J

IRREGULAR SEAWAY

5.1 CLASSIFICATlON

f In the preceding chapters ship motions due to long-crested waves of regular sinusoidal form have been tudied because these were all that could be handled -by analytical methods. However, a treatment of hip motion purely for regular waves is, to a consider-

able extent, academic in nature, since an actual sea does no1 have a regularly corrugated surface, and the wave paltern is rather complex and extremely irregular.

Although an actual sea surface is very irregular, oceanographers have been able to predict by statistical means how often various wave heigh1s may occur over a certain period of time for a particular sea urface of a given amount of energy. Sincc in an

irreglllar seaway the sea surface constantly changes, a few new terms regarding wave height will be defined in this chapter in order to perform a statistical tudy of the seaway.

The irregular wave surface varies from time to time and place to place, depending on the wind speed or the Beaufort llumber, which is a means of estimating and reporting wind speeds. This system was devised in the early nineteenth century by Admiral Beaufort of the British Navy. The sea state is the condition 0 1' the surface of the seas water, white caps). By observmg the indivdual can derive a Beaufort number and hence a WiN\ Pctllred in Fig. 5.1 are some of the various sea states and the wind speeds that ac-company these states. These sea states arc a ls described in Table 5. 1. ln the case of sinusoidal waves the and so on remain the same, but for an irregular seaway these properties

Chapter

Five

cons1antly change from time to lime and place to place. For an irregular seaway the dislurbance of the sea sllrface a1 any given point as a func1ion of 1ime is shown in Fig. 5.2a, where the following ymbols are used:

r = wave eleva1ion (i.e., ins1antaneous displacement of the sea surface from the position of rest or the reference line). apparent wave amplitllde d iSlance of 1 he crest of the wave from the posilion of rest or the reference line)

i", = apparent wave height (i.e. , vertical distance between a successive crest and a trough)

= apparent zero-crossing period elapsing between two successive upward crossings of zero in

Figure 5.1 lIIustration of sea ndilions for various wind speeds. Hurricane ght-rough s knolS. (b) Twelve knots. (c) Eighteen knots. (d) Thirty knots. (e) Forly knots. Forty-five knots. (g) Fifty knots. (h) Sixty-five knots. (1) Seventy 1cnots. U) Ninety-five knots. (k) One hundrcd dtwenty knots.

lQ3

TABLE5.1 DEFINITIONS OF SEA CONDITIONS: WAVE AND SEA FOR FULLY ARISEN SEAu

-Sc. - G.n.r.1 W,nd Sc.

Sca iphon (Beou. D Rang. W,nd W.. fI.,gh, Sign ficQ PC'mS A 10 35 24.1 17.2 whllc with dnvon8 spray. Vib h.y canc. 64-11 >46.6 74.S 94,6 ry Kriously atcd.

For hurncanc wlnds (and oftcn whotc gale and 'lorm ndsl rld durauons and rerls art barely 8tt .oncd. SS 8rc therdorc oot rully.rnI Rc'OOea:mbc l by L. MoskoWIIZ and W Plerson Uscd y of Th. Nraphlc 104

105 CLASSIFICATrON OF SEAS

Time

~-r.

1/ l

t

a) Irregular seaway plotted 10 the base 1 lime at a given polnt )(.

f

x

b) Irregular seaway plotted to tne base 01 )( at a given instanl 01 time.

Fgure 5.2 (a) IrreguJar seaway pJots to a base of time at a given point x. (b) IrreguJar seaway pJotteda base of x at a given instant of time.

of the heights of the one-hundredth highest wave for the same wave rccord.

The Wave Height Characteristic!: from Records are obtained as shown in Table 5.2.

The a verage height, the significant height, the

Wave

Example 5.1

or signijcant wave IIeigh~ (h...) 1/3' of an irregular seaway at any given time and in a given location in the sea is the arithmetic mean of the heights of the one-third highes1. w!Yes for a given record. Ag"in, waves less than 1 ft in heit are igno~. Sirnilarly, the average of the 10% highest waves, (11 ,.,) 1/ 10' is the average of the heights of the one-tenth highest waves for the same observation, and the average of the 1% highest waves, (h ,.,) I/IOO' is the average

Q) x @ for One-Hundredth Highesl Waves'

Q) x @for One-TeOl h

Highest Waves

Q) x @for One-Third

Highesl Waves

Q) x @ Cumulative Number of

Waves observed Reading from

Top of2 @

Approximate Percent

Number of Waves Having Heighls of Q)

TABLE5.2 Wave Heighl

@

5( = 5 x 1)

5

32(=4x8) lO

42

@

21( = 3 x 7) 100

lO

131

@

4 80 93

i lO

287

4 44 75

l 102

@

4 39 30 25 2

l

@

4 40 31 25 2

SUM 102

G

-E

-da

3

106 IRREGULAR SEAWAY

average of the one-tenth and one hundredth highest waves can be determined as fo l1ows:

Solution: From Table 5.2 the fo l1owing data are obtained:

Average height = 287/ 102 = 2.81 Signilicant height = 131 / 34 = 3.85

(considering 1- x 102 or 34 waves) Average of the one-tenth highest waves = 42/ 10 =

4.2 ft (consider x 1020r 10 waves) Average of the one-hundred highest waves = 5/ 1 = (considering x 102 or 1 wave on1y)

Height of the highest wave = at 1east 5 ft Note: in calculating the average of the one-third

highest waves of the total we need to consider only 102/3 (i34) highest waves. The results of the last two rows plus waves from the third row of column 0> wi l1 be considered in the evaluation of the column for the signilicant height, that is, f,., @. Similarly in linding the average of the one-tenth highest waves, only 102/ 10 (i.e., 10 out of 102 waves) should be consiered for column (J). The value of the lifth row 0f colurnn wi l1 give only 0> wa and therefore another eight waves from the fourth row of the same column are to be considered in the evaluation of (J).

Although. the bliliayij)r of a shio in Irr~S!l~rseawax.-shQ.uI_r~lat-.!9 .....the__signilicant wave !ght of the irregular seaway in a statistical studyis standard practice in oceanography, I! more severe SW (h ..)I/IO is often considered for the seakpingdesign.

5.2 IRREGULARITY OF THE SEA WA Y AND THE mSTOGRAM

The degree of irregularity of a seaway can be deter-

, I f Mean f. = 4'

60 C

TA8LE 5.3

Mcan Numbcr or PcrentageElevalion Groups or Occurrencc [rl]

1.5 10 2.5 26 13 2.510 3.5 22 11 3.5104.5 20 10 1.510 - 2.5 30 15 2.5 10 - 3.5 24 12

CIC.

mined by the shape o( a histogram, that is, a frequency function for the individual wave characteristics at a given time or in a given place. For the preparatioll of the histogram the wave record is grouped into equally spaced intervals of time, say 60 se and the )Iave ele.vations for all of these intervals are tabulated in increasing order. (S Fig. 5.3). Next , the values of C are dividto groups and the number in each elevation group is divided by the total number in the sample. The percentage. value in each group is then plotted a shown in the histogram in Fig. 5.4.

Assume that we have an actual record of wave elevations for 200 min, similar to Fig. 5.3. From the wave record there are 26 intervals with mean eleva-tions of-h5 intervals with mean elevation of 2.5 to 3.5 20 intervals with mean elevations of 3 to 4. and so on.

There are 200 one minute intervals. Experience has shown that the histcgram of the wave elevation takes the shape of a Gaussian or normal distribution as shown by the dotted line in Fig. 5.4. Table 5.3 has been made for the histogram ofthe wave elevation.

Mean f. 3' Mean f. -3'

60 sec

Figure 5,3 Re of wave clevations over a period of time , . Ij " (p' f f h g (1Ah -:h

-) "J

11.

IRREGULARITY OF THE SEAWAY AND THE HISTOG RAM 107

Pernt of occurren

- 10 - 8 6 - 4 - 2 2 10

Figure 5.4 Frequency function ofwave elevatlons.

Similarly, a histogram or frequency function of the apparent wave periods of an irregular seaway can be prepared from a table in which all the t values from the w.ave record are tabulated in order of i!,1creasing period. The values of apparent period, arc then grouped between 0.5 and 1.5 sec, 1.5 and 2.5 sec, and so on, showing the number of zero up-crossings in each group. By dividing thc number of zero up-crossings for each group by the total number of zero up-crossings available from the record, the percentage is calculated and drawn as in Fig. 5.5a.

It has been found by taking innumerable wave records that a comparatively regular seaway wi lJ yield a high, narrow histogram, whereas an irregular seaway will produce a low, wider histogram. The locat.on of the center of gravity of the histogram in relation to the y-axis ~ives lhe average vallle of the apparent wave period T, that is,

T= JpjpdR

'.vhere p is the percentage of occurrence. Anolher way of using the histogram is to plot

the cumulative distribution diagram (see Fig. 5.5b) by taking the cumulative number of observalio~ that fall b~low a series of increasing values of T, h.." or ra , converting them to percentages, and drawing them against the base values. The significanl value or lheaverage value ofthe one-tenth or one-hundredth highest values can be readily obtained by using the cumulative distribulion.

lt has been found by expcrience that the theoretical Rayleigh curve fits the histograms for the wave height (double amplitude) very wel l. The Rayleigh

distribulion is expressed by the following equation:

2H p(He (5.1)

where p(Hi) is the probability density per foot or the percentage of times that any particular wave height Hi will appear, with 0 < p < 1. If p = 0, Hi will never ocur whereas p = 1 means that H; will occur with every experimen t. Furthermore, FI2 is the average of all the wave heights squared (i .e., lhe square of wave heights), defined by

;.2 L[(Hy xf(Hi)] -

L [J(Hi)] where f(Hi) is the number of 0urrencf Hi'

This is very easily illustrated in Example 5.2.

Example 5.2

The following wave heights were recorded over a 24 hr period:

Hcighl (0-5 5- 10 10- 15' 15- 20 20- 25 umbcr of wavcs 5600 72 1910 960 320

Plot the wave height histogram along with the theoretical Rayleigh distribution.

Solution: The hislogram data are shown in Table 5.4. The values of column (3), obtained by dividing

@) by 5, which is the interval of wave height record, determine the ordinates for the wave histogram. Note, however, that, when the wave records are

108 IRREGULAR SEAWAY

20% T

10')(,

6 7 8 9 10 11 To-F

a) Hgram 01 apparem wave periods

100 ')(, C

50')(,

12 0 sec T A

o 2 3 4 E 5 6 7 8 9 10 11 b) Cumulative distribution diagram

Figure 5.5 (0) Histogram of apparent wave periods. ( Cumulative djstribution diagram.

grouped into 1 ft intervals, column G> wiII have the TABLE 5.4 IUSTOGRAM DATA same values as @.

A sample calculation for Rayleigh distribution Wave Mean Occur- Percent Percent

is as fo lIows. The mean square of wave heights is Record Height rence Occurrence Occurrence

H, tt1 ;. ) per Foot fj2 = I: [(Ht~!(!fi)J fWaveHeight

I:[j(H;)J @=~ x I (from reco)(2.W x 5600 + (7.5)2 X 72+ (12.5? @ @ SUM x 1920 + (17.5)2 x 960 + (22.5)2 x 320 -5 2.5 56 35 7

160 5- )0 7.5 72 45 9

= 73.72 ft 2 10- 15 12.5 1,920 12 2.4 15- 20 17.5 960 6 1.2 Thus the root mean sque (rms) of wave heights 20- 25 22.5 320 2 0.4

= H = 8.59 SUM )()Now, according to the theoretical Rayleigh distri-

r.

'

4~

4

IRREGULARITY OF THE SEAWAY AND THE HISTOGRAM 109

TABLE 5.5

Wave Record

-55- 10

10- 15 15- 20 20-25

Mean Heighl, H,

[]

2.5 7.5

12.5 17.5 22.5

Rayleigh Ord. p(H,)

0.0623 O.49

O.7

O.75 O.6

bution in (5.1), the probabily for a wave height of 2.5 ft is

2(2.5) _ p {2 =-==- e

73.72 = 0.0623 or 6.23%

Similarly, the other probability values are calcu-lated as shown in Table 5.5, and plotted in Fig. 5.6.

Note: The total area under the Rayleigh distribution curve should equal 1.0 (i.e., the total probability should be 1.0).

From 5.6 we have

Area from 2.5 to 22.5 = t hL Product, using the Simpson rule for integration

=(5)(0.0623 x 1 + O.49

Area from 0 to 2.5

Total area

x 4 + 0.0407 x 2 + O.75 x 4+0.6 x 1)

=0.92 =(2.5) [5(0) + 8(0.0623) 0.0966] using the 5 8 lrule fo[ integration

= 0.08 = 0.92 + 0.08 = 1.0

which should be the probability for all waves.

30. '" g 20

From its definition one can see that being the average over the entire area of the sea, should very close\y represent the average energy of the sea, that is

gfj2 g = (h~" + h:'1 + .:.) 8 8 ,. ...', . .....

= Energy (5.2) Therefore, if the area under the histogram curve i known, one can directly relate H to the Rayleigh distribution formula and determine from this the probability of occurrence of different wave heights. Thus the probability that h> Hj is

fllj 2H I _ 1 .:.'.'J ,, - H/ln' I TJ 2 JO

p{h> Hj

= e- m 1fl1

For example, if Hj = 10 ft , then p = e- (I00J7 3.72) = 0.258. There is a probability of 0.258 that the wave height will be greate"r than H j or 10 In other wordout of a number of waves N , Ne - /HIH' waves will be higher than H i'

From such a formulation one can find the average wave height, or the average height of the one-third highest waves, or the average height of the one-tenth highest waves, and so on. For example, in the preced-ing example the average wave height is

(H)O = 0.89(jj 2)1 12 (5.3a) the average height of the one-third highest waves i

(H)I /3 = 1.4J(jj2) 1/2 (5.3b)

and the average height of the one-tenth highest wave IS

(H)I /1 0 = 1.80(iP}1 /2 (5.3c)

These expressions are vaJid, however, only if the Rayleigh distribution is correcl. Some correction are required if the actual distribution differs from

Rav!eigh distribution E B C

10 2 u s c u ...

10 15 Wave height [ ft )

Figure 5.6 Wave bistogram and lheorelical Rayleigh dislribulio t1 .

'

110 IRREGULAR SEAWAY

TABLE 5.6 Heighl Number [fl] of Waves

Cumulalive Numberof

Waves @

Number x Heighl

:: Q) x0 Q) 0

2.5 56 7.5 72

12.5 1,920 17.5 960 22.5 320

56 1 28 14,720 15,680 16

14

54

24)()

168 720

SUM 16)() 116)()

A verage wa ve height is 116'/ 160= 7.25 According to the Rayleigh distribution , average wave height is 0.89 x 8.59 = 7.65 . One can see that thc theoretical Rayleigh distribution fits the example rather wel l.

The histograms as described above can define only one characteristic of the irregular seaway, namely, either period , height , or amplitude. Only the wave energy spectrum method of describing a seaway takes into account both the frequency and the wave amplitude. This is discussed in the following sect lOn.

the Rayleigh one, as will be shown later. Table 5.6 has been compiled in accordance with 5.3 WA VE SPECTRUM

Table 5.4 in order to compare the actual values of the average wave heights with those pricted by the An irregular wave pattern can be generated if a large Rayleigh law. number of sinusoidal waves of different wavelengths

::(\~/

w.= loh = 1ftlf V Vvy V V

= 0.6

r. = 2.5 ft

w = 0.8 r. = 2 h

L = 1264.53 ft

~ X

1. .. = 562.02 ft

.-

L w = 316.13 ft

x

L = 202.33 ft

~X

A combination f AII Four Wave$

x

\ , \ I (c )

.

',

1 .. ,.

',, ',

/

1 .EE--...

Figure 5.7 Addition of four sinusoidal waves.

WAVE SPECTRUM 111

and heights are superimposed on each other. The resulting wave shows no definite pattern for either wave height, wavelength, or wave period. This is i1Iustrated in Fig. 5.7 by considering four sinusoidal waves, each having its own particular wavelength and wave heigh t. The combination of four waves, shown in Fig. 5.7e, is of extremely irregular shape in regard to both wavelength and wave heigh t.

Not only does the superposition of many sinusoidal waves create an extremely irregular seaway, but also the pattern of the seaway is never repeated from one time to another. There is, however, only one way to take into account the irregularity of the waves, and that is to determine the total energy. This is obtained by adding together t he energies of all of the small, regular sinusoidal waves that produce the eaway by their superposition. The severity of the

seaway is then measured by the total energy content of all the waves presen t.

As mentioned in Section 3.7, the energy of a sinusoidal wave is given a!:

1P9(:

per square foot of sea surface. Therefore the total energy per square foot of surface of all the waves, with amplitudes (0" ( 02' ... '(0.' is given by

ET = g((: + (;2 + ... +{;J (5Thus any given seaway n be described by the

energy distribution versus the different frequencies (or wavelengths or wave periods) for various wave components. The frequency distribution of energy

is called the energy speclrum for the particular seaway, as illustrated in Example 5.3.

Example 5.3

Find the energy distribution of an irregular seaway composed offour di fTerent waves having the following characteristics :

,a Lw

d

urL

U

nu.

LIt

e--in

vgob i-

-l

WUU

234 J265 562 316 202 354 2

SolUlion : The circular frequency expression is

"-from which

1 =0.4 2 =0.63 =0.8

and 4 = 1.o sec - I

Thus the total energy per square foot of the wave surface

'J

.

1 +

4

J

4.l

ZM

3

)CJ

++

a+'

-F

LI-

-)H 3

5lh

HH1

.

za2

',

v

s

BJ'

"

J1

dOJmH

I

-1.1

-U+

HIF

A

-

L

ti

g-22.

-2nuU

AY-

ro-

A

Ta E

1o

9

800

ZB S6 :l 5

4

300

200

1

o

S"' = bandwidth = 0.2

624 Ib-s~'C

MJoub

\\Area = 31 .2 Ib/ft

0.6 Circular frequency.[sec- t )

Figure 5.8 Energy spectrum for four waves.

0.2 0.4 1.0

112 JRREGULAR SEAWAY

Il----u

u"-

w [sec I J

Figure 5.9 Approach toward fina[ spectrum.

where pg = 62.4 1b/ ft 3 for fresh water. The distribution of this total energy according to

the frequency of waves is given in Fig. 5.8, where the ordinates are obtained by dividing the individual energy content by the bandwidth, which is 0.2 in this particular example.

waves, having all different wavelengths and very small amplitudes, as Fig. 5.9 shows.

The energy spectrum in Fig. 5.10 covers the entire range of frequency; the bandwidth w is decreased, and the number of individual wave trains increases until it reaches inrmity. At the same time the energy content of each individual wave component is also decreased; however, the total amount of energy available in the seaway remains the same. This continuous curve between w = 0 and '" actually represents the energy spectrum of the waves.

It is to be noted that for a given wind speed the waves that are first generated are short; the longer wavelengths are generated when the wind continues to blow. U1timately aJully deueloped sea is produced which is stable and does not change as the wind continues to blow. Thus the energy spectrum also changes continuously until a fully developed sea is formed. During its growth longer waves are produced (i .e., the contributions from the shorter

Note that the dimension of energy (see Table 4.5) is Ib-ft. Since the area under the total curve should give this dimension , the ordinates represent Ib-sec/since the abscissa has the dimension of sec - 1.

The total area under the energy spectrum gives the total energy of a l1 the wave components. Note that the energy due to each frequency has been given a small bandwidth w in order to obtain a continuous curve, as shown by the dotted line of Fig. 5.8. Since the real sea is made up of all frequencies and the wave pattern itself is never repeat:d the energy sptrum has to be a continuous curve, composed of the contributions of 1n infinite nlimber of regular

| [sec-I

J

Figure 5.10 Final energy spectrum.

"pmC

Wave frequency.(sec- '

Energy build-up of partially and rully developi seas.

of the new figure, generally denoted as mo , is Jater multpLed by. pg to obtan the energy. The new figure s called the wave spectrum, and the ordnate are represented by the symbol S(...) whch s called the spect/'al density ofwave energy. The wave spectrum for example 5.3 s shown in Fig. 5. 13.

In Example 5.3

((; + (;1 + (;J + (;) =[(1.52) + (2.5)2 + 22 + 12J = 6.755 2

Therefore the total area under the wave spectrum in Fig. 5.1 3 should have a value of 6.755, whch when multpLied by pg, that is, about 62.4 lb/ft 3 (for fresh water), gives the energy, which s 42 1.2 lb/ft as before. As in the case of the energy spectrum, the ordinates of the wave spectrum are obtained by dividing the ndividual t(amplitude)2 .values by the bandwidth, which is 0.2 n this particular example.

In conc\usion t sbould be noted that there is a lmit on the number of waves to be consdered in obtanng the maxmum wave height from the Rayleigh distrbuton. A1though we may obtain a very hgh wave fthe wave record is made for a very long time, the probabi Lity of occurrence for the extremely high wave is very low. Therefore often a record of 1000 waves is considered to be sumciently representative for the determination of the wave spectrum, and the most probable" value of the one-thousandth. highest waves is taken to be the most probable largest value [152].

Sometimes it is not the number or observation but the period of time that is considered io obtaioing a wave record for statistical evaluation, for example,

frequencies become predominant), as is shown in Fig. 5.1 1. This figure also shows that, along with the generation of longer waves, the max.imu valueof the energy spectrum shifts ' toward the lower frequency sde. This is also the case for a fully deve-loped sea when it is experiencing increasing wind speed. See Fig. 5.12.

We have seen that (5.4) repr

Figure 5.11 .

HC;, + C;1 + ... + C;.l Note: 9 is divided out, and the area under the curve

(5.5)

AU au u"'

"HAva" UHe F

S

Au

uasPE

Wave frequency. (sec-'

Figure 5.12 Energy 5tra of fully develo!d seas ror various wind speeds.

114 IRREGULAR SEAWAY

20

3 10

"'~

U>mE- 5

Frequency. .. ($.- IJ .~

Figure 5.13 Wave spectrum for.four waves.

the total number of waves passing a point within an hour, or the total number passing a point during a longer period of time. The total number of waves is obtained by dividing 1 hr by the significant wave period.

5.4 PREDICfION OF AN IRREGULAR SEAWAY

To be able to define a seaway it is nessry to take sample records of the wave heights and the frequency of the particular seaway concerned over a limited period of time. Although the wave pattern wiU never be repeated, the statistical characteristics of the sea state, that is, the energy spectrum or wave spectrum, wi11 remain the same. This is the advantage of statis-tical investigations. In other words, the sinusoidal components that approximate a record for a parti-cular sea state are th same regardless of time and place and differ from one record to another only in the phase orientation, thereby keeping the energy of the wave system constant.

Although a spectral density curve may be drawn from just one wave record, it is often preferred to obtain the average wave cbaracteristics for any given area by taking many sarnples of wave records. The spectral density curve may also be approximated

by an analytical expression bas on probability tbeory.

The procedure .for plotting a spectral density curve to a base of wave frequency is illustrated below. Let us suppose that a rord has been made of four component waves of wave frequen w' between 0.75 and 0:85 (i.e. w = 0.8 ::t 0.05); artd that w o. 0.75 = 0.1 is the bandwidth.e wave ampli-tudes of the wave record are

(

PREDICTION OF AN IRREGULAR SEAWAY 115

Energy density S () [ft2_secJ f

Aegu lar wave

rms

61

f

Irregular wave

0.2 0.4 0.6 Circular frequency.lradlcJ

J (b) Figure 5.14. (a) Energy density for a particular frequency band. (b) Surface elevation at equal intervals of time. which is the area under the energy density curve for cw between 0.75 and 0.85 sec-

1. Since w",=

0.1 sec- 1 that is. 0.85 0.75 in our records

2.25 S(J= = 22.5 ftZ-sec

0.1 This value obtained for energy density has been plotted in Fig. 5.14a.

Now the total energy E = gmo per square foot of wa ve surfa where mo is the under the energy density spectrum, or, dimensionally, E = (MC 3 ) (LT - 2)(L2 ) or E = (ML T - 2) for the total wave surface. This 1s the dimensional expression for energy as given in Table 4.6.

Nte that it is not the energy spectrum of the seaway that has a Gaussia,n form; it is the seaway record (histogram) for wave elevation that is Gaussian. The histogram for the \vave heights of an irregular seaway js considered to be more or less a Rayleigh distribution. The energy spectrum may have any functional form.

To use the principle of the energy spectrum for eng~neering studies, the quantitative values oJ the wave spectrumJor diJJerent sea regions andJor diJJerent c1imatic .conditions should be known.

Altbough there are differences between different wave spectrum formulas, the ordinate of the curve iS" generally taken as th spectrar density, which is used to represent the full energy of the component waves. Since the energy of the component waves is directly related to the square of their amplitudes, the spectral density can be reJerred directly to the square oJ the amplitude oJ the waves. The notation

fr the spectral density is S(w)' and the area under the curve is equivalent to the statistical variance. Furthermore the significant height is otained by the relation

(h"')1 /3 = 4.0ji or (hJ1 /3 = 4.0 .Jmo where

mo = I $(Jd'" :t (5.6) J 0

is the area under the curve of the wave spectrum. The factor 4.0 is obtained on the basis that the hjstogram for wave height follows the mathematical approximation given by the Rayleigh distribution, wh.ich characterizes a rather narrow wave spectrum. Ag

S(")/w = H(~ + (;2 + ...) ~5.7) for the particular value of bandwidth ww ' as men-tioned before. The wave 5peclrum so obtained not only gives the various average values (the significant amplitude, average amplitude ofthe ooe-tenth highest waves, etc.) but is also us for the extreme values, as will be shown later.

The following useful information can be derived from a wave spectrum:

a. The range of frequencies that are important for the contribution of energy to the seaway. b. The frequency at which the maximum energy is supplied.

c. The content of energy at different frequency bands.

d. The existence of a swell at low frequencies.

116 lRREGULAR SEAWAY

TABLE 5.7

Amplilude Heighl

Average wave : 1.25 J 2. 50 JiioAverage of one-lhird highesl waves : 2. Jio 4. Average of one-lenlh higheSI wavcs : 2.55 foAverage of one-hundredlh highesl wa ves: 3.34 fo

From a known wave spectrum one can obtain the wave amplitudes or heights by using the simple formuJas given in TabJe 5.7. The area under a wave spectrum, as described earJier, is

j S

which is lhe average lime belween or lroughs in lhe record.

In an irregular seaway, CreSls and lroughs occur from lime 10 lime below and above, respeclively lhe mean waler level; lherefore lhe Crest-lO-Cresl period 1; has a value less lhan lhal of T: , which is lhe zero-crossing period. Since ~ does nol include the effect of impOrlanl ripples, il is lherefore 'tonsi-dered 10 be more closely relaled to visual eSlimates of period than is

Anolher impOrlant charaClerislic for lhe irregular seaway is lhe average apparenl wavelenglh (L.J: which is also defined in lerms oflhe speclral moments:

2gJE (5.15)

Standard Wave Sptrum (Recommend by the International Towing Tank Conference)

When the wave spectrum of a particular sea is not a vailable, Inlernational Towing Tank Conference (lITC) spectral formulation should be used as follows:

a. S~)= e w

(5.16)

Here w is thircular frequency in radians per second, and A = 8.10 X 1O- 3g2, where g is the acleration of gravity in appropriate units.

Also, S(.J is in cm2-sec units when B = 3.11 x 104/H~ where H 1/3 is the significant wave height in centimeters, or S(w) is in fe-sec units when B = 33. 56/ (H) /3 ' where H 1/ 3 is the significant wave heighl in feet.

b. If Slalistical information is available on both the characteristic wave p"eriod and the significant wave heighl, then

A = 173(H)J3 / TI4, B = 691/ T14

where lhe significant wave period T1 is given as

=2The data suggest that trus period can be taken

as the observed period. Furthermore, the signi f1cant wave height is (H)I /3 =

4.0Jmo , that is, the significant height is 4.0 x variance. c. Although a wide. variation in significant wave heights exists for a given wind speed, the approximate relationship between wind speed" and significant

PREDICTION OF AN IRREGULAR SEAWAY 117

heighl in the open ocean, to be used when only wind speed is known, is tentalively defined by a curve (Fig. 5.15) having the following ordinates :

Wind Speed [knols]

20 30 40 50 60

Significanl Wave Height [

10 17.2 26.5 36.6 48.0

The wind speed" is taken to be that which is sensed" by personnel on board ship.

Example 5.4

a. Using the 11C formulation , plot a wave spectrum for a wind speed of 31 knots. b. Find the significant wave height from this spectrum if the assumption that the wave height histogram follows the Rayleigh distribution is not valid.

Solution : Since wind speed. = 31 knots, the significant wave

height H 1/3 is 18.5 ft from Fig. 5.1 5. The spectral density is

S(J=r w

where A.= 8.10 x 1O- 3g2 = 8.385 ft 2-sec, and B = 33. 56/(H) /3 = 0.09806.

Substituting the values of A and B, we obtain Table 5.8 for the wave spectrum (see Fig. 5.16); Table 5.9 has been prepared for the calculation of the correction factor (CF) to be used for the broadness ofthe wave spectrum.

The variance is then

mo= xw x SUMo = t x 0.1 x 646.92 ft 2 = 2 1.564 2

Furthermore,

(H)J /3 = 4.0J = 4= 18.50 ft

(H).ver.sc = 2.506J= 2.506 x 4.65 = 11.67 ft

and

(H) 1/ 10 = 5.090J= 5.090 x 4.65 = 23.65

l. N

'IJ VS. wind speed:

ITTC standard wave spectrum

118

10

IRREGULAR SEAWAY

50

45

40t:!

:t: g35

3 30m .. .t:

~ 25 z c '" i 20

18.5

15

60 20 3 40 50 Wind speed [knots]

Significant wave height versus wind Spe [10.Figure 5.15

~verafl.C =;== 8.83

= 2=2= 7.2 sec

119 IRR EGULAR SEAWAY PREDlCTIO

from which the correction factor is

CF = (1 - e2)1 /2 =(1 0.4 1 / 2

= 0.76

S(J [ l_sec]

@

B

TABLE 5.8

W., (sec-1J

@

Finally, the significant wave a-mplitude is

((0)1 /3 = 2jm; x CF = 2 x x 0.76 = 7.5 ft so that the significan t wave height is 15

The average, significant, and other wave heights are' obtained statistically by applying the correction factor. However, the CF is taken to be 1.0 when standard ITIC formulation is used for wave spectra for a given significant wave height , since it is assumed by ITIC that the wave height (double amplitude) histogram follows the Rayleigh distribution.

5.5 MOST PROBABLE LARGEST WA VE AMPLITUDE

The expected value of the most probable largest amplitude in a record of n waves is obtained statisti-l as given by Ref. 139, using (5.1 7).. The most probable largest amplitude is

o 0.02

17.80 55.97 50.68 33.22 20.17 12.25 7.61 4.88 3.22 2.19 1.52 1.09 0.79 0.59 0.44 .340.26

o O 0.022 0.208 0.469

O.5 0.787 0.861 0.907 0.935 0.954 0.966 0.975 0.981 0.985 0.988 0.99 1 0.993 0.994

square foot of sea surface). Thus

@

6 1.286 12.106 3.830 1.569 0.757 0.408 0.239 0.149 0.098 0.067 0.047 0.034 0.026 0.019 0.015 0.012 O.9 O.8 O.6

J-; x J~xCF (5.1 7)

where n is the total number of observations, mo is

mnm. - m~ 2 = "'0'''4 '''2

mnm 0'''4 -

).

i-

ny

---

nu-

-l

-qL

'-qJ

.

. 9

4

'J

-

.?o ny-J

.

. X

l-L

ro

zJe-

----

=

mwmmmMNMWmNmmw

AVAUnuhunUAnvnu'S

E

E1.shtaaL1hgEU'L

TABLE 5.9 WAVE CHARACTERISTICS FROM ITTC STANDARD SPECTRUM, (11")1 /3 = 18.5 ft.

W .. (SCC- 1]

ProdUCl (!l v (

Simpson's Mulliplier

U

@

dw@@ ProdUCl @)x(!)

@

Simpson's Multiplicr

~S(.)Z w .. ProdUCl Q) x0

@

Simpson's Multiplier

@

S(!J

@

mmNMmmm

mNmmwmwm

nnUAUnUAUAUAUAU'

aa

-2

EE.E-

8

s

't

-E

&

O 1.06

13.44 13. 18 31.88 16.54 32.36 15.22 28.28 13.34 25.04 11.68 22.0 10.34 19.72 9.24

17.72 4.16

142424242424242424

l

.

o O. 0.53 3. 36 6.59 7.97 8.27 8.

7.6 1 7.12 6.67 6.26 5.84 5.52 5.17 4.93 4.62 4.43 4.16

s

l3634l606764655030

UnUAUAU

'

h

AUTrOAUAU

AUJ

Jnunu

aaaaaonual.-

-2.Z1168036

327. 14 ft 2-sec - 1

5.70

55.96 36.50 65.12 25.82 39.68 15.22 23.60 9.28

14.80 5.96 9.80 4.04 6.84 2.86 4.92 1.04

SUM 2

-E4

-A-AU-AMT4A

'ALaaTA'A

O. 2.85

13.99 18.25 16.28 12.91 9,92 7.6 1 5.90 4.64 3.70 2.98 2.45 2.02 1.7 1 1.43 1.23 1.04

@

MWMWM5MAMW

Mm"

aa--

-

oooaooool-

---22234

646.92 ft l-sec

0.08 35.

233.8R 101.36 132.88 40.34 49. 15.22 19.52 6.44 8.76 3.04 4.36 1.5 2.36 0.88 1.36 0.26

SUMo

-tAUT7.A-au-AU-AUTLAU

hAa-A7.AU-g

@

0.02 170

55.97 50.68 33.22 iO.17 12.25 7.61 4.8 3.22 2.19 1.52 1.09 0.79 0.59 0.44 0.34 0.26

120 IRREGULAR SEAWAY

TABLE 5.10

10 l l)Q

10.000 l)Q

the area under the wave spectrum (variance), and CF is the correction factor. (J - e2)1 !2

For di fTerent values of n. Table 5.10 shows thc resu lts using (5.17).

For example, if the wave height (double amplil\histogram follows the ideal ' Rayleigh distributioD (which means that the correction factor is unity~ then, for a value of mo = 80.70 ft 2, out of 50 waone would have an amplitude of 25.s. or out ora total of 10 waves one should atlain an amplitudc of 3 3. 6 .

Statistical estimation of the most probable largest values of wave amplitudes is useful in practical desiconsideration.