x

x

x

x

x

x

xbxa

dxbxadxxf

0

00

2

21

⋅+⋅⋅=

⋅+⋅=⋅ ∫∫

with:( ) ( )

( ) 00

0

0

xaxfb

xx

xfxfa

p

p

⋅−=

−−

=

Integration over two intervals results into:

( ) ( ) ( ) ( ) ( ) ( ) ( )2

000 ppmpmmx

x

xfxxxfxxxfxxdxxf

p

m

⋅−+⋅−+⋅−=⋅∫

14.2.2 Second Degree Integration

Second-degree integration - carried out by Simpson's rule, see Figure 14.1–1-b – has to becarried out over a set of two intervals. At each of the two intervals, the integrand is describedby a second-degree polynomial:

( ) cxbxaxf +⋅+⋅= 2


320

Then the integral becomes:

( ) ( )p

m

p

m

p

m

x

x

x

x

x

x

xcxbxa

dxcxbxadxxf

⋅+⋅⋅+⋅⋅=

⋅+⋅+⋅=⋅ ∫∫

23

2

21

31

with:( ) ( ) ( ) ( )

( ) ( ) ( )

( ) 02

00

00

0

0

0

0

0

xbxaxfc

xxaxx

xfxfb

xx

xxxfxf

xx

xfxf

a

pp

p

mp

m

m

p

p

⋅−⋅−=

+⋅−−−

=

−−−

−−−

=

Some algebra leads for the integration over these two intervals to:

( )

( )

( )( ) ( ) ( )

( )

3

2

2

2

0

00

000

2

0

00

mp

pp

mp

pm

mp

mm

pm

x

x

xx

xfxx

xxxx

xfxxxx

xx

xfxx

xxxx

dxxfp

m

−⋅

⋅−

−−−

+⋅−⋅−⋅

−

+⋅−

−−−

=⋅∫

14.2.3 Integration of Wave Loads

The total wave loads can be written as:( )

( ) ( )tFtF

tFF

eFwaeFwa

Fewaw

ww

w

⋅⋅⋅−+⋅⋅⋅=

+⋅⋅=

ωεωε

εω

ζζ

ζ

sinsincoscos

cos

The in-phase and out-phase parts of the total wave loads have to be obtained fromlongitudinal integration of sectional values. Direct numerical integration of bFwa dxF

w⋅⋅

ζε 'cos' and bFwa dxF

w⋅⋅−

ζε 'sin' over the ship length, L , require integration

intervals, bx∆ , which are much smaller than the smallest wave length, 10minλ≤∆ bx . Thismeans that a large number of cross sections are required.

This can be avoided by writing the integrands in terms of ( ) dxxxf ⋅⋅ cos1 and ( ) dxxxf ⋅⋅ sin2 ,

in which the integrands ( )xf 2,1 vary much slower over short wave lengths as the harmonicsitself.


321

The functions ( )xf 2,1 can be approximated by a second degree polynomial:

( ) cxbxaxf +⋅+⋅= 2

When making use of the general integral rules:

( ) xxxxdxxx

xxxdxxx

xdxx

sin2cos2cos

sincoscos

sincos

22 ⋅−+⋅⋅+=⋅⋅

⋅+=⋅⋅

+=⋅

∫∫

∫

and:

( ) xxxxdxxx

xxxdxxx

xdxx

cos2sin2sin

cossinsin

cossin

22 ⋅−−⋅⋅+=⋅⋅

⋅−=⋅⋅

−=⋅

∫∫

∫

the following expressions can be obtained for the in-phase and out-phase parts of the waveloads, integrated from mx through px , so over the two intervals mxx −0 and 0xxp − :

( ) ( )

( )

( )( ) ( )[ ]

( ) ( )

( )

( )( ) ( )[ ] p

m

p

m

p

m

p

m

p

m

p

m

p

m

p

m

p

m

p

m

p

m

p

m

p

m

p

m

xx

x

x

x

x

x

x

x

x

x

x

x

x

xx

x

x

x

x

x

x

x

x

x

x

x

x

xbxaxaxf

dxxcdxxxbdxxxa

dxxcxbxa

dxxxfdxxF

xbxaxaxf

dxxcdxxxbdxxxa

dxxcxbxa

dxxxfdxxF

sin2cos2

sinsinsin

sin

sin

cos2sin2

coscoscos

cos

cos

2

2

2

2

⋅+⋅⋅+⋅−−=

⋅⋅+⋅⋅⋅+⋅⋅⋅=

⋅⋅+⋅+⋅=

⋅⋅=⋅

⋅+⋅⋅+⋅−+=

⋅⋅+⋅⋅⋅+⋅⋅⋅=

⋅⋅+⋅+⋅=

⋅⋅=⋅

∫∫∫

∫

∫∫

∫∫∫

∫

∫∫

with coefficients obtained by:


322

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) 02

00

00

0

0

0

0

0

xbxaxfc

xxaxx

xfxfb

xx

xxxfxf

xx

xfxf

a

pp

p

mp

m

m

p

p

⋅−⋅−=

+⋅−−−

=

−−−

−−−

=

With this approach, the wave loads on a barge, for instance, can be calculated by using twosection intervals only for any length of the barge.


323

14.3 Derivatives

First and second degree functions, of which the derivatives have to be determined, have beengiven in Figure 14.3–1-a,b.

Figure 14.3–1: Determination of Longitudinal Derivatives

14.3.1 First Degree Derivative

The two polynomials - each valid over two intervals below and above 0xx = - are given by:

for 0xx < : ( ) mm bxaxf +⋅=for 0xx > : ( ) pp bxaxf +⋅=

The derivative is given by:

for 0xx < :( )

madx

xdf=

for 0xx > :( )

padx

xdf=

It is obvious that, generally, the derivative at the left-hand side of 0x - with index m (minus) -

and the derivative at the right-hand side of 0x - with index p (plus) - will differ:

zero ofright or (plus zero ofleft or (minus 00 pxxmxx dxdf

dxdf

≠

==

A mean derivative dxdf at 0xx = can be obtained by:

( ) ( )

mp

pxxp

mxxm

xx xx

dxdfxx

dxdfxx

dxdf

−

⋅−+

⋅−

=

==

=

00

0

00

14.3.2 Second Degree Derivative


324

The two polynomials - each valid over two intervals below and above 0xx = - are given by:

for 0xx < : ( ) mmm cxbxaxf +⋅+⋅= 2

for 0xx > : ( ) ppp cxbxaxf +⋅+⋅= 2

A derivative of a second degree function:( ) cxbxaxf +⋅+⋅= 2

is given by:( )

bxadx

xdf+⋅⋅= 2

This leads for 0xx < to:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )201021

212

10

102

21

101021

201021

212

10

102

21

201021

212

10

102

21

211021

2

2

0

1

2

mmmm

mmm

mmm

mmmm

xx

mmmm

mmm

mmm

xx

mmmm

mmm

mmm

mmmmm

xx

xxxxxx

xfxfxx

xfxfxx

xfxfxxxx

dxdf

xxxxxxxfxfxx

xfxfxx

dxdf

xxxxxxxfxfxx

xfxfxx

xfxfxxxx

dxdf

m

m

−⋅−⋅−

−⋅−−

−⋅−+

−⋅−⋅−⋅+

=

−⋅−⋅−

−⋅−+

−⋅−+

=

−⋅−⋅−

−⋅−+

−⋅−−

−⋅−⋅−⋅−

=

=

=

=

and for 0xx > to:


325

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )021201

012

12

122

01

121201

021201

012

12

122

01

021201

012

12

122

01

011221

2

2

0

1

2

xxxxxx

xfxfxx

xfxfxx

xfxfxxxx

dxdf

xxxxxx

xfxfxx

xfxfxx

dxdf

xxxxxx

xfxfxx

xfxfxx

xfxfxxxx

dxdf

pppp

ppp

ppp

ppppp

xx

pppp

ppp

ppp

xx

pppp

ppp

ppp

pppmp

xx

m

m

−⋅−⋅−

−⋅−−

−⋅−+

−⋅−⋅−⋅+

=

−⋅−⋅−

−⋅−+

−⋅−+

=

−⋅−⋅−

−⋅−+

−⋅−−

−⋅−⋅−⋅−

=

=

=

=

Generally, the derivative at the left-hand side of 0x - with index m (minus) - and the

derivative at the right-hand side of 0x - with index p (plus) - will differ:

zero ofright or (plus zero ofleft or (minus 00 pxxmxx dxdf

dxdf

≠

==

A mean derivative dxdf at 0xx = can be obtained by:

pm

pxxp

mxxm

xx dd

dxdfd

dxdfd

dxdf

+

⋅+

⋅

=

==

=

00

0

with:

( )

( )

( )( )01

0212

01

10

2021

10

32

32

xx

xxxx

xxd

xx

xxxx

xxd

p

ppp

p

p

m

mmm

m

m

−⋅

−⋅

−−−

=

−⋅

−⋅

−

−−=


326

14.4 Curve Lengths

Discrete points, connected by a first degree or a second degree polynomial, are given inFigure 14.4–1-a,b.

Figure 14.4–1: First and Second Order Curves

The curve length follows from:

( ) ( )∫

∫

+=

=

p

m

p

m

x

x

x

xmp

dydx

dss

22

14.4.1 First Degree Curve

The curve length of a first degree curve, see Figure 14.4–1-a, in the two intervals in the region

pm xxx << is:

( ) ( ) ( ) ( )20

20

20

20 yyxxyyxxs ppmmmp −+−+−+−=

14.4.2 Second Degree Curve

The curve length of a second degree curve, see Figure 14.4–1-b, in the two intervals in theregion pm xxx << is:

++−+⋅−

++++⋅+

⋅=2

112

11

200

200

2

1ln1

1ln1

pppp

pppppsmp

with:


327

( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )( ) ( )

( ) ( )4

sincos

sincossincos

sincossincos

sincos

sincos2

sin

cos

00

00

00

00

001

000

22

22

αα

αααααααα

αααα

π

α

α

⋅−+⋅−=

⋅−+⋅−⋅−+⋅−

−

⋅−+⋅−⋅−−⋅−

=

⋅−+⋅−⋅−+⋅−

⋅+=

−+−

−=

−+−

−=

yyxxp

yyxxyyxx

yyxxxxyy

p

yyxx

yyxxp

yyxx

yy

yyxx

xx

pp

pp

mm

mm

mm

pp

mpmp

mpmp

mp

mpmp

mp


328


329

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