On the family of wave spectra for long term prediction of ship bending moments

HIEF

Lab. V. Sc

Tec&ce tp'.schooI

With the Compliments

ofIjtAuCthor.

On the Family of Wave Spectra for Long-Term

Prediction of Ship Bending Moments

By

Jun-ichi FIJKUDA

Reprinted from the Memoirs of the Faculty of Engineering

Kyushu University, Vol. XXVI, No. 2

Contribution to

3rd International Ship Structures Congress, 1967, Oslo

FUKUOKA, JAPAN

1. Infroduetion

Prediction of the longterm distribution of wave bending, moments acting upon a ship hull has become possible lately by using the statistic wave data collected on the world main sea areas. The long-term distribution of ocean waves can be obtained by accumu-lating the numerous data of short-term dis-tribution of those for a long period, and in order to obtain acurate data of short-term distribution, it is necessary to carry out the continuous measurements of ocean waves and find out the wave spectrum, significant wave height, average wave period etc., by

analyz-ing the obtained records of wave

Accumu-lating, numerous data of such 'a short-term distribution, we can obtain the reliable data of long-term distribution of sea waves. There

is gradual increase in the number of such perfect records of short-term distribution in

On the Family of Wave Spectra for Long-Term

Prediction of Ship Bending Moinents*

By

Jun-ichi FIJKUDA

Assistant Professor of Naval Architecture

(Received October 26, 1966)

Summary

Available data on statistics of wave height and period have been collected on the world sea areas and routes The purpose of this paper is to establish tentatively the standard family of wave spectra for short term and long term prediction of wave bending moments in rough seas based on such wave

data.

Theoretical wave spectra originally proposed by Neumann, Darbyshire, Roll-Fischer and Pierson-Moskowitz were modified realistically so as to fit' the observed sea states and compared with one

an-'other. Further, the comparative 'investigation was carried out on the standard deviations of wave

bending moments of two ships in rough seas which were obtained from those modified wave spectra and response operators.

There is not so significant difference among the results obtained from those modified wave spectra.

The result derived from the modified Darbyshire spectrum however has a tendency rather different

from the others. The modified Pierson-Moskowitz spectrum proposed by I. S. S. C. wodld be appropri-ate for the purpose of short term and long term prediction of wave bending moments and acceptable as an interith proposal.

This paper is rewritten in English from the

pa-per published in Japadese in "Journal of ZOSEN KIOKAI (The Societyof Naval Architects of Ja-pan), Vol.. .120, 1966".

recent years, 'but it may be said that they are still far from being sufficient at the pre-sent stage.

The most part of the. long-term wave data available at present is consisted of the vi-sually estimated average wave height (whih is believed to be equivalent to the significant wave height)and the i5ual1y estimated aver-age wave period which are obtained by the observation on weather ships and trade route ships. For example, according to the repo±t of Committee on Environmental ConditiOns, I. S. S. C., 1964', the long-term distributions of ocean waves in the world sea areas are

given in tables, where the both of wave

height 'and wave periód are classiñed 'in six ranks respectively and the frequencies of those are indicated for the divisions of these 36 combinations. _{Beside these, in the wave'} data of the 'Notth Atlantic Ocean by Roll2 as well as of the North Pacific Ocean and' the seas near Japan by Yamanouchi et al.3, wave height and wave period are classified into more number of ranks.

Thus, the long-term distribution of such a sea state as expressed by the combination

of significant wave height and average wave period has been available, though it is not accurate enough being based on the data of visual observation. Accordingly, it is possible to estimate the long-term distribution of wave bending moment by utilizing the long-term distribution of sea state (which is expressed with significant wave height and average were period), provided that the short-term

distribution of wave bending moment in such a sea state can be known. Hence, it becomes a subject how to estimate the short-term dis-tribution of wave bending moment among the sea waves expressed with significant wave height and average wave period.

There are two kinds of method to evaluate the short-term distribution of wave bending moment; one is to know the short-term dis-tribution of wave bending moment among a

sea state by analyzing the results of full

scale measurements and the other is to do

thè same by using the wave spectrum cor-responding to a sea state and the response operator of wave bending moment obtained by model tests or theoretical calculations. In .order to carry out the long-term predicti-on of wave bending moment by the former method collecting numerous data of short-term distribution, we have to conduct the

measurements on actual ship for a long

peri-od and at a great expense. Whereas, if we

can express sea states adequately with wave spectra and adopt the latter method, we shall be able to attain our object within a compa-ratively short time and at less expense.

Accordingly, let, us consider the method to

exprss a short-term sea state with a wave

spectrum. The available data of ocean waves are given, as mentioned above, in the form of frequency distribution of sea states expres-sed by the combination of significant wave height ,and average wave period. It is, there-fore, necessary to express the sea state defin-ed by the significant wave height and the

ave-rage wave period with a appropriate wave spectrum enough to satisfy such conditions. The wave spectra corresponding to actual sea states are so different from one another that absolutely the same spectrum appears never again. Only with the case of a certain sea

state, it is impossible to give the theoretical wave spectrum expressing exactly the sea' state merely by the parameters of significant wave height and average wave period. There will possibly appear a great number of sea

state having the given significant wave height and averagewave period and having different shapes of wave spectra. Since it is impossi-ble to take up for consideration all wave spectra that can exist countlessly, we may have tentatively such a sea state as having the given significant wave height and aver-age wave period represented by an assumed wave spectrum.

If such an assumption is

allowed, - that is, if it can be assumed that no serious error will arise from such an

as-sumption in the estimated results of long-term distribution of wave bending moment - we can use such an assumed wave spec-trum in order to estimate the short-term dis-tribution of wave beading moment and also the long-term distribution consequently. The short and long-term data on waves now ava-ilable are not perfect, because most of them lack accumulation for a long period and be-ing based onthe visual estimation. For

inst-ance, Cartwright4 points out that there are

considerable differences between 'the signifi-cant wave height and the average wave pe-riod which are obtained from instrumental

wave records and those due to visual estima-tions. It will be, however, a practically use-full method as a means for such a macro-scopic study as the prediction of long-term distribution of wave bending moment, to 'as-sume the family of wave spectra at the view-point as mentioned above. Essentially there is no conflict in the treatment of the problem, even if there is some incompleteness in the' wave data to be availed. Defects in the wave data may be corrected according to the reli-able data to 'be accumulated in the near fu-ture.

2. Formulae of wave spectra

Among the theoretical formulae of the wave spectrum hitherto proposed by oceangraphers, there are those by Neumann5, Darbyshire6, RoIl-Fischer7 and Pierson-Moskowitz8. In these original formulae, the wave spectra are defined as the functions of wind velocity, and they are equated not only based on the

1966) _{Family of Wave Spectra for Prediction of Ship Bending Moments}

tical ground but also adapted to the empiri-cal facts of wind-generated sea waves. There are introduced "fully developed storm seas" and "not fully devoloped storm seas" taking duration and fetch into account, as fbund in Neumann Spectrum, and in those theoretical wave spectra it is assumed that the shape of wave spectrum is mainly dependent upon the wind velocity. On the actual sea area, the sea state is not determined only by the

wind velocity but largely dependent upon the degree of wave development, the existence of

past history of swell and the changes of

wind velocity and wind direction, even under the same wind velocity.

The author will therefore consider here the expressions of wave spectra representing the sea states actually observed or measured, apart from the theoretical Spectra of ideal

wind-generated sea waves. In order to represent

such sea states approximately with wave

spectra, these original formulae will be modi-fied so as to fit the sea states defined by the significant wave height and the average wave period. The wave spectra to be assumed here have the shapes resembled to those due to the original theoretical formulae having the forms of "fully developed spectrum ". The expression of such wave spectra will be generally written as follows, except the spe-ctrum of Darbyshire type.

[f(co)J2=PPexp(_QwQ) (1)

where [f(w)]2 spectral density of waves F, Q : constant defined by the sea

state

p, q : positive integer

In the theoretical formulae of original

wave spectra, P is constant and Q the func-tion of wind velocity, but here P and Q will be assumed so as to satisfy the sea state. Significant wave height and average wave period will be used as the parameters repre-senting the sea state.

If appropriate positive integers are given to p and q, the type of theoretical formula of original wave spectrum will be determin-ed. Namely,

p=5, q=4 : Pierson-Moskowitz type

p-6, q=2 Neumann type

p=5, q=2 : Roll-Fischer type The average wave height (which is believ-ed to be equivalent to the significant wave height) observed on the sea surface of short-term will be assumed as H and the average wave period as T. Assuming that the wave spectrum on sea surface can be expressed with (1), the author will difine m as follows.

m=

(2)

The integrated value of (2) can be obtain-ed as follows.

P r(Plfl /Q(P-1-n)f

q

' q

_ii

(3)

If the elevation of surface wave from the average level is assumed to follow the Gaus-sian law of distribution, its variance or mean square S2 is equal to a half of so-called E-value and given as follows, which is equal to the area of wave spectrum.

m= J [f(w)]2do)=S2=E/2 (4)

If the width of wave spectrum is assume'd, to be narrow enough and wave height distri. bution approximately follows the Rayleigh distribution, there will hold the following relation between the significant wave height and m0.

H0=4S=4v'(=2.83/E5

(5)

Though the approximation of Rayleigh distribution does not hold in the case when a wave spectrum has a comparatively wide

width, the approximation of (5) gives no

se-rious error if its width is not so wide.

There is, however, no authorized evidenece

on the assumption that the average wave height due to the visual estimation comes

closer to the significant wave height obtained from the measured wave record, and it is said sometimes that the average wave height due to the visual estimation is somewhat smaller than the instrumental significant wave height. If this is true, it will mean to underestimate a wave spectrum by depending upon the

as-sumption of (5). For the present, however,

Since there are no sufficient data to be obtain-ed on these points, the assumption of (5) will be tentatively accepted.

In the next place, if the average wave p0-nod is accepted despite of its uncertainty

[f(w) ] 2/H02w =P1 (w/w) ' exp (- Qi(co/w0)

w=2ir/T, P1=P/H,2co, Q== Q/w,

and P1 and Q are determined so that they may satisfy the equations of (5) and (6) or (5) and (7), and they are obtained as

fol-lows.

In the case due to (5) and (6): Pj= (q/16) tr('' 1) } L°7 {(P3)} (p-1)12 (1o)

fr(P l)1(P_3)}I

₍₁₁₎

w/w=

(q/p)IIq{p

(P l)1(P_3) }/

(12) [f(wo) ] 2/H2cor1 =Pi (w0/w0) Pexp (- (p/q)) (13)

peak frequency

In the case due to (5) and (7)

1 p-2

' 2 pI

Pi=(q/16){f(!q )}

/{fPq

)} (14) QI

{r("1)/r(.°2)}°

(15) w/&= (q/p)I/g

f(Pl)/(P_2)

(16) [1(wo)] 2/H2w ' =Pi (coo/we) Pexp ( - (p/q)) (17) Then, ö° of (8) and (1O)(13) or (14).(17) were evaluated in each case of wave spectra: Modified Pierson-Moskowitz type (p= 5, q=4)

Modified Neumann type (p=6, q=2) Modified Roll-Fischer type (p=5,q=2).

The results are shown in Table 1.

-In the case of (a) or (b), the value of ö2 is comparatively small and the wave spect-rum may be looked upon as a narrow banded

one, but in the case of (c) the value of a2 seerns to be rather large and the assumption

when the width of wave spectrum is com-paratively wide, though (7) is also adoptable in approximatioU. When considering at the practical viewpoint, however, (7) seems to be more suited for use because it is less de-pendent on the attenuated part in theregion of high frequency of wave spectrum.

Now, if the formula of (1) is transformed into the non-dimensional form, the followings

are obtained.

-of narrow band is less suitable than in the cases of (a) and (b). When a narrow band is assumed and the equation of (7) adopted, the peak frequency of the maximum energy density becomes a little larger than the peak frequency based on the equation of(6).. The peak value of energy density is larger when it is based on the assumption of narrow band.

The shape of each wave spectrum in non-dimensional form evaluated under the

as-sumption of narrow band is shown in -Fig. 1

The peak of spectrum is the sharpest and highest in the case of (a), followed by the cases of (b) and (c). There is, however, no significant difference among the three wave spectra.

As an expression of wave spectrum- that has a form different from what is generally expressed with the fOrmula of (1), there is an formula of Darbyshire type pectrum.

There is a method adopted by the British Towing Tank Panel (BTTP)'""21 in which the original Darbyshire wave spectrum is modified and the wave spectrum is assumed based on the average wave height (signifi-cant wave height) and the average wave period due to the visual estimation. That is,

(d) Modified Darbyshire spectrum [f(o.)]2=O.214H2 exp((wwo)/ /O.065(wwo±O.26)) (18) for O.26<(cocoo)<l.65 [f() ]2 = 0 elsewhere w0(3.15/T) + (8.98/T2) : peak frequency (19) In this case, the peak frequency w0 where the maximum value of energy density is

ge-Jun-ichi FEXUDA (Vol. XXVI,

due to the visual estimation, the following will be equated according to Rice9.

T=2ir i/mo/rn2 (6)

And, in the case when the width of wave spectrum is narrow, the following

approx-imation is given by Longuet-Higgins'°. T=2ir(mo/mi) for a2 <K 1 (7)

= (mom - m12) /m0m2 (8) It is, therefore, necessary to depend on (6)

q)

1966) _{Family of Wave Spectra for Prediction of Ship Bending Moments 89}

nerated has been obviously expressed by the quadratic equation of w(==27r/T0), different

from the cases of (a)'(c). Again, the max-imum value of energy density is independent of w and constant as follows,

[f(wo)]2/H2-0 214 (sec) (20)

In Fig. 2, the values of coo and {f(w0)]2JH2

derived from the wave spectra of (a), (b) and (c) under an assumption of narrow band and depending on (16) and (17),are compar-ed with those of the wave spectrum of (d). Also in Fig. 3, examples of the wave spectra of (a) and (b) are compared.

As understood from Figs. 1-3, there is

a resembled tendency among the wave

spec-tra of (a), (b) and (c) with no significant difference, but the wave spectrum of (d) shows a tendency somewhat different from the others.

Thus, the obtained wave spectra are more or less different, if we attempt to assume a wave specrum on the basis of average wave height (significant wave height) and aver-age wave period, according to which

ex-pression of (a), (b), (c) and (d) we choose

and which assumption of

(6) or (7) we

adopt for the average wave period. Now, the object of our attempt to express a certain

sea state with a wave spectrum lies in the prediction of long-term distribution of wave bending moment based on the short-term

dis-tribution of that which has been evaluated

among such a sea state. Accordingly, even

if there were some differences between the assumed wave spectra, no significant objec-tion would be caused so long as there is found

no significant difference in the results of

short-term distribution of wave bending

mo-ment obtained by using these wave spectra. There would be little objection for practical use, too, if there is found no serious differ-ence in the estimated results of long-term distribution of wave bending moment based on the results of short-term distribution thus obtained, despite of its little difference. In the next section, comparisons are made on the results of evaluation conducted on the

short-term distribution of wave bending mo-ment by the aid of these wave spectra, and the differences in the results due to the dif-ferent type wave spectra are checked. 3. _{Comparisons of the calculated results of}

short-term distribution

When we consider the oscillatory ship

res-ponse in irregular seas as a stochastic pro-cess, its variance R2 or standard deviation

R is used as the parameter of short-term dis-tribution of response. Expressing a sea state with each wave spectrum mentioned in the preceding section, that is,

Modified Pierson-Moskowitz spectrum (ISSC spectrum)

Modified Neumann spectrum Modified Roll-Fischer spectrum Modified Darbyshire spectrum (BTTP spectrum)

and evaluating standard deviation R of the wave bending moment at midship acting on a 150 meter long cargo ship and a 300 meter long oil tankes', we get the results as follows. For each wave spectrum of (a), (b) and (c), however, here is adopted the one in which the equation of (7.) is used according to the

assumption of narrow band spectrum. The

wave spectrum expressions for these are as follows.

[1(w)] 2/H2 = 0.11wr (w/w0) exp ( - 0.44 (w/co) ) (21)

[f(w) ] 2/H2= 0.39co01(w/w,) ° exp (- l.77(co/w) 2) ₍₂₂₎

[1(w)] 2/H2= 0.20w01 (co/w,) _{exp ( - l.27(w/w) _2) (23)} [f(w) ] 2/H2=0.214 exp ( - (wwo)/V0.065(w w0+ 0.26)) for 0.26<(w--coo)<1.65 =0 elsewhere . (24) w0= (3.15/Tn) + (8.98/T°) =O.5Olco-j-0.227w2

The main particulars of the cargo-ship and the tanker adopted in the examples of calcula-tion are given in Table 2.

In Fig. 4, thete are giveh the evaluated

results of the response operator of wave bend-ing moment at midship actbend-ing on these ships

in fuji load among regular oblique waves, where the response operators in the case of 0.15 Froude number are expressed as func-tions of v'ship length/wave length. They have been obtaind by the theoretical calcu-lations based on the linear strip theory'3. In

the figure, only the results for the heading

angles of 30' interval are given, though calcu-lations have been performed for the heading angles of 15' interval from head waves (=

0') to following waves (c1=l8O').

The notations used in the figures are as

follows:

M0 : amplitude of vertical wave bending moment at midship

p : density of sea water g : acceleration of gravity

where

R : variance of wave bending momentin the case of heading angle 0

Mo(w)/ho] : response operator of bending moment

0 heading angle to the average wave direction (0 0' head waves) x : angle between a component wave

di-rection and the average wave direc-tion

In the case of short-crested irregular seas, it is assumed that the directional energy

den-sity of component wave presents cos2x

dis-tribution in the range. of ± 90' from the aver-age wave direction.

Examples of calculation made on energy spectra of wave bending moment in

long-crested irregular head seas are shown in Fig.

5, where the results obtained by using (a)

modified Pierson-Moskowitz wave spectrum (ISSC spectrum) and (d) modified Darbyshire wave spectrum (BTTP spectrum) are com-pared.

In Fig. 6, there are given the results of

standard deviation of wave bending moment in long-crested irregular seas obtained by

us-ing fOur kinds of wave spectrum (a)--'(d),

where the heading angle to waves(0 = 0' : head seas) is employed as the parameter, and the dimensionless values of R/ pgL2BH, are expres-sed as functions of average wave period T. Also, with the results obtained by using (a) Modified Pierson-Moskowitz spectrum (ISSC spectrum) as standard, the results obtained from the wave spectra of three other kinds are compared as shown in Table 3.

As understood from Fig. 6 and Table 3, in either case of the 150 meter long cargo ship and the 300 meter long tanker, no significant

difference is generally seen in. the results obtained by using four kinds of wave spec-trum, though more or less difference found

in them.. The value of standard deviation is large in the case of head seas or following seas, and decreases as the heading angle is changed from head seas or following seas to beam seas, - but if the value of standard de-viation is large, the difference due to four

90 Jun-ichi FUKUDA (Vol. XXVI,

L length between perpendiculars B breadth of ship

h0 amplitude of wave elevation

A wave length

heading angle to waves (cl=O' head waves)

Fr. Froude number

C block coefficient

The results of standard deviation of wave bending moment in long-crested irregular seas arid short-crested irregular seas evalu-ated by using these response operators are shown in the diagrams after Fig. 6.. The standard deviation of wave bending moment R is obtained by evaluating the variance of wave bending moment R2, using the wave spectra of (21).(24) and applying the meth-od of linear superposition. That is,

(i) In the case of long-crested irregular

waves

[Mo(w)/ho]2,p=o[f(o-')Pdw (25)

(ii) In the case of short-crested irregar wave

.i'._,r/2 I

kinds of wave spectrum is small, and the dif-ference is comparatively large when the value of standard deviation is small. It is suppos-ed, therefore, that the difference due to four kinds of wave spectrum will have no serious effect on the estimated results of long-term distribution.

When the results given in Fig. 6 are ex-amined more carefully, however, the results obtained from the wave spectra of (a), (b) and (c) have a tendency comparatively well resembled to one another, whereas only the results obtained from the wave spectrum of (d) show a tendency somewhat different from the others. That is, while the value of aver-age wave period that gives the maximum

value of R/pgL2BH, when the value of

stand-ard deviation is large in the case of head seas or sollowing seas, does not so differ in any case of (a), (b) or (c), it is somewhat smaller in the case .of wave spectrum of (d) and the curve of R/pgL2BH has a rather sharp peak. The reason for this is because that the peak frequency and the peak value

of each wave spectrum of (a), (b) and (c)

are not so different but those of wave spect-rum of (d) are somewhat different from the other&

As mentioned so far, the results obtained by using three kinds of wave spectrum of (a), (b) and (c) have a tendency comparatively resembled to one another, except the results obtained from the wave spectrum of (d), which have a tendency somewhat different from the others. Accordingly, the author has

carried out the comparisons on the results

obtained by using (a) modified

Pierson-Mos-kowitz spectrum (ISSC spectrum) and (d)

modified Darbyshire spectrum (BTTP spect-rum) as follows.

InFig 7, the results of calculation in short-crested irregular seas are shown. Similarly as in the case of Fig. 6, the heading angle to waves is used as the parameter and the

values of R/pgL2BH, are expressed as func-tions of average wave period.

In Fig. 8, the values of RJ pgL2BH in

long-crested irregular seas and short-long-crested ir-regular seas are represented as functions of heading angle to waves with the parameter of average wave period.

From the results shown in Figs. 6.8, we can find out the following facts.

Either in long-crested irregular seas

or short-crested irregular seas, the dimension-less standard deviation of wave bending mo-ment RJpgL2BH is large in head seas or fol-lowing seas, but it is small in beam seas.

In the case in head seas or following seas where the value of RJpgL2BH is large, the short-crested irregular seas give some-what smaller dimensionless R-values than that in long-crested irregular seas, and in

the case of beam seas where its value is

small the short-crested irregular seas give a little larger one. Hence, the change of the value of R/pgL2BH,, due to the heading angle to waves is more gentle in the case of short-crested irregular seas.

When the modified Pierson-Mosko-witz spectrum (ISSC spectrum) is adopted, in either case in long-crested irregular seas or short-crested irregular seas, the value of R/pgL2BH becomes maximum in the case of head waves or following waves among the sea state of average wave period of about 8 seconds by the 150 meter long cargo ship and about 12 seconds by the 300 meter long tanker. Similar tendency is seen also in the case of modified Darbyshire spectrum (BTTP spect-rum), but such a value of average wave pe nod as giving the maximum value of R/pg L2BHV is smaller than in the former case, being about 65 seconds by the 150 meter long cargo ship and about 9 seconds by the 300 meter long tanker. And, in the neighborhood of average wave period where the value of R/ pgL2BH becomes maximum, the curve of

the dimensionless R is more sharp than in the case of the former. Except these points there

is no great difference in the value

of R/pgL2BH due to the wave spectra of the two.

Now, as stated in the above (iii), the dif-ference in the value of average wave period that gives the maximum value of RJpgL2BH due to the modified Pierson-Moskowitz

spec-trum (ISSC specspec-trum) and the modifed

Darbyshire spectrum (BTTP spectrum) is dependent on the difference in the peak fre-quency of wave spectra of the two (see Fig. 3). If the modified Pierson-Moslcowitz

92

trum (ISSC spectrum) of (21) is modified with w5=(3.l5/T) +(8.98/T02) adopted in the

formula of modified Darbyshire spectrum

The evaluated results of

RJpgL2BH by

using the wave spectrum thus modified are compared with those by using the modified Darbyshire spectrum (BTTP spectrum) of (24), as shown in Fig. 9. The difference tween the two is much smaller than that be-tween the cases depending upon the wave

spectra of (21) and (24) (see Fig. 6). It is un-derstood from the above that the difference between the results obtained by using the

modified Pierson-Moskowitz spectrum (ISSC spectrum) of (21) and the modified Darby. shire spectrum (BTTP spectrum) of (24) is mainly caused by the difference in the peak frequency of wave spectrum and little

depend-ent on the shape of wave spectrum. In the

case where the modified Pierson-Moskowitz spectrum is used, however, there is a t

end-ency giving larger.value of R/pgL2BH than

in the case where the modified Darbyshire pectrum is used, and such a tendency is remarkable in the case of a long ship. 4. Conclusion

The author investigated the method to re-present the short-term sea states by means of wave spectra, which is necessary for the estimation of short-term and long-term dis tribution of wave bending moment. Namely, by modifying practically the original expres-sions of theoretical wave spectrum propos-ed by some oceanographers so as to fit to the short-term sea states defined by the visually estimated average wave height (significant wave height) and average wave period, he assumed the wave, spectra representing such

short-term sea states. With regard to the

original expressions of theoretical wave spec-trum, four kinds of formula by (a)

Pierson-Moskowitz, (b) Neumann, (c) Roll-Fischer,

and (d) Darbyshire are taken up. And, by

modifying these original wave spectra, he has assumed such formulae of wave spectra as satisfying the visually estimated average

Jun-ichiFtrKUDA (Vol. XXVI,

(BTTP spectrum) of (24) So that the peak frequency may coincide with the case of wave spectrum of (24), the transformed for-mula will be written as follows

wave height (significant wave height) and

average wave period.

The standard deviations of wave bending moment were evaluated on the 150 meter long

cargo ship and the 300 meter long tanker in

short-term sea states by' using these wave spectra and the obtained results were com-pared and investigated. There is no great difference among the results obtained from' four kinds of wave spectrum. The results ob-tained from the wave spectra of (a), (b) and

(c) types show a tendency almost resembled,

whereas only the result obtained from the wave spectrum of type (d) gives a somewhat different tendency. That is, when the dimen-sionless standard deviation of wave bending moment R/ pgL2BH which is obtained by us-ing (d) modified Darbyshire spectrum (BTTP spectrum) is compared with those which are obtained from other wave spectra, for

ins-tance, (a) modffied Pierson-Moskowitz

spect-rum (ISSC spectspect-rum), the former shows a tendency to give a maximum of dimension-less R-value in a sea state of rather small average wave period. The cause for that is more dependent on the fact that the peak frequency of the modified Darbyshire spec-trum (BTTP specspec-trum) is lower than that of other wave Spectra and less dependent on

the difference in the shape of wave spectrum. If we modify further the modified Pierson-Moskowitz spectrum (ISSC spectrum) and make its peak frequency agree with, that of the modified Darbyshire spectrum (BTTP spectrum), 'the difference in dimensionless R-value obtained by the two wave spectra

will become smaller.

It is therefore most important, when as-suming thew ave spectra for the estimation of short-term and long-term distribution of wave bending moment, how to equate the, peak frequency of wave spectrum with the average wave period, and the shape of wave, spectrum has little significance. There are;

[1(w)]2/H2 = O.11w'(w/) exp (-0.44 (w/äi) _4) ₍₂₇₎

1966) Family of Wave Spectra for Prediction of Ship Bending Moments 93'

however, few reliable data at present about the relation between the visually estimated average wave period and the peak frequency

of wave spectrum. At present, it seems to

be too early to admit the equation expressing the relation between the peak frequency and the average wave period in the modified Dar-byshire spectrum (BTTP spectrum). More-over,we considering that the value of standard deviation of wave bending moment obtained by the modiñed Pierson-Moskowitz spectrum (ISSC spectrum) is somewhat larger and more on safe side than those obtained from other wave spectra, it seems to be appropriate at, the present stage to adopt the modified Pier-son-Moskowitz spectrum (ISSC spectrum) as

the family of wave spectra for the predic-tion of wave bending moment. When the* Iatiön between the visually estimated average wave period and the peak frequency of wave spectrum is made clear in the near future, we may modify, according to it, the abscissa T0 of the figure showing the relation between the average wave period T0 and the dimensionless standard deviation R/ pgL2BH (for example, Figs. 6 and 7). Also in the future when we are necessitated to modify the relation be-tween the visually estimated average wave height H,, (which is believed to be equal to the significant wave height) and the area of wave spectrum by some new findings, we may modify the vertical axis R/pgL2BH ac-cording to them. The pressing need at present is to clarify such relations between the vis-ually estimated average wave period and the peak frequency of wave spectrum and also the relation between the visually estimated average wave height and the area of wave spectrum. Under the present circumstances, therefore, the modified Pierson-Moskowitz wave spectrum (ISSC spectrum) would be appropriate for the purpose of short-term and long-term prediction of wave bending mo-ment and might be acceptable as an interim family of wave spectra.

Acknowledgment

Before ending this paper, the author wishes

to express his deep thanks to Mr. I. Hata and Mr. S. Tsutsumi for their cooperation in the numerical calculations of.this.work..

Reférénëés -

-W. H. Warnsinck: "Report of Committee 1 on

Environmental Conditions" Proceedings of ISSC, 1964, Deift

H. U. Roll: "Height, Length and Steepness of Seawaves in the North Atlantic and Dimensions of Seawaves as Functions of Wind Force" (Eng-lish Translation) Technical and Reeearch

Bul-letin No. 1-19 SNAME (1958)

Y. Yamanouchi, S. Unoki, T. Kanda: "On the

Winds and Waves on the Northern North Pacific Ocean and South Adjacent Seas of Japan as the Environmental Conditions for the Ship" Papers of Ship Reserch Institute (Tokyo) No. 5 (1965) D. E. Cartwright: "A Comparison of

Instrumen-tal and Visually Estimated Wave Heights and

Periods Recorded on Ocean Weather Ships"

Ap-pendix of Ship Rep. 49, NPL Ship Division (1964)

G. Neumann: "On Ocean Wave Spectra and a

New Method of Forecasting Wind Generated Sea"

Technical Memorandum No. 43, Beach Erosion,

Board (1953)

J. Darbyshire: "A Further Investigation of Wind

Generated Waves" Deutsche Hydrographische

Ze-itschrift, Band 12; Heft 1 (1959)

H. U., Roll, G. Fischer: "Eine Kritische

Bermer-kung zum Neumann-Spectrum des Seegangs"

Deutsche Hydrographische Zeitschrift, Band 9,

Heft 1 (1956)

W. J. Pierson Jr., L. Moskowitz: "A Proposed Spectral Form for Fully Developed Wind Seas

Based on the Similarity Theory of S. A.

Kitai-gorodski" New York University G.S. Report

63-12 (1963)

S. 0. Rice: "Mathematical Analysis of Random Noise" The Bell System Technical Journal, Vol.

23 (1944), Vol. 24 (1945)

M. S. Longuet-Higgins "On the Intervals

be-tween Successive Zeros of a Random Function" Proceedings of Royal Society of London, Series

A, Vol. 246, No. 1244 (1958)

G. J. Goodrich: "Discussion to the Report of

Com-mittee 1 on Environmental Conditions"

Proceed-ings of ISSC 1964, Delft

J. R. Scott: "A Sea Spectrum for Model Tests

and Long-Term Ship Prediction" JSR, Vol. 9, No. 3 (1965)

J. Fukuda: "Computer Program Results for

Response Operators of Wave Bending Moment

in Regular Oblique Waves" Journal of the Society

Nomenclatures

B : breadth of ship

C2 : block coefficient

C, water plane area coefficient

C : midship sectional area coefficient

E : twice of variance of wave

eleva-tion co

Fr. Froude number

H visually estimated average wave coo

height (significant wave height)

L : length between perpendiculars A

M0 : amplitude of midship bending mo- p

ment

R : standard deviation of wave bending

moment 0

R2 : variance of wave bending moment

S : standard deviation of wave ele- z

vation

52 _{: variance of wave elevation}

W : weight of displacement

Table 1. Comparison of various type wave spectra

m

drought

spectral density of waves acceleration of gravity wave amplitude

n-th moment of wave spectrum parameter representing the width of spectrum

circular frequency of component

wave

peak circular frequency of wave

spectrum wave length

density of sea water

heading angle to regular wave di-rection

heading angle to the average di-rection of irregular waves

angle between a componet wave direction and the average direc-tion of irregular waves

(a) p=r5, q=4 (b)p=6, q=2 (c) p=5, q=2

82 0. 1803 0.1781 0.2732

Tv/2ir (=-1) mo/mi i/mo/rn2 mo/mi i/mo/rn2 mo/rn1 v'rno/m2

p1 0.1109 0.0796 0.3904 Q. 2591 0.2026 0. 1250

Q1 0.4435 0.3183 1.7672 1.5000 1.2732 1.0000

0.7718 0.7104 0.7675 0.7071 0.7137 0.6325

[f(wo)] 2/H2w0_l 0. 1f60 0. 1260 0.0951 0. 1032 0.0899 0.1014

Table 2. Main Particulars of thesship

Cargo Ship Oil Tanker

Length between perpendiculars (L) 150. 000 m 300. 000 m

Length/Breadth (L/B) 7.000 6.000

Length/Drought (Lid) 17. 500 17.500

Breadth/Drought (B/d) 2.500 2.917

Block coefficient (C2) 0.700 0.830

Water plane area coefficient (Cm) 0.787 0.891

Midship sectional area coefficient (C) 0.986 0.993

Centre of buoyancy from midship (forward) 0.005L 0.034L

Longitudinal gyradius 0.250L 0.245L

Weight of displacement (W) 19,766 ton 218,127 ton

Afterbody weight 0.4883W 0.4825W

Forebody weight 0. 5117W 0. 5175W

Afterbody moment about midship -0. 1025WL -0. 0876WL Forebody moment about midship 0. 1075WL 0. 1212WL

Afterbody 2nd. moment about midship 0. 0305WL 0. 0223 WL2

Forebody 2nd. moment about midship 0. 0320WL2 0. 0389WL2

Still water bending moment (hog) 0. O1O8WL -0. 003OWL

1966) Family of Wave Spectra for Prediction. of Ship Bending Moments 95

Ra : Estimated in modified Pierson-Moskowitz wave spectrum

Rb : Estimated in modified Neumann wave spectrum Rc: Estimated in modified Roll-Fischer wave spectrum Rd: Estimated in modified Darbyshire wave spectrum

Table 3. Comparison of results in various type wave spectra (long crested irregular seas)

150 m Cargo Ship 300 m Oil Tanker

Rb/Ra Rb/Ra 8 (deg.) T (sec.) T (sec.) 4 6 8 10 12 6 8 10 12 14 0 1.356 0.993 0.928 0.960 0.997 1.200 1.040 0.959 0.954 0.967 30 1.386 0.966 0.934 0. 973 1. 009 1.278 0.954 0.954 0.959 0. 975 60 1.129 0. 940 0. 976 0.992 1.043 1.104 0.954 0.963 0.978 0.995 90 1.016 0.948 1.025 1.064 1.070 1.059 0.934 0.960 1.005 1.038 120 1. 126 0.942 0. 965 1.008 1.036 1.104 0.953 0.953 0.979 1.006 150 1.516 0.980 0. 932 0.966 1.001 1.420 1.024 0. 948 0.951 0.974 180 1.586 0.997 0.929 0.953 0.987 1.454 1.076 0. 957 0.946 0.963 Rc/Ra Rc/Ra 9 T0 (sec.) T (sec.) (deg.) _{4 6 8 10 12 6 8 10 12 14} 0 1.750 1.027 0.885 0.901 0.944 1.465 1.109 0.950 0.913 0. 914 30 1.748 0.976 0.883 0.913 0.961 1.540 1.041 0.933 0.914 0.921 60 1.256 0.893 0.915 0.952 1. 036 1.218 0.943 0.921 0.926 0. 945 90 1.067 0.885 0. 969 1.049 1.106 1. 142 0.899 0.897 0.944 0.994 120 1.264 0. 904 0.905 0.961 1.018 1.228 0.942 0.905 0.922 0.953 150 1.970 1.001 0.886 0.907 0.950 1.790 1.085 0.928 0.902 0. 915 180 2.156 1.047 0.892 0.895 0.932 1.895 1.175 0. 954 0.905 0.907 Rd/Ra Rd/Ra 8 T (sec.) T (sec.) (deg.) ₄ 6 8 10 12 6 8 10 12 14 0 0.597 1.172 0.896 0.777 0.805 0.994 1.274 0.971 0.805 0.748 30 0.449 1.185 0.865 0.782 0.836 1.034 1.215 0.926 0.794 0.756 60 0.348 1.007 0.816 .0.838 1.012 1.023 1.010 0.863 0.792 0.798 90 0. 441 0.954 0.835 0.972 1. 189 1. 108 0.968 0.772 0.787 0.874 120 0.287 1.070 0.820 0.838 0.967 1.092 1.041 0.827 0.773 0.803 150 0.219 1. 166 0.885 0.781 0.817 1.002 1.255 0.931 0.774 0.742 180 0.279 1.132 0.931 0.780 0.787 1.015 1.334 0.994 0.794 0.735

'I,

'I

:1

'I

:(a) Pierson-Moskowitz type. (b) Neumann type Ic) Roll-Fischer type

assumed with Tv 2 irIme/m,) Hv W 0.10 0.05 0 0

Pjerson-Moskowlfz type (I.S.S.C-I964) Darbyshire type (B.T.T.P.)

Tv

I2sec

Tv

Lsec

0.10 0.05 0

(scales to be used for ujo)

2015 I2

6 Tv 4

0 0.5 1.0 1.5 0 2 4 6 8 10 I2 Tv I6(uc) 2D 15 1.0 08 0.7 0.6 0.5 ..iIls_ O4(oeJ

(scales to be used for (f(w)]2/H)

Fig. 2.

Peak frequency and peak value of various type wave spectra

0 0.5 1.0 ., 1.5 20 Fig. 3.

Comparison of ISSC and BTTP wave spectra

0.5

1.0

15

w/Wv 20

Fig. 1.

Various type wave spectra in dimensionless form

O;25 0.20

0.3

1966) Family of Wave Spectrafor Prediction of Ship Bending Moments 97 0 0 0 0.5 1.0

1.5 .L/A 2.

05 10

15 Vi7 2.0

Fig. 4. Response operators of vertical wave bending moment at midship

(Cb =0.83)Tanker

Fr=0.I5

1= o

'=30

= 90°:

=60°:----

---"=120°: ,=160°.

-!(L9 \ \

-/

\

i'

(Cb =0.70)-Ccrgo Ship Fr=0.15 (j'= _0°

30°:--=

60°.----

90° I 20° :

-"-150°:

°=!8O°.

P

\\

/1"

0.03 ID 0 0.02 0.0! 0 0.03 0 ID 0.02 0.0!

U 0 U,

'0

OC N Tanker. : L300m,Cb O.83, Fr 0.15 8 0°

_.4

II I'

I'

: I.S.S.C. spect,um o : BTT.P. spectrum - A (w) Mo/pL2Bho 0.2 0.4 0.6 0.8 1.0 1.2 I 4 - w

(sec)

-0 -N

4-

3- 2-in longcresfed sea Tv' Bsec 8 = 0° Fig. 5.

Energy spectra of vertical wave bending moment at midship

Cargo Ship: L I 50m, Cb=0.70, Fr =0.15 + : :l.S.S.C. spectrum o :B.T.T.P. spectrum-

_\

w)=Mo/pfBho 0.2 0.4 0.6 0.8 1.0 1.2 a) (sec') 14 C-, 0) 0,

'0

in long-crested TvoI2sec seas + -U 0

_'O

0.0 03 0.002 O.00l aD 0.003 0.002 0.00 0 0.003 0.002 0.002 0.001 0

Fig. 6. Standard deviations of vertical wave bending moment in long-crested irregular seas as functions of visual average wave period

Cargo ship: L l50m , Cb-0.70, Fr =0.15 (in long-crested seas)

6' -oo - '-I /

,..

-

;:-/

/

-.--.---

-. :..

'I

-_

(a) Pierson -Moskowitz type (C) Roll-Fischer type

-/ 6'O° 150° - - _d00 / I .

-1;'

:

300

-

/",/ 90°

(b) Neumann type , Cd) Darbyshire type

Tanker L = 300rn ,Cb =0.83, Fr 0.15 (in long-crested seas)

iec°

59-

-(a) Person-Moskowitz type (c) RotI-Fischer type

1800

°_----i:

=o°

-Cd) Dorbyshire type (b) Neumann type

1966) Family of Wave Spectra for Prediction of Ship Bending Moments 99

6 8 10 12 4 6 l0 (2

- Tv (sec)

6 8 l0 l2 14 6 8 (0 (2 '4

0.0 03 O0O2 0.0OI 0.00 3 0.002 0.001 0

Cargo ship: LI50m, Cb= 070

Fr0.15 (11.2k?) in short-crested seas

Wave spectra : Pierson-Moskowitz type

-/500

Wave spectra: Darbyshpre type

4 6 8 tO 12

Tv (sec)

Fig. 7. Standard deviations of vertical wave bending wave period 0.0 03 0.002 .0.00l c'JJ

I0

0.003 0.002 0.001 0 6 8 10 12 14 - Tv (sec)

moment in short-crested irregular seas as functions of visual average

Tanker L300m, CbO.83

Fr =0.15 (15.8kt)

in short-crested seas

Wave spectra : Pierson-Moskowltz type

-900

Wave spectra Darbyshire type

/

'

6 0d

/20° 900

0 0.001 J -.2: 0.003 0.00 2 0.003 0.002 0.00I 0 0 30 60

-0

90 120 150 l80 0 30 60 90 120 150 180c -.-0

Fig. 8. Standard deviations of vertical wave bending moment as functions of heading angle to waves

Cargo ship :.L 150m, C0.7O

Fr = 0.15 (I I.2 kt ).

Wave spectra : Oarby shire type

(B.T.T.P.)

(in Ionq-crested seas)

'a /

-. sn _,/

-(in short-crested seas)

- - - 4 .nc.

Cargo ship: L 150m /Cb 0.70 Fr = 0.15 (11.2 kt

Wove spectra P ersonMosowitz type

(l.S.S.C.- 964)

7.$--(in long-cresled seas)

\-J

(in short-crested seas)

----

-Tanker L 300m, Cb = 0.83 Fr =0.15 (I5.8k1)

Wave spectra Dorbyshire type

(8.T.T.P.) (in long-crested seas)

miii:

0.00!

Tanker : L=300m, Cs = 0.83 Fr = 0.15 (15.8 kf)

Wove spectra : PiersonMOskOWitZ type

CI. S. S.C.-! 9 64 (in

-

---longcrested seas)

\.,

/

-, .z_' /2sçc_ &.

- (in shortcrested seas)

-0.003 0.002 0.003 0.00 2 0.00! 0 30 60 90 I20 50 180° 0 30 60 90 120 150 (800

,1966) Family of Wave Spectra for Prediction of Ship Bending Moments 101

-8

0 0.003 0.002 0.00! 0.00 3 0.002 0.00 I 0 0.003 0.002 : 0.001 J 0.003 0.002 0.00I

0.003 0.002 > 0.00 I

I

cn cJ 0.003 0.00 2 0.001 0 Cargo ship: L = 150 m, Cb = 0.70 Fr 0.15

in long-crested wave spectra modified with (W0o 3.I5/Tvi-8.98/Tv2)

Person-Moskowitz type Darbyshre type 0.00 3 0.002 0.001 CD 0.003 0.0 02 0.00l 0 Tanker L=300rn,Cb=0.83,Fr =0.15

in long-crested wave spectra modified

with (t'o3.l5 /Ty+ 8.98/Tv2)

1200 1500 Person-Moskowitz type -Darbyshire type. 4 6 JO 12 6 8 JO 12 14 - Tv (sac) - Tv (sec)

Fig. 9. Comparison of results in modified Darbyshire wave spectra (BTTP spectra) and further modified Pierson-Moskowitz wave spectra