Chapter 3 Small-Amplitude Water Wave Theory Formulation and Solution

3

Small-Amplitude WaterWave Theory Formulation

and Solution

Dedication

PIERRE SIMON LAPLACE

Pierre Simon Laplace (1749-1827) is well known for the equation thatbears his name . The Laplace equation is one of the most ubiquitousequations of mathematical physics (the Helmholtz, the diffusion, andthe wave equation being others ); it appears in electrostatics , hydrody-namics, groundwater flow, thermostatics , and other fields.

As had Euler, Laplace worked in a great variety of areas , applyinghis knowledge of mathematics to physical problems . He has beencalled the Newton of France.

He was born in Beaumont-en-Auge , Normandy, France, andeducated at Caen (1765-1767). In 1768 he became Professor of Mathe-matics at the Ecole Militaire in Paris . Later he moved to the EcoleNormale , also in Paris.

Napoleon appointed him Minister of the Interior in 1799, and hebecame a Count in 1806 and a Marquis in 1807, the same year that heassumed the presidency of the French Academy of Sciences.

A large portion of Laplace 's research was devoted to astronomy.He wrote on the orbital motion of the planets and celestial mechanicsand on the stability of the solar system. He also developed the hypothe-sis that the solar system coalesced out of a gaseous nebula.

In other areas of physics , he developed the theory of tides whichbears his name , worked with Lavoisier on specific heat of solids,studied capillary action , surface tension , and electric theory, and withLegendre, introduced partial differential equations into the study ofprobability. He also developed and applied numerous solutions (poten-tial functions) of the Laplace equation.

41

42 Small-Amplitude Water Wave Theory Formulation and Solution Chap. 3

3.1 INTRODUCTION

Real water waves propagate in a viscous fluid over an irregular bottom ofvarying permeability. A remarkable fact, however, is that in most cases themain body of the fluid motion is nearly irrotational. This is because theviscous effects are usually concentrated in thin "boundary" layers near thesurface and the bottom. Since water can also be considered reasonablyincompressible, a velocity potential and a stream function should exist forwaves. To simplify the mathematical analysis, numerous other assumptionsmust and will be made as the development of the theory proceeds.

3.2 BOUNDARY VALUE PROBLEMS

In formulating the small-amplitude water wave problem, it is useful toreview, in very general terms, the structure of boundary value problems, ofwhich the present problem of interest is an example. Numerous classical

Z 'f' dB

B.C.

...) specs .eBoundarycondrtIons (

Governingng

differentialequation

Region of interest (in general,can be any shape)

B.C. specified

(a)

Lateral boundary condition(LBC) (pseudo boundary)

Kinematic free surfaceboundary condition

Dynamic free surfaceboundary condition

X

w V20 = V2 0 = 0 (Governing differentialf LBC

u equation) f

Velocity components

Bottom boundary condition(kinematic requirement)

(b)

Figure 3.1 (a) General structure of two-dimensional boundary value problems.

(Note: The number of boundary conditions required depends on the order of the

differential equation .) (b) Two-dimensional water waves specified as a boundary

value problem.

Sec. 3.2 Boundary ValueProblems 43

problems of physics and most analytical problems in engineering may beposed as boundary value problems; however, in some developments, thismay not be apparent.

The formulation of a boundary value problem is simply the expressionin mathematical terms of the physical situation such that a unique solutionexists. This generally consists of first establishing a region of interest andspecifying a differential equation that must be satisfied within the region (seeFigure 3.1a). Often, there are an infinite number of solutions to the differen-tial equation and the remaining task is selecting the one or more solutionsthat are relevant to the physical problem under investigation. This selectionis effected through the boundary conditions, that is, rejecting those solutionsthat are not compatible with these conditions.

In addition to the spatial (or geometric) boundary conditions, there aretemporal boundary conditions which specify the state of the variable ofinterest at some point in time. This temporal condition is termed an "initialcondition." If we are interested in water waves, which are periodic in space,then we might specify, for example, that the waves are propagating in thepositive x direction and that at t = 0, the wave crest is located at x = 0.

In the following development of linear water wave theory, it will behelpful to relate each major step to the general structure of boundary valueproblems discussed previously. Figure 3.lb presents the region of interest, thegoverning differential equations, and indicates in a general manner theimportant boundary conditions.

3.2.1 The Governing Differential Equation

With the assumption of irrotational motion and an incompressiblefluid, a velocity potential exists which should satisfy the continuity equation

V.u=0 (3.1a)

or

V•V4=0 (3.1b)

As was shown in Chapter 2, the divergence of a gradient leads to the Laplaceequation, which must hold throughout the fluid.

V2(k =a2^ +a2(k +2O_ =0 (3.2)axe aye az2

The Laplace equation occurs frequently in many fields of physics andengineering and numerous solutions to this equation exist (see, e.g., the bookby Bland, 1961), and therefore it is necessary to select only those which areapplicable to the particular water wave motion of interest.

In addition, for flows that are nondivergent and irrotational, theLaplace equation also applies to the stream function. The incompressibility


or, equivalently, the nondivergent condition for two dimensions guaranteesthe existence of a stream function , from which the velocities under the wavecan be determined . Substituting these velocities into the irrotationalitycondition again yields the Laplace equation, except for the stream functionthis time,

awau0

ax az

or

v2W=a?w+ V=0ax aZ

(3.3a)

(3.3b)

This equation must hold throughout the fluid . If the motion had beenrotational, yet frictionless , the governing equation would be

VV = w (3.4)

where w is the vorticity.A few comments on the velocity potential and the stream function may

help in obtaining a better understanding for later applications . First, asmentioned earlier, the velocity potential can be defined for both two andthree dimensions , whereas the definition of the stream function is such that itcan only be defined for three dimensions if the flow is symmetric about anaxis (in this case although the flow occurs in three dimensions, it ismathematically two-dimensional ). It therefore follows that the stream func-tion is of greatest use in cases where the wave motion occurs in one plane.Second, the Laplace equation is linear ; that is, it involves no products andthus has the interesting and valuable property of superposition ; that is, if01 and 02 each satisfy the Laplace equation , then 03 = A41 + B02 also willsolve the equation , where A and B are arbitrary constants . Therefore, we canadd and subtract solutions to build up solutions applicable for differentproblems of interest.

3.2.2 Boundary Conditions

Kinematic boundary conditions . At any boundary, whether it is fixed,such as the bottom, or free, such as the water surface, which is free to deformunder the influence of forces, certain physical conditions must be satisfied bythe fluid velocities. These conditions on the water particle kinematics arecalled kinematic boundary conditions. At any surface or fluid interface, it isclear that there must be no flow across the interface; otherwise, there wouldbe no interface. This is most obvious in the case of an impermeable fixedsurface such as a sheet pile seawall.

The mathematical expression for the kinematic boundary conditionmay be derived from the equation which describes the surface that consti-tutes the boundary. Any fixed or moving surface can be expressed in terms of

Sec. 3.2 Boundary Value Problems 45

a mathematical expression of the form F(x, y, z, t) = 0. For example, for astationary sphere of fixed radius a, F(x, y, z, t) = x2 + yZ + z2 - a2 = 0. If thesurface varies with time , as would the water surface , then the total derivativeof the surface with respect to time would be zero on the surface . In otherwords , if we move with the surface, it does not change.

DF(x, y, z, t) aF OF OF OF=0=-+u-+v-+w- (3.5a)

Dt at ax ay az o^ F(X,y,z,O = 0

or

aF--u• VF = u. nJ VFat

(3.5b)

where the unit vector normal to the surface has been introduced asn=VF/IVFI.

Rearranging the kinematic boundary condition results:

u•n =-aF/aton F(x, y,z,t)=0

IVFI

where

(3.6)

IVFI =aF + - + -

ax ay az

This condition requires that the component of the fluid velocity normalto the surface be related to the local velocity of the surface . If the surface doesnot change with time, then u • n = 0; that is, the velocity component normalto the surface is zero.

Example 3.1

Fluid in a U-tube has been forced to oscillate sinusoidally due to an oscillatingpressure on one leg of the tube (see Figure 3.2). Develop the kinematic boundarycondition for the free surface in leg A.

Solution. The still water level in the U-tube is located at z = 0. The motion of the freesurface can be described by z = i7(t) = a cost, where a is the amplitude of the variationof n.

If we examine closely the motion of a fluid particle at the surface (Figure 3.2b),as the surface drops, with velocity w, it follows that the particle has to move with thespeed of the surface or else the particle leaves the surface. The same is true for a risingsurface. Therefore, we would postulate on physical grounds that

w=dndt -Mt)


Oscillating pressure

(b)

(a)

z=0

Figure 3.2 (a) Oscillating flow in a U-tube; (b) details of free surface.

where dry/dt = the rate of rise or fall of the surface. To ensure that this is formallycorrect, we follow the equation for the kinematic boundary condition, Eq. (3.6), whereF(z, t) = z - ?Kt) = 0. Therefore,

where n = Oi + Oj + 1k, directed vertically upward and u = ui + vj + wk, and carryingout the scalar product, we find that

w= anat

which is the same as obtained previously, when we realize that dry/dt = ark/at, as q isonly a function of time.

The Bottom Boundary Condition (BBC). In general, the lower bound-ary of our region of interest is described as z = -h(x) for a two-dimensionalcase where the origin is located at the still water level and h represents thedepth. If the bottom is impermeable, we expect that u • n = 0, as the bottomdoes not move with time. (For some cases, such as earthquake motions,obviously the time dependency of the bottom must be included.)

The surface equation for the bottom is F(x, z) = z + h(x) = 0. There-fore,

u•n=0 (3.7)

where

Ai+ lkVF _ dx

n_

I of I (dh/dx)2 + 1(3.8)

Sec. 3 .2 Boundary Value Problems 47

Carrying out the dot product and multiplying through by the square root, wehave

uA+w=0 onz=-h(x)

or

(3.9a)

w = -u dx on z = -h(x) (3.9b)

For a horizontal bottom , then , w = 0 on z = -h. For a sloping bottom, wehave

w dh(3.10)

u dx

Referring to Figure 3.3, it is clear that the kinematic condition states that theflow at the bottom is tangent to the bottom. In fact, we could treat the bottomas a streamline, as the flow is everywhere tangential to it. The bottomboundary condition, Eq. (3.7), also applies directly to flows in three dimen-sions in which h is h(x, y).

Kinematic Free Surface Boundary Condition (KFSBC). The free sur-face of a wave can be described as F(x, y, z, t) = z - tl(x, y, t) = 0, where?(x, y, t) is the displacement of the free surface about the horizontal plane,z = 0. The kinematic boundary condition at the free surface is

u • n =ark/at on

z = ri(x, y, t) (3.11a)(a,71ax) +

(017lay)2 + 1

z

Figure 3.3 Illustration of bottom boundary condition for the two- dimensionalcase.


where

-ai-anj+lkn - ax ay

(3.llb)(an/ax) + 017/8y) + 1

Carrying out the dot product yields

an 17 aw = + u a+ v (3.llc)at ax ay on ==Mx,y,r)

This condition, the KFSBC, is a more complicated expression than thatobtained for (1), the U-tube, where the flow was normal to the surface and(2) the bottom, where the flow was tangential. In fact, inspection of Eq. (3.1lc)will verify that the KFSBC is a combination of the other two conditions,which are just special cases of this more general type of condition.'

Dynamic Free Surface Boundary Condition . The boundary condi-tions for fixed surfaces are.relatively easy to prescribe, as shown in thepreceding section, and they apply on the known surface. A distinguishingfeature of fixed (in space) surfaces is that they can support pressure varia-tions. However, surfaces that are "free," such as the air-water interface,cannot support variations in pressure2 across the interface and hence mustrespond in order to maintain the pressure as uniform. A second boundarycondition, termed a dynamic boundary condition, is thus required on anyfree surface or interface, to prescribe the pressure distribution pressures onthis boundary. An interesting effect of the displacement of the free surface isthat the position of the upper boundary is not known a priori in the waterwave problem. This aspect causes considerable difficulty in the attempt toobtain accurate solutions that apply for large wave heights (Chapter 11).

As the dynamic free surface boundary condition is a requirement thatthe pressure on the free surface be uniform along the wave form, theBernoulli equation [Eq. (2.92)] with p,, = constant is applied on the freesurface, z = n(x, t),

ao +I(u2+w2)+p""+gz=C(t) (3.12)P

where p,, is a constant and usually taken as gage pressure, p,, = 0.

Conditions at "Responsive" Boundaries. As noted previously, an addi-tional condition must be imposed on those boundaries that can respond tospatial or temporal variations in pressure. In the case of wind blowing across

'The reader is urged to develop the general kinematic free surface boundary condition for a wavepropagating in the x direction alone.

Neglecting surface tension.


a water surface and generating waves, if the pressure relationship wereknown, the Bernoulli equation would serve to couple that wind field with thekinematics of the wave. The wave and wind field would be interdependentand the wave motion would be termed "coupled." If the wave were driven by,but did not affect the applied surface pressure distribution, this would be acase of "forced" wave motion and again the Bernoulli equation would serveto express the boundary condition. For the simpler case that is explored insome detail in this chapter, the pressure will be considered to be uniform andhence a case of "free" wave motion exists. Figure 3.4 depicts various degreesof coupling between the wind and wave fields.

Wind

Surface pressure distributionaffected by interaction of

wind and waves

(a)

Translating pressure field(not affected by waves)

(b)

p = atmospheric everywhere

(c)

Figure 3.4 Various degrees of air-water boundary interaction and coupling toatmospheric pressure field: (a) coupled wind and waves; (b) forced waves due tomoving pressure field; (c) free waves-not affected by pressure variations at air-water interface.


The boundary condition for free waves is termed the "dynamic freesurface boundary condition" (DFSBC), which the Bernoulli equationexpresses as Eq. (3.13) with a uniform surface pressure pn:

z zr ^- ^ P+ -n +

l- + ^ + gz = Qt), z = ri(x, t ) (3.13)

at p 2 L ax azwhere pn is a constant and usually taken as gage pressure, pn = 0.

If the wave lengths are very short (on the order of several centimeters),the surface is no longer "free." Although the pressure is uniform above thewater surface, as a result of the surface curvature, a nonuniform pressure willoccur within the water immediately below the surface film. Denoting thecoefficient of surface tension as a', the tension per unit length T is simply

T = a' (3.14)

Consider now a surface for which a curvature exists as shown in Figure3.5. Denoting p as the pressure under the free surface, a free-body forceanalysis in the vertical direction yields

T [-sin a I x + sin a I x+nx] + (p - pn) Ax + terms of order Axe = 0

in which the approximation an/ax -- sin a will be made. Expanding byTaylor's series and allowing the size of the element to shrink to zero yields

z17P= pn - a - (3.15)

axe

Thus for cases in which surface tension forces are important, thedynamic free surface boundary condition is modified to

CIOp n

0.'a2^- + - -- -+ - +

(a,,)2]+ gz = C(t), z = r7(x, t) (3.16)

at p p axe 2 ax az

which will be of use in our later examination of capillary water waves.

Lateral Boundary Conditions. At this stage boundary conditionshave been discussed for the bottom and upper surfaces. In order to completespecification of the boundary value problem, conditions must also be speci-

T

pOfFigure 3.5 Definition sketch forsurface element.

Sec. 3.2 Boundary Value Problems 51

fied on the remaining lateral boundaries. There are several situations that must be considered.

If the waves are propagating in one direction (say the x direction), conditions are two-dimensional and then "no-flow" conditions are appropriate for the velocities in the y direction. The boundary conditions to be applied in the x direction depend on the problem under consideration. If the wave motion results fiom a prescribed disturbance of, say, an object at x = 0, which is the classical wavemaker problem, then at the object, the usual kinematic boundary condition is expressed by Figure 3.6a.

Consider a vertical paddle acting as a wavemaker in a wave tank. If the displacement of the paddle may be described as x = S(z, t), the kinematic boundary condition is

where

Outgoing waves only __t

Figure 3.6 (a) Schematic of wavemaker in a wave tank; (b) radiation condition for wavemaker problem for region unbounded in x direction.


fled on the remaining lateral boundaries . There are several situations thatmust be considered.

If the waves are propagating in one direction (say the x direction),conditions are two -dimensional and then "no-flow" conditions are appropri-ate for the velocities in the y direction . The boundary conditions to beapplied in the x direction depend on the problem under consideration. If thewave motion results from a prescribed disturbance of, say, an object at x = 0,which is the classical wavemaker problem , then at the object, the usualkinematic boundary condition is expressed by Figure 3.6a.

Consider a vertical paddle acting as a wavemaker in a wave tank. If thedisplacement of the paddle may be described as x = S(z, t), the kinematicboundary condition is

u • n _ 8S(z, t) / 1 + TZ

at

n=

z

XN

1 + (as/az)Z

(a)

Outgoing waves only

(b)

Figure 3.6 (a) Schematic of wavemaker in a wave tank; (b) radiation conditionfor wavemaker problem for region unbounded in x direction.


or, carrying out the dot product,

as asl- w-=- (3.17)

U az at on x=S(zj)

which, of course, requires that the fluid particles at the moving wall followthe wall.

Two different conditions occur at the other possible lateral boundaries:at a fixed beach as shown at the right side of Figure 3.6a, where a kinematiccondition would be applied, or as in Figure 3.6b, where a "radiation"boundary condition is applied which requires that only outgoing waves occurat infinity. This precludes incoming waves which would not be physicallymeaningful in a wavemaker problem.

For waves that are periodic in space and time, the boundary conditionis expressed as a periodicity condition,

O(x, t) = di(x + L, t) (3.18a)

f(x, t) _ 4(x, t + T) (3.18b)

where L is the wave length and T is the wave period.

3.3 SUMMARY OF THE TWO -DIMENSIONAL PERIODICWATER WAVE BOUNDARY VALUE PROBLEM

The governing second-order differential equation for the fluid motion undera periodic two-dimensional water wave is the Laplace equation, which holdsthroughout the fluid domain consisting of one wave, shown in Figure 3.7.

Vet=0, 0<x<L, -h <z<r1 (3.19)

v2m=vz0 =o

BBC

I-I Periodic lateral boundary

condition (PLBC)

Figure 3.7 Boundary value problem specification for periodic water waves.

Sec. 3 . 4 Solution to Linearized Water Wave Boundary Value Problems 53

At the bottom , which is assumed to be horizontal , a no-flow condition applies(BBC):

W=0 on z = -h (3.20a)

or

- 4=0 onz=-h (3.20b)az

At the free surface, two conditions must be satisfied . The KFSBC, Eq. (3.11c),

-4 =an anon z= n(x,t) (3.11c)

az at ax axThe DFSBC, Eq. (3.13), with p,, = 0,

LO) + ( )2] + gn = C(t)ax azon z=n(x,t) (3.13)

Finally, the periodic lateral boundary conditions apply in both time andspace, Eqs. (3.18).

O(x, t) _ 4(x + L, t) (3.18a)

(k(x, t) = 0(x, t + T) (3.18b)

3.4 SOLUTION TO LINEARIZED WATER WAVEBOUNDARY VALUE PROBLEM FOR A HORIZONTALBOTTOM

In this section a solution is developed for the boundary value problemrepresenting waves that are periodic in space and time propagating over ahorizontal bottom. This requires solution of the Laplace equation with theboundary conditions as expressed by Eqs. (3.19), (3.20b), (3.11c), (3.13), and(3.18).

3.4.1 Separation of Variables

A convenient method for solving some linear partial differential equa-tions is called separation of variables . The assumption behind its use is thatthe solution can be expressed as a product of terms , each of which is afunction of only one of the independent variables . For our case,

O(x, z, t) = X(x) • Z(z) • T(t) (3.21)

where X(x) is some function that depends only on x, the horizontal coordi-nate , Z(z) depends only on z, and T(t) varies only with time . Since we know


that 0 must be periodic in time by the lateral boundary conditions, we canspecify T(t) = sin at . To find a, the angular frequency of the wave , we utilizethe periodic boundary condition, Eq. (3.18b).

sin at = sin a(t + T)

or

sin at = sin at cos aT + cos at sin aT

which is true for aT = 27r or a = 27r/T. Equally as likely, we could have chosencos at or some combination of the two : A cos at + B sin at. Since theequations to be solved will be linear and superposition is valid, we can defergeneralizing the solution in time until after the solution components havebeen obtained and discussed . The velocity potential now takes the form

/(x, z, t) = X(x) • Z(z) • sin at (3.22)

Substituting into the Laplace equation , we have

Z )•Z(z)•sinat +X(x). (z)• sinat=0dZ2

Dividing through by 0 gives us

l a2X l a2z-+--=0 (3.23)

X axe Z az2

Clearly, the first term of this equation depends on x alone , while the secondterm depends only on z. If we consider a variation in z in Eq . (3.23) holding xconstant, the second term could conceivably vary, whereas the first termcould not . This would give a nonzero sum in Eq . (3.23) and thus the equationwould not be satisfied. The only way that the equation would hold is if eachterm is equal to the same constant except for a sign difference , that is,

d2X(x)/dx2 = -k2(3.24a)

X(x)

d2Z(z)/dz2 = +k2(3.24b)

Z(z)

The fact that we have assigned a minus constant to the x term is not ofimportance , as we will permit the separation constant k to have an imaginaryvalue in this problem and in general the separation constant can be complex.

Equations (3.24) are now ordinary differential equations and may besolved separately. Three possible cases may now be examined depending onthe nature of k; these are for k real, k = 0, and k a pure imaginary number.Table 3 . 1 lists the separate cases . (Note that if k consisted of both a real and animaginary part, this could imply a change of wave height with distance,which may be valid for cases of waves propagating with damping or wavegrowth by wind.)

Sec. 3 .4 Solution to Linearized Water Wave Boundary Value Problems 55

TABLE 3.1 Possible Solutions to the Laplace Equation , Based on Separationof Variables

Character of k, the Ordinary DifferentialSeparation Constant Equations

Real

k2>0

Solutions

dx2 + k2X = 0 X(x) = A cos kx + B sin kx

d2Z_ k2Z=0 Z(z)=Ce" +De'' 'dz2

k=0

Imaginary

d2X0dx2

d-d?- Z=0Z(z)=Cz+D

k2<0,k=iIkI 2 2 - IkI2X=0 X(x) =Ae I k I x + Be I kI

I k I =magnitudeofk d2Z Ik12Z=0 Z(z)=Ccos Iklz+Dsin Iklzdz2

3.4.2 Application of Boundary Conditions

The boundary conditions serve to select, from the trial solutions inTable 3.1, those which are applicable to the physical situation of interest. Inaddition, the use of the boundary conditions allows determination of some ofthe unknown constants (e.g., A, B, C, and D).

Lateral periodicity condition . All solutions in Table 3.1 satisfy theLaplace equation; however, some of them are not periodic in x; in fact, thesolution is spatially periodic only if k is real3 and nonzero. Therefore, we haveas a solution to the Laplace equation the following velocity potential:

O(x, z, t) = (A cos kx + B sin kx) (Ce'z + De-k) sin at (3.25)

To satisfy the periodicity requirement (3.18a) explicitly,

A cos kx + B sin kx = A cos k(x + L) + B sin k(x + L)

=A (cos kx cos kL - sin kx sin kL)

+ B(sin kx cos kL + cos kx sin kL)

which is satisfied for cos kL =1 and sin kL = 0; which means that kL = 27r ork (called the wave number) = 2ir/L.

Using the superposition principle, we can divide 0 into several parts.Let us keep, for present purposes, only 0 = A cos kx(Ce"C2 + De '`) sin at. Lest

3For k = 0, A is zero. This ultimately yields 0 = B sin at.


this be thought of as sleight of hand, the B sin kx term will be added back in later by superposition.

Bottom boundary condition for horizontal bottom. Substituting in the bottom boundary condition yields

W=- -=- * A cos kx(kC? - sin at = 0 on z = -h (3.26) az

-Ak cos kx(Ce-kh - Dekh) sin at = 0

For this equation to be true for any x and t, the terms within the parentheses must be identically zero, which yields

The velocity potential now reads

4 = A cos kX(DeZkh p- + ~ e - ~ ~ ) sin at

or, factoring out ~ e k h , 4 AD^^^ cos kx(ek(h+z) + e-k(h+z)) sin at

or

4 = G cos kx cosh k(h + z) sin at

where G = 2ADekh, a new constant.

Dynamic free surface boundary condition. As stated previously, the Bernoulli equation can be used to specify a constant pressure on the surface ofthe water.Yet the Bernoulli equation must be satisfied on z = tl(x, t), which is a priori unknown. A convenient method used to evaluate the condition, then, is to evaluate it on z = Mx, t ) by expanding the value of the condition at z = 0 (a known location) by the truncated Taylor series.

(Bernoulli equation),,, = (Bernoulli eq~a t ion ) ,~

a (3.28) + q - (Bernoulli e q ~ a t i o n ) , ~ + - . .

az


this be thought of as sleight of hand, the B sin kx term will be added back inlater by superposition.

Bottom boundary condition for horizontal bottom . Substituting in the

bottom boundary condition yields

w = - ` = -A cos kx(kCe' - kDekz) sin at = 0 on z = -h (3.26)az

or

-Ak cos kx(Cek' - Dekh) sin at = 0

For this equation to be true for any x and t, the terms within the parenthesesmust be identically zero, which yields

C = De2kh

The velocity potential now reads

di = A cos kx(De2khekz.

+ De-k) sin at

or, factoring out Dekh,

0 = ADekh cos kx(ek(h+z) + ek(h+z))

sin at

or

0 = G cos kx cosh k(h + z) sin at (3.27)

where G = 2ADekh, a new constant.

Dynamic free surface boundary condition . As stated previously, the

Bernoulli equation can be used to specify a constant pressure on the surfaceof the water. Yet the Bernoulli equation must be satisfied on z = tXx, t), which

is a priori unknown. A convenient method used to evaluate the condition,then, is to evaluate it on z = rj(x, t) by expanding the value of the condition atz = 0 (a known location) by the truncated Taylor series.

(Bernoulli equation),-, = (Bernoulli equation),(3.28)

+ n a (Bernoulli equation), +az

or2 Z 2 2

9z-+u 2w gz - a^+u+ wC )^ at 2 ^^C a2O 1 a U2 + W) Qt)

az at 2 az z=a

where p =0 onz=r1.

Sec. 3.4 Solution to Linearized Water Wave Boundary Value Problems 57

Now for infinitesimally small waves , n is small, and therefore it isassumed that velocities and pressures are small ; thus any products of thesevariables are very small: rt << 1, but r' << r1, or url << ?l. If we neglect thesesmall terms, the Bernoulli equation is written as

a + g1l = C(t)at Z=

This process is called linearization. We have retained only the terms that arelinear in our variables.' The resulting linear dynamic free surface boundarycondition relates the instantaneous displacement of the free surface to thetime rate of change of the velocity potential,

?1 10,01 + C(t)

(3.29)gat Z-o g

If we substitute the velocity potential, as given by Eq. (3.27),

r1= Ga cos kx cosh k(h + z)cos at 1,=o + C(t)g g

(3.30)1 coshkh1 cos kx cos at + C(t)

L g J gSince by our definition >J will have a zero spatial and temporal mean,C(t) = 0.5 The terms within the brackets are constant ; therefore, 11 is given asa constant times periodic terms in space and time plus a function of time. Wecan rewrite r1 as

r= 2 coskxcosat (3.31)

The last substitution came about by comparing the analytical representationof r1 to the physical model, as shown in Figure 3.7. G can now be obtainedfrom

G= Hg2v cosh kh

The velocity potential is now

_ Hg cosh k(h + z)cos kx sin at

2a cosh kh(3.32)

The velocity potential is now prescribed in terms ofH, a, h, and k. The

"Linear in the sense that variables are only raised to the first power.5Had we not used p ( ,) = 0, how would C(t) be changed?


fmt three of these would be available from the data or alternatively the wave length might be known and a unknown.

Kinematic free surface boundary condiion. The remaining free surface boundary condition will be utilized to establish the relationship between aand k. Using theTaylor series expansion to relate the boundary condition at the unknown elevation, z = q(x, t ) to z = 0, we have

Again retaining only the terms that are linear in our small parameters, q, u, and w, and recalling that q is not a hnction of z, the linearized kinematic free surface boundary condition results:

Substituting for 4 and q gives us

H gk sinh k(h + z) --- cos kx sin at ),a 2 a cosh kh

Rewriting this equation as dhlgkh = tanh kh and plotting each term versus kh for a particular value of d h / g yields Figure 3.8. The solution is determined by the intersection of the two curves. Therefore, the equation has only one solution or equivalently one value of k for given values of a and h.

Noting that by definition a propagating wave will travel a distance of one wave length L, in one wave period T, and recalling that a = 2n/T and k = 2n/L, it is clear that the speed of wave. propagation C can be expressed fiom Eq. (3.34) as


first three of these would be available from the data or alternatively the wavelength might be known and a unknown.

Kinematic free surface boundary condition . The remaining free sur-face boundary condition will be utilized to establish the relationship betweena and k. Using the Taylor series expansion to relate the boundary condition atthe unknown elevation, z = rI(x, t) to z = 0, we have

(w- an -u anat ax).' all =(w- a-u a'')

x z.

+,1 a I w --- u -I +... =0az \ at ax z=O

Again retaining only the terms that are linear in our small parameters, rl, u,and w, and recalling that n is not a function of z, the linearized kinematic freesurface boundary condition results:

W L17at

or

00az

z-O

__ anz=O at

Substituting for 0 and n gives us

- H gk sinh k(h+ z) cos kx sin at I z=o

2 a cosh kh

=-Hacos kxsin at2

or

a2 = gk tanh kh

(3.33a)

(3.33b)

(3.34)

Rewriting this equation as a2h/gkh = tanh kh and plotting each term versuskh for a particular value of a'h/g yields Figure 3.8. The solution is deter-mined by the intersection of the two curves. Therefore, the equation has onlyone solution or equivalently one value of k for given values of a and h.

Noting that by definition a propagating wave will travel a distance ofone wave length L, in one wave period T, and recalling that a = 2n/T and

k = 2n/L, it is clear that the speed of wave propagation C can be expressedfrom Eq. (3.34) as

2

^ T - g L tanh kh


Figure 3 .8 Illustrating single root to dispersion equation.

or

C2=T2=ktanhkh (3.35)

A similar algebraic manipulation of Eq . (3.34) will yield a relationship for thewave length,

L = g T2 tanh2nh

(3.36)2ir L

In deep water, kh is large and tanh 27th/L = 1.0; therefore, L = LO = g7/27r,where the zero subscript is used to denote deep water values. In general, then,

L = Lo tanh kh (3.37)

Thus the wave length continually decreases with decreasing depth for aconstant wave period.

Equations (3.34), (3.35), and (3.37), which are really the same equationexpressed in slightly different variables, are referred to as the "dispersion"equation, because they describe the manner in which a field of propagatingwaves consisting of many frequencies would separate or "disperse" due to thedifferent celerities of the various frequency components.

The wave speed, or celerity, C, has been defined as C = LIT. Therefore,

C = T° tanh kh (3.38a)

10°

10-1

10-

sinh khcosh kh

sink kh

`

CG

"00 0 1 / Co

ko L C

k Lo' Co

1000, zLo = 2 = 5.12 T2(ft) = 1.56 T2(m)

A ll

- CO = 2T = 5.12 T(ft/s) = 1.56 T(m/s)

Deep water waves

Shallow water waves Intermediate waves

10-zh/L0

10-,

Figure 3 .9 Variation of various wave parameters with h/Lo.


or

C= Cotanh kh (3.38b)

since, as will be shown later, the wave period does not change with depth.Waves of constant period slow down as they enter shallow water. Figure 3.9presents , as a function ofh/Lo, the ratio C/Co (= L/Lo = ko/k) and a numberof other variables commonly occuring in water wave calculations . This figureprovides a convenient graphical means to determine intermediate and shal-low water values of these variables.

3.4.3 Summary of Standing Waves

One solution of the boundary value problem for small-amplitudewaves has been found to be

_ H g cosh k(h + z)cos kx sin at

2 a cosh kh

rI(x, t) = g Hcos kxcos atZ=0 2

(3.39)

where a2 = gk tanh kh.The wave form is shown in Figure 3.10. At at = it/2, the wave form is

zero for all x, at at = 0, it has a cosine shape and at other times, the samecosine shape with different magnitudes. This wave form is obviously a"standing wave," as it does not propagate in any direction. At positionskx = it/2, and 3ir/2, and so on, nodes exist; that is, there is no motion of thefree surface at these points. Standing waves often occur when incomingwaves are completely reflected by vertical walls. At which phase positionwould the wall be located? See Figure 4.6 for a hint.

L/4 \ J/ 3L/4

n(x, T/2)

Figure 3.10 Water surface displacement associated with a standing water wave.


3.4.4 Progressive Waves

Consider another standing wave,

H g cosh k(h + z)sin kx cos at (3.40)^(x, z, t) = Q

cosh kh2

This velocity potential is also a solution to the Laplace equation and all theboundary conditions, as may be verified readily. It is, in fact, one of thesolutions that we discarded. It differs from the previous solution in that the xand t terms are 90° out of phase. The associated water surface displacement is

Ir^(x, t) =1

g 8t4

Z-0

_ _ 2H sin kx sin at (3.41)

as determined from the linearized DFSBC. Remembering that the Laplaceequation is linear and superposition is valid, we can add or subtract solutionsto the linearized boundary value problem to generate new solutions. If wesubtract the present velocity potential in Eq. (3.40) from the previoussolution we had, Eq. (3.32), we obtain

H g cosh k(h + z)(cos kx sin at - sin kx cos at)

Q cosh kh2

--Hgcoshk(h+z)sin(kx-Qt)

2 a cosh kh

(3.42)

This new velocity potential has a water surface elevation, given as

n(x , t)=g ao H cos (kx - at) (3.43)

Z^ 2

Had we just subtracted the two >q(x, t) corresponding to the two velocitypotentials, we would have had

n(x,t)= H cos kx cos at+H sin kx sin at= H cos (kx - at)

which is the same result . This should not have been a surprise , as the totalboundary value problem has been linearized and superposition is valid for allvariables in the problem.

Examining the equation for the water surface profile , it is clear that thiswave form moves with time . To determine the direction of movement, let usexamine the same point on the wave form at two different time values,t, and t2. The x location of the point also changes with time . In Figure 3.11,the locations of the point at time t, and t2 are shown . The speed at which the


Figure 3.11 Characteristics of a propagating wave form.

wave propagated from one point to the other is C, given as

C x2 -x1=

t2 - ti

We further point out that the same point on the wave crest implies that we areexamining the wave at the same phase, that is , at constant values of theargument of the trigonometric function of x and t. Therefore, we expect that

?1(x1, t1) = n(x2, t2)

or, in fact,

kx1-atl=kx2-ate

k(xi - x2) = a(t1- t2)

or

a27r/T-C-XI -X2 X2-XIk 27r/L t1-t2 t2-t1

as before . Therefore, if t2 > t1, x2 > xi, the wave form propagates from left toright . Had the argument of the trigonometric function been (kx + at), thewaves would propagate from right to left (i.e., in the negative x direction).

Simplifications for shallow and deep water . The hyperbolic functionshave convenient shallow and deep water asymptotes, and often it is helpful touse them to obtain simplified forms of the equations describing wavemotion. For example, the function cosh kh, which appears in the denomina-tor for the velocity potential, is defined as

ek'+ek'cosh kh =

2

For a small argument , the exponential function ez can be expanded to z = khin a Taylor series about zero as

e(o+w,) = e° +dez kh + - (kh )2 + .. .

dz =-o dz2 z=o 2!


or

zek''=1+kh +(k2) +•••

Of course, a-k' would then equalz

ekk= 1-kh +(kh) +2

Therefore , for small kh,

cosh kh I [(1 +kh +(2`)2 +(1 -kh +(k2)2...A

1 + (kh)2

2

For large kh, cosh kh = e' /2 as akh becomes quite small . Table 3.2 presentsthe asymptotes.

TABLE 3.2 Asymptotic Forms of Hyperbolic Functions

Function Large kh Small kh

cosh kh ew'/2 1sinh kh e''/2 khtanh kh 1 kh

It is worthwhile to distinguish the regions within which these asymp-totic approximations become valid. Figure 3 . 12 is a plot of hyperbolicfunctions together with the asymptotes , f, = kh, f2 = 1.0, f3 = e' /2. The per-centage values presented in Figure 3.12 represent, for particular ranges of kh,the errors incurred by using the asymptotes rather than the actual value of thefunction . The largest error is 5%. The lower scale on the figure is the relativedepth. Note that due to this dimensionless representation a 200-m-long wavein 1000 m of water has the same relative depth as a 0.2-m wave in 1 m ofwater. Limits for three regions are denoted in the figure: kh < 7l/10,is/10 < kh < n, and kh > it. These regions are defined as the shallow water,intermediate depth , and the deep water regions, respectively. It may bejustified to modify the limits of these regions for particular applications.

The dispersion relationship in shallow and deep water . The disper-sion relationship for shallow water reduces in the following manner:

aZ = gk tanh kh = gk2h

0. 19%(Both sinh and cosh)

h kh i h khcos s n

Of,

I3(kh)ekk

= 2

-

5%

- - / /-eO

000,

000,^^- ft (kh) = kh

0000 -Arf2 (kh) = 1.0 0.4%

tank kh

1.5%3.3%

7

6

5

4

3

2

1

0 it/ 10

1/20

Shallow water waves(long waves)

1

kh

h/L

2

Intermediate depthwaves

3 a

1/2

7Deep water waves

(short waves)

Figure 3.12 Relative depth and asymptotes to hyperbolic functions.

65


or

and

-=C2=ghk2

C= (3.44)

The wave speed in shallow water is determined solely by the water depth.Recall that the definition of shallow water is based on the relative depth. Forthe ocean, where h might be -_ 1 km, a wave with a length of 20 km is inshallow water. For example, tsunamis , which are waves caused by earth-quake motions of the ocean boundaries, have lengths much longer than this.The speed in the ocean basins for long waves would be about 100 m/s(225 mph).

For deep water, kh > 7C,

a2 = gk tanh kh ,gk

L = Lo

where

Lo =g

T2-_ 5.127' (English system of units, ft)

-27n 1.56T2 (SI units , meters)

and (3.45)

Co =g _ (5.12T (English system of units, ft/s)-T-2ic 1.56T (SI units, m/s)

3.4.5 Waves with Uniform Current Uo

As an example of the procedure just followed for the solution forprogressive and standing waves, it is instructive to repeat the process for adifferent case: water waves propagating on a current. For example, for wavesin rivers or on ocean currents, a first approximation to the waves andcurrents is to assume that the current is uniform over depth and horizontaldistance and flowing in the same direction as the waves.

An assumed form of the velocity potential will be chosen to representthe uniform current Uo and a progressive wave, which satisfies the Laplace

equation.

0 = -Uox + A cosh k(h + z) cos (kx - at) (3.46)

The form of this solution guarantees periodicity of the wave in space andtime and satisfies the no-flow bottom boundary condition . It remains neces-


sary to satisfy the linearized form of the KFSBC and the DFSBC. Yet wecannot just apply the forms that we arrived at earlier, as errors would beincurred because the velocity Uo is no longer necessarily small; we mustrederive the linear boundary conditions.

The dynamic free surface boundary condition. Again, we will expandthe Bernoulli equation about the free surface on which a zero gage pressure isprescribed.

[!(u2+w2)+gz_]1(u2+w) +gz- 12 at [2 at Jz^

(3.47)

+r^ (u2 +w2)+gz- + • • • =C(t)az 2 at I=.Now the horizontal velocity is

u = -.= Uo + Ak cosh k(h + z) sin (kx - at)ax

Therefore, the u2 term is

u2= U+2AkUo cosh k(h +z)sin (kx-at)

+ A2k2 cosh2 k(h + z) sine (kx - at)

For infinitesimal waves , it is expected that the wave-induced horizontalvelocity component would be small (i.e., Ak small), and therefore (Ak)2would be much smaller. We will then neglect the last term in the equationabove.

The linearized Bernoulli equation [i.e., dropping all terms of order(Ak)2], evaluated on z = 0, is now

I[ U01 + 2AkUo cosh kh sin (kx - at)]

- Aa cosh kh sin (kx - at) + gn = C(t)

or

rj(x, t) = - Zg + gal 1 - U°k I cosh kh sin (kx - at) + C(t) (3.48)

To determine the Bernoulli term C(t), we average both sides of Eq. (3.48)over space. Since the space average of rt(x, t) is taken to be zero, it is clear thatC(t) = constant = UZ0/2g. Also, if we define a water surface displacement,rI(x, t) = H/2 sin (kx - at), then

A gH (3.49)2a(l - UoIC) cosh kh


The kinematic free surface boundary condition . The remainingboundary condition to be satisfied is the linearized form of the KFSBC.

an ao an = - 010 Zat ax ax az'

Expanding about the still water level, we have

an 010 an a (a'7 a0 anat ax ax + n az at ax ax +

a- - n a Cadaz az azz=0

or, retaining only the linear terms,

an+Uoan=-!, z=0 (3.50)at ax az

Substituting for n and 0 yields the following dispersion equation for thecase of a uniform current Uo:

oz - gktanhkh

(1 - Uo/C)2

or, another form can be developed by using the relationship a = kC:z

a2 1 - UUok = gk tanh kh

or

(3.51)

a = Uok + gk tanh kh (3.52)

The second term on the right-hand side is the angular frequency formulaobtained without a current.

In terms of the celerity, the dispersion relationship can be written as

(C - Uo)z =k

tanh kh (3.53)

It is worthwhile noting that it is possible to solve the preceding problemof a uniform current simply by adopting a reference frame which moves withthe current Uo. With reference to our new coordinate system , there is nocurrent and the methods , equations , and solutions obtained are thereforeidentical to those obtained originally for the case of no current.

When relating this moving frame solution for a stationary referencesystem , it is simply necessary to recognize that (1) the wavelength is the samein both systems ; (2) the period T relative to a stationary reference system isrelated to the period ' relative to the reference system moving with the


current U° by

T_ T' Uo (3.54)1+U°/C'=T 1-C

where C is the speed relative to the moving observer ; and (3) the total waterparticle velocity is Uo + uW , where u. is the wave-induced component. It isnoted that in the case of arbitrary depth , when T and h are given, it isnecessary to solve for the wave length from Eq. (3.54) by iteration.'

For shallow water, we have , from Eq. (3.53),

C=T,L= Uo+ Igh (3.55)

That is, since the celerity of the wave is independent of wave length, it issimply increased by the advecting current Uo. For deep water, thecorresponding result is determined by solving Eq. (3.53) for C using thequadratic solution and replacing k with Q/C, that is,

C=1 U " + U-°$+l g (3.56)2G) 4 Q

For small currents with respect to C (i.e., Uo < g/Q),

C^ +2UoQ

Capillary waves . As indicated in Eq. (3.16), the surface tension at thewater surface causes a modification to the dynamic free surface boundarycondition. To explore the effects of surface tension, we proceed as before bychoosing a velocity potential of the form

0=Acosh k(h+z)sin(kx-at) (3.57)

which is appropriate for a progressive water wave, satisfies the Laplaceequation, and all boundary conditions except those at the upper surface. Thesurface displacement associated with Eq. (3.57) will be of the form

n=H2 cos(kx - at) (3.58)

Substituting Eqs. (3.57) and (3.58) into the linearized form of Eq. (3.16), andemploying the linearized form of the kinematic free surface boundarycondition, Eq. (3.33a), the dispersion equation is found to be

zCZ = g 1 + a'k f tanh kh (3.59)

k pg/

'This technique has been applied to nearly breaking waves by Dalrymple and Dean (1975).


and it can be seen that the effect of surface tension is to increase the celerityfor all wave frequencies. The effect of surface tension can be examined mostreadily by considering the case of deep water waves.

CZ=k+k (3.60)p

That is, the contributions due to the speed of short waves (large wavenumbers) is small due to the effect of gravity and large due to the effect ofsurface tension. There is a minimum speed C,n at which waves can propagate,found in the usual way:

(3.61)

which leads to

km = Y d (3.62)

C,= Pg+ pg=2 r (3.63)

That is, the contributions from gravity and surface tension to CZ„, are equal.For a reasonable value of surface tension , a= 7.4 x 10-2 N/m, C= 23.2 cm/s,which occurs at a wave period of approximately 0.074 s. Figure 3.13 presents

C2

c2M

0 1 2

k

L.

3 4

Figure 3.13 Capillary and gravitational components of the square of wave celer-ity in deep water.

Sec. 3 .5 Appendix : Approximate Solutions to the Dispersion Equation 71

the relationship

C2 1 1 k ll

C„, 2 k/km + km

3.4.6 The Stream Function for Small-AmplitudeWaves

(3.64)

For convenience, the velocity potential has been used to develop thesmall-amplitude wave theory, yet often it is convenient to use the streamfunction representation . Therefore, we can use the Cauchy-Riemann equa-tions, Eqs. (2.82), to develop them from the velocity potentials.

Progressive waves.

H g cosh k(h + z)^(x, z, t) sin (kx - at) (3.65)

2 a cosh kh

v./(x, z, t)H g sinhk(h + z) )

cos (kx - at) (3.66)a cosh kh

It is often convenient for a progressive wave that propagates withoutchange of form to translate the coordinate system horizontally with the speedof the wave , that is, with the celerity C, as this then gives a steady flowcondition.

w=Cz-Hgsinhk(h+z) cos kx(3.67)

2 a cosh kh

Standing waves . From before,

H g cosh k(h + z)cos kx sin at (3.68)

2 a cosh kh

Hgsinhk(h+z)sin kx sin at (3.69)W

a cosh kh2

The streamlines and velocity potential for both cases are shown in Figure3.14. The streamlines and potential lines are lines of constant v and 0.

3.5 APPENDIX : APPROXIMATE SOLUTIONS TO THEDISPERSION EQUATION

The solution to the dispersion relationship, Eq. (3.34), fork is not difficult toobtain for given a and h. However, since the relationship is a transcendental


---- Streamlines

Progressive wave,stationary reference

frame

Velocity potential

Progressive wave,reference frame moving

with speed of wave

Standing wave,stationary reference

frame

Figure 3.14 Approximate streamlines and lines of constant velocity potential for vari-ous types of wave systems and reference frames.

equation, in that it is not algebraic, graphical (see Figure 3.8) and iterativetechniques are used (see Problem 3.15).

Eckart (1951) developed an approximate wave theory with a corre-sponding dispersion relationship,

a' =gk tanh hg

This can be solved directly for k and generally is in error by only a fewpercent. This equation therefore can be used as a first approximation to k foran iterative technique or can be used to determine k directly if accuracy is nota paramount consideration.

Recently, Hunt (1979) proposed an approximate solution that can besolved directly for kh:

(kh)2=Y2+

y6

1 + I dnynn=1

where y = a2h/g = koh and di = 0.666 ..., d2 = 0.355 .... d3 = 0.1608465608, d4= 0.0632098765 , d5 = 0.0217540484 , and d6 = 0 .0065407983. The last digits ind1 and d2 are repeated seven more times . This formula can be convenientlyused on a programmable calculator.

The wave celerity was also obtained

C2- = [y + (1 + 0.6522y + 0.4622y2 + 0.0864y4 + 0.0675y5) '] 'gh

which is accurate to 0.1% for 0 < y < oo.

Chap .3 Problems 73

REFERENCES

BLAND, D. R., Solutions of Laplace's Equation, Routledge & Kegan Paul , London,1961.

DALRYMPLE, R. A., and R. G. DEAN, "Waves of Maximum Height on UniformCurrents," J. Waterways, Harbors Coastal Eng. Div., ASCE, Vol. 101, No. WW3,pp. 259-268,1975.

ECKART, C., "Surface Waves on Water of Variable Depth," SIO 51-12 , Scripps Instituteof Oceanography, Aug. 1951.

HUNT, J. N., "Direct Solution of Wave Dispersion Equation ," J. Waterways, Ports,Coastal Ocean Div., ASCE, Vol. 105, No. WW4, pp. 457-459,1979.

PROBLEMS

3.1 The linearization of the kinematic and dynamic free surface boundary condi-tions involved neglecting nonlinear terms. Show , for both the conditions, thatthis linearization implies that

kH<<12

3.2 Near the bow of a moving submarine, the hull can be represented as a movingparabola,

D(z-A)2=-(x- Ut)

where U is the speed of the submarine , A represents the depth of the centerlineof the submarine below the free surface , and D is a constant.(a) Plot the hull shape at t = 0 and t = 1 s if the submarine is moving at 2 m/s.(b) Determine the kinematic boundary condition at the hull.

3.3 The equation for the stationary boundary C(x) of an incompressible fluid is

C(x) = Acr"

The horizontal velocity component may be regarded to be approximatelyuniform in the z direction. If u(x = 0) = 40 cm/s, A = 30 cm, andK = 0.02 cm , calculate w at the upper boundary for x = 50 cm.


3.4 The equation for the upper moving boundary C(x, t) of an incompressible fluid is

Cu(x, t) = ~ e ~ ~ - ~ ' )

The lower boundary C is expressed by

C(x, 1) = 0

(a) Sketch the boundaries for t = 0. (b) Discuss the motional characteristics of the upper boundary (i.e., speed and

direction). (c) The horizontal velocity component (u) may be regarded to be approxi-

mately uniform in the z direction. If

calculate w at the upper boundary for x = 50 cm and t = 10 s.

3.5 Using separation of variables, solve in cylindrical coordinates the problem of steady flow past a cylinder. Given Laplace's equation

1 1 4, + - 4, + - 4- = 0 in two dimensions r ?

in which the subscripts denote partial differentiation with respect to the subscripted variable. The boundary conditions are

4 = Ur cos 8 at r large

and

brIm=O

3.6 A two-dimensional horizontal flow is described by

44x9 Y) = 10(x2 - y2)

Find the point of maximum pressure ifp = 0 at (x, y) = (1, 1).

3.7 A wave field is observed by satellite. The wave lengths are determined to be 312 m in deep water and 200 m over the continental shelf. What is the shelf depth?

3.8 Formulate the boundary value problem for the situation below, which represents a model to study the effects of waves on a harbor with a narrow entrance. The stroke S of the wavemaker is considered to be small compared to the depth h.


3.4 The equation for the upper moving boundary C„ (x, t) of an incompressiblefluid is

C„ (x, t) = Ae-(kx-Mt)

The lower boundary Cr is expressed by

C,(x, t) = 0

A 30 cm

k 0.02 cm'

M=0.1 s'

(a) Sketch the boundaries for t = 0.(b) Discuss the motional characteristics of the upper boundary (i.e., speed and

direction).(c) The horizontal velocity component (u) may be regarded to be approxi-

mately uniform in the z direction. If

u(x = 0, t = 10 s) = 40 cm/s

calculate w at the upper boundary for x = 50 cm and t = 10 s.

3.5 Using separation of variables , solve in cylindrical coordinates the problem ofsteady flow past a cylinder . Given Laplace's equation

0' + 1 0' + 1r12

Oee = 0 in two dimensions

in which the subscripts denote partial differentiation with respect to thesubscripted variable. The boundary conditions are

0 = Ur cos 9 at r large

and

OrI r-c =0

3.6 A two-dimensional horizontal flow is described by

O (X, y) = 10(x2 - y2)

Find the point of maximum pressure ifp = 0 at (x, y)

3.7 A wave field is observed by satellite . The wave lengths are determined to be312 m in deep water and 200 m over the continental shelf . What is the shelfdepth?

3.8 Formulate the boundary value problem for the situation below, whichrepresents a model to study the effects of waves on a harbor with a narrowentrance . The stroke S of the wavemaker is considered to be small compared tothe depth h.

Chap.3 Problems 75

-4-i-►

Flae wavemakermonic(simple harmonic

motion) -

Plan view

- 11 12

1 // h

Elevation view

3.9 Set up, but do not solve, the complete two-dimensional (x, z, t) boundary valueproblem as illustrated, which was designed to simulate earthquake motions ofthe continental shelf. The sloping bottom oscillates with a period T and has anamplitude a. State all assumptions.


3.10 A horizontal cylindrical wavemaker is oscillating vertically in the free surface.Examining the two-dimensional problem shown below, develop the kinematicboundary condition for the fluid at the cylinder wall. Discuss the results.

t=0T

t=-4

3Tt=-

4

where T is period of oscillation.3.11 The stream function for a progressive small-amplitude wave is

w--H9sinhk(h+z)cos(kx-at)2 o cosh kh

Draw the streamlines for t = 0, when T = 5 s, h = 10 m, and H = 2.0 m.

3.12 You are on a ship (100 m in length) on the deep ocean traveling north. The(regular) waves are propagating north also and you note two items of informa-tion: (1) when the ship bow is positioned at a crest, the stern is at a trough, and(2) a different crest is positioned at the bow every 20 s.(a) Do you have enough information to determine the ship speed?(b) If the answer to part (a) is "no," what additional item(s) of information are

required?(c) If the answer to part (a) is "yes," what is the ship speed?

3.13 A tsunami is detected at 12:00 h on the edge of the continental shelf by awarning system. At what time can the tsunami be expected to reach theshoreline?

Chap .3 Problems 77

3.14 A rigid sinusoidal form is located as shown in the sketch . The form is forced tomove in the +x direction at speed V.(a) Derive an expression for the velocity potential for the water motion

induced by the moving form.(b) Evaluate p, - p, for the following cases:

(1) V2 < 9 tanh kh

(2)V2=ktanhkh

(3) V2 >

k

tanh kh

where p. and p, denote the pressure just below the form at the crest andtrough, respectively.

(c) Discuss the special significance of b(2).

L

Rigid form

HrvX

3.15 Develop an iterative technique to solve the dispersion relationship fork givena and h. Note: It is somewhat easier to first solve for kh. (Hint: A Newton-Raphson technique could be used.)

3.16 Determine the celerity of a deep water wave on a current equal to 50 cm/s andT = 5 s. What is the wave period seen by an observer moving with the current?

3.17 Develop the boundary value problem for small-amplitude waves in terms ofthe pressure, assuming that Euler's equations are valid and the flow is incom-pressible.

Chapter 3 Small-Amplitude Water Wave Theory Formulation and Solution

Documents

Transcript of Chapter 3 Small-Amplitude Water Wave Theory Formulation and Solution