Modelling of the Probabilistic Input for a Nonlinear Wave-Load Simulation Model

MODELLING OF THE PROBABILISTIC

INPUT FOR A NONLINEAR WAVE-LOAD

SIMULATION MODEL

L.J.M. Adegeest

Report No. 933

Jijiie 22, 1992

Dalft University of T e c h n o l o g y

Ship Hydromechanics Laboratory Mekelweg 2

Modelling of the Probabilistic Input for a

Nonlinear Wave-Load Simulation Model

L.J.M. Adegeest*

June 22, 1992

A b s t r a c t

For the calculation of long-term fatigue and extreme loads i n vessels, a statistically correct description of the environmental conditions is re-quired. I n order to take into account nonlinear response effects, t i m e domain methods are apphed. These methods require a reahstic descrip-tion of an irregular wave surface i n the time domain satisfying spectral and statistical properties resulting f r o m oceanographical research. I t is shown that i n case of nonlinear simulation models, a random wave amplitude model should he applied.

Commonly used parameters i n the statistical description of the waves are the significant wave height, a characteristic wave period and a wave spectrum. Usually i n f o r m a t i o n ahout the spectral shape is not available. For that reason a modification of the Pierson-Moskowitz spectrum or a J O N S W A P spectrum is usuahy applied. B o t h spectral shapes have their limitations. The use of a Wallops spectrum eliminates the problem of the choice of a suitable spectrum since the formulation is applicable to general sea states. The spectral shape is a f u n c t i o n of the variance and the peak period and has a variable band w i d t h . I t is shown that this spectrum satisfies theoretical and measured wave height distributions very well.

1 I n t r o d u c t i o n

For t h e design a n d c o n s t r u c t i o n o f ships t h e e x t r e m e responses o f the s t r u c t u r e m u s t be k n o w n . These extreme responses occur i n c e r t a i n ocean wave c o n d i t i o n s f r o m w h i c h t h e p r o b a b i l i t y o f occurrence a n d severeness m a y be p r e d i c t e d o n t h e base o f l o n g t e r m ocean wave s t a t i s t i c s . I n an o p t i m u m s t r u c t u r a l design process b o t h t h e s h o r t - t e r m and the l o n g - t e r m response s t a t i s t i c s are needed. T h e s h o r t - t e r m s t a t i s t i c s can be d e r i v e d f r o m the response o n a s t a t i o n a r y r a n d o m seastate. U s u a l l y a c e r t a i n seastate can be considered t o be s t a t i o n a r y

o n l y f o r a f e w hours, c o r r e s p o n d i n g w i t h the s h o r t - t e r m response p e r i o d . T h e l o n g - t e r m response s t a t i s t i c s are r e l a t e d t o t h e expected l i f e t i m e o f the ship a n d can be considered as a s u m m a t i o n o f s h o r t - t e r m responses.

L i n e a r s h i p responses i n i r r e g u l a r waves have succesfully been described i n the f r e q u e n c y d o m a i n . U n d e r t h e a s s u m p t i o n of a Gaussian e x c i t a t i o n b y t h e i r -regular waves, t h e l i n e a r response s p e c t r u m is g i v e n b y t h e p r o d u c t of the square of t h e f r e q u e n c y t r a n s f e r f u n c t i o n a n d the wave s p e c t r u m . T h e s t a t i s t i c a l p r o p -erties of these processes have been s t u d i e d by m a n y researchers, especially i n t h e f i e l d o f oceanography. P a r a m e t e r s d e s c r i b i n g a s h o r t - t e r m seastate are f o r i n s t a n c e t h e s i g n i f i c a n t wave h e i g h t a n d a peak p e r i o d o f t h e wave s p e c t r u m . I t has been observed t h a t due t o the v a r i a b i l i t y i n s h o r t - t e r m e n v i r o n m e n t a l c o n d i t i o n s t h e s p e c t r a l shape varies as w e l l . T y p i c a l s p e c t r a l shapes c o m m o n l y used i n t h e ship response statistics are t h e P i e r s o n - M o s k o w i t z - s p e c t r u m [21] f o r f u l l y developed seas a n d t h e J O N S W A P s p e c t r u m [9] f o r f e t c h - l i m i t e d seas. T h e I T T C proposed a m o d i f i c a t i o n o f the P - M s p e c t r u m i n t e r m s o f t h e sig-n i f i c a sig-n t wave h e i g h t a sig-n d peak p e r i o d . A s s h o w sig-n b y S t . D e sig-n i s [23] t h i s s p e c t r a l f o r m u l a t i o n involves some c o n t r a d i c t i o n s on t h e assumed narrow-bandedness. T a y f u n [24] showed t h a t t h e s p e c t r a l b a n d w i d t h effects the d i s t r i b u t i o n o f wave heights a n d p e r i o d s . For t h a t reason the W a l l o p s s p e c t r u m [11] is p r o p o s e d f o r t h e s t u d y o f t h e s h o r t t e r m responses as w e l l as o f l o n g t e r m responses. L o n g -t e r m p r e d i c -t i o n s s h o u l d be based o n -the j o i n -t p r o b a b i l i -t y o f s i g n i f i c a n -t wave heights, peak periods a n d s p e c t r a l shapes. T h e W a l l o p s s p e c t r u m is a p p r o p r i -ate t o a general sea s t a t e a n d t h e shape o f the s p e c t r u m is d e t e r m i n e d b y t h e peak p e r i o d a n d t h e s i g n i f i c a n t wave slope.

T h e o r e t i c a U y the s h o r t t e r m d i s t r i b u t i o n of wave heights as w e l l as of l i n -ear responses are R a y l e i g h d i s t r i b u t e d . A l t h o u g h m a n y d i s t r i b u t i o n f u n c t i o n s have been suggested at times there is considerable evidence t h a t t h e p r o b a b i h t y d i s t r i b u t i o n o f the wave heights of Gaussian waves is q u i t e w e l l described b y a R a y l e i g h d i s t r i b u t i o n as proposed b y L o n g u e t - H i g g i n s [17]. T h i s has b e e n p o i n t e d o u t b y V i n j e [26] w h o suggested a m u l t i p l i c a t i o n f a c t o r i n f r o n t o f t h e o r i g i n a l d i s t r i b u t i o n a n d gave a n e s t i m a t e o f the R a y l e i g h coefficients f o r Gaus-sian waves as w e l l as i n the range o f s t r o n g l y n o n h n e a r effects. These char-acteristics can also be a p p l i e d t o t h e statistics o f linear responses. E x t r e m e s can be e s t i m a t e d f o r the c a l c u l a t i o n o f the desired m a x i m u m s t r e n g t h a n d t h e d i s t r i b u t i o n o f loads can be d e t e r m i n e d a n d used f o r f a t i g u e c a l c u l a t i o n s .

I f the responses are s t r o n g l y n o n l i n e a r another s t r a t e g y has t o be f o l l o w e d . I n t h i s s t u d y t h e m o s t a t t e n t i o n is p a i d t o a specific n o n l i n e a r p h e n o m e n o n o c c u r r i n g at slender vessels i n waves, i.e. t h e r a t i o of t h e sagging b e n d i n g m o -m e n t s a n d t h e h o g g i n g b e n d i n g -m o -m e n t s w h i c h appears t o be u n e q u a l t o one. I t has been s h o w n t h a t t h i s effect is m a i n l y caused by a change of i m m e r s e d v o l u m e i.e. t h e r e l a t i v e m o t i o n of fiaring sections, see f o r instance [12]. A f o r -m u l a t i o n f o r t h e e s t i -m a t i o n o f the f a t i g u e da-mage s u b j e c t e d t o q u a d r a t i c wave loads o n slender vessels has been presented b y Juncher-Jensen [13]. I t was f o u n d t h a t due t o the n o n l i n e a r d e s c r i p t i o n o f the loads the f a t i g u e damage increased

w i t h 5 0 - 1 0 0 % d e p e n d i n g o n the speed of the vessel. A s i m i l a r m e t h o d is ap-p l i e d t o offshore s t r u c t u r e s by M a d s e n [18]. T h e i m ap-p o r t a n t advantage o f these m e t h o d s is the reduced c a l c u l a t i o n t i m e c o m p a r e d t o t i m e d o m a i n s i m u l a t i o n s . T h e m e t h o d is l i m i t e d t o s h g h t l y n o n h n e a r systems. P h e n o m e n a such as s l a m -m i n g , b o w f l a r e s l a -m -m i n g a n d deck wetness can n o t be dealt w i t h w h i c h -means t h a t e x t r e m e loads can n o t be p r e d i c t e d u s i n g t h i s approach. Besides t h a t the p h y s i c a l i n t e r p r e t a t i o n o f the results is n o t as easy as o f t i m e records.

For the reasons m e n t i o n e d above n o n l i n e a r t i m e - d o m a i n s i m u l a t i o n models have become very p o p u l a r over the last years f o r the c a l c u l a t i o n of t h e n o n l i n e a r b e h a v i o u r of f l o a t i n g s t r u c t u r e s . A n a l y s i s o f t h e s i m u l a t e d t i m e - r e c o r d s give us the s h o r t - t e r m s p e c t r a l a n d s t a t i s t i c a l i n f o r m a t i o n w i t h a c e r t a i n degree o f accuracy. For p r a c t i c a l reasons i t is i m p o s s i b l e t o p e r f o r m l o n g - t e r m l i f e - t i m e s i m u l a t i o n s i n order t o i n v e s t i g a t e t h e l o n g - t e r m f a t i g u e a n d e x t r e m e loads. A second p r o b l e m is the d e s c r i p t i o n o f the i r r e g u l a r waves w i t h respect t o the realistic s h o r t - t e r m s t a t i s t i c a l p r o p e r t i e s of the i n p u t as w e l l as of t h e r e s u l t i n g responses. A desired accuracy o f t h e s t a t i s t i c a l p r o p e r t i e s o f t h e waves a n d of t h e responses has t o be o b t a i n e d a f t e r a l i m i t e d n u m b e r o f s i m u l a t i o n s of finite l e n g t h .

I n t h e f o l l o w i n g p a r a g r a p h s the d e s c r i p t i o n of realistic e n v i r o n m e n t a l c o n d i -tions w i l l be discussed. T h e r e a l i z a t i o n of wave records w i l l be discussed w i t h respect t o the spectral shape, r a n d o m n e s s a n d accuracy as well as the l o n g - t e r m d i s t r i b u t i o n of sea states, defined b y s p e c t r a l shape, s i g n i f i c a n t wave heights a n d peak periods.

2 The l o n g - t e r m d i s t r i b u t i o n of environmental

conditions

T h e m a i n e n v i r o n m e n t a l c o n d i t i o n s as experienced b y a vessel i n an i r r e g u l a r seaway are d e t e r m i n e d b y a s p e c t r a l shape S a n d a characteristic wave p e r i o d Tp or f r e q u e n c y w,, a n d wave h e i g h t i / 1 / 3 . E v e n t u a l l y a d i r e c t i o n a l s p r e a d i n g f u n c t i o n D m a y be used. Parameters such as c u r r e n t and w i n d can n o t be neglected i n case o f o f f s h o r e s t r u c t u r e s b u t are o f m i n o r i m p o r t a n c e f o r a s h i p .

These stochastic q u a n t i t i e s i n c o m b i n a t i o n w i t h the ship's speed, r e l a t i v e heading, g e o m e t r y a n d w e i g h t d i s t r i b u t i o n d e t e r m i n e the e x c i t a t i o n c o n d i t i o n . T h e r e l a t i v e h e a d i n g o f a vessel is d e t e r m i n e d b y the p r i m a r y wave d i r e c t i o n a n d the course o f t h e vessel. For a given vessel w i t h k n o w n g e o m e t r y a n d mass d i s t r i b u t i o n , denoted w i t h the stochastic vector M, saihng w i t h a speed U a n d a r e l a t i v e h e a d i n g fi, t h e l o n g - t e r m p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n o f t h e wave-i n d u c e d loads, d e f wave-i n e d as L, can n o w be w r wave-i t t e n as t h e wave-i n t e g r a l o f a c o n d wave-i t wave-i o n a l s h o r t - t e r m p r o b a b i l i t y d e n s i t y f u n c t i o n o f the loads, m u l t i p l i e d w i t h t h e j o i n t p r o b a b i l i t y density f u n c t i o n o f t h e o p e r a t i o n a l a n d c l i m a t o l o g i c a l p a r a m e t e r s M , 7 Ï 1 / 3 , Tp, S a n d D : PL{L) = J ... J PL{L I M , [ƒ, ^L, Hif3,Tp,S, D)dMdUdndHdTdSdD ( 1 ) T h e c o n d i t i o n a l s h o r t - t e r m p r o b a b i l i t y density f u n c t i o n s are o b t a i n e d b y means of statistics of Gaussian processes i n t h e range o f linear responses or b y means of a n o n l i n e a r s i m u l a t i o n p r o g r a m [1]. I n a c o m p l e t e p r o b a b i h s t i c m o d e l the effect o f v o l u n t a r y changes o f speed a n d h e a d i n g have t o be i n c o r p o r a t e d t o o , w h i c h makes i t v e r y c o m p l i c a t e d t o d e t e r m i n e the j . p . d . f u n c t i o n o f t h e op-e r a t i o n a l a n d c l i m a t o l o g i c a l p a r a m op-e t op-e r s . A s i m i l a r op-exprop-ession has bop-eop-en d op-e r i v op-e d b y D a l z e l l [6] b u t i n h i s expression, an e x t r a stochastical v a r i a b l e occurs t a k i n g i n t o account the fiexible r i g i d i t y a n d s t r u c t u r a l d a m p i n g p r o p e r t i e s o f the ves-sel. D e p e n d i n g o n the t y p e o f s h i p , e q u a t i o n (1) can be s i m p l i f i e d w i t h respect t o t h e j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n of the c l i m a t o l o g i c a l a n d o p e r a t i o n a l p a r a m e t e r s .

T h e p r i m a r y wave d i r e c t i o n f o r instance is o n l y o f i m p o r t a n c e i f a vessel has a n o n u n i f o r m l y d i s t r i b u t e d course. G e n e r a l l y i n case of navyvessels, a u n i -f o r m l y d i s t r i b u t e d course m a y be assumed. A q u a h t a t i v e s t u d y o n the i n -f i u e n c e of u n c e r t a i n t i e s i n t h e e x c i t a t i o n c o n d i t i o n s has been p e r f o r m e d b y Guedes-Soares [8]. B y some c a l c u l a t i o n examples p e r f o r m e d f o r d i f f e r e n t ship types, i t is s h o w n t h a t the i n f l u e n c e of t h e ship's speed o n t h e d i s t r i b u t i o n o f wave-i n d u c e d loads can wave-increase the characterwave-istwave-ic response value w wave-i t h a b o u t .30%. F u r t h e r m o r e i t is s h o w n t h a t t h e use o f d i f f e r e n t wave c l i m a t e databases leads t o results w i t h differences u p t o 25%. T h e i n f l u e n c e of t h e w a v e - d i r e c t i o n a l i t y was s h o w n t o be o f m i n o r i m p o r t a n c e , a m a x i m u m d e v i a t i o n w i t h long-crested

waves o f 10% lias been m e n t i o n e d . T h e s h o r t - t e r m responses however are be-lieved t o be reduced b y m o r e t h a t 10% b y a p p l i c a t i o n o f a s p r e a d i n g f u n c t i o n . T h i s has also been p o i n t e d o u t by Bales et al. [2]. I t was shown t h a t especially t h e r o l l m o t i o n s are v e r y sensitive t o angular spreading o f the waves. T h e v e r t i -cal m o t i o n s t o w h i c h s l a m m i n g , deck-wetness and the v e r t i c a l b e n d i n g m o m e n t s a n d shear forces are r e l a t e d , are m u c h less sensitive t o angular spread. For t h a t reason, t h e d i r e c t i o n a l spreading of t h e waves w i l l be neglected i n t h i s s t u d y .

U s i n g these conclusions, t h e l o n g - t e r m d i s t r i b u t i o n o f the e n v i r o n m e n t a l c o n d i t i o n s i n case of a u n i f o r m l y d i s t r i b u t e d course is d e t e r m i n e d b y t h e j o i n t p r o b a b i l i t y o f a s i g n i f i c a n t wave height ^ ^ 1 / 3 , s p e c t r a l peak p e r i o d Tp a n d spec-t r a l shape S. For spec-t h e l o n g - spec-t e r m j o i n spec-t p r o b a b i l i spec-t y d i s spec-t r i b u spec-t i o n o f s i g n i f i c a n spec-t wave heights a n d peak p e r i o d s , p u b l i s h e d wave c l i m a t e d a t a can be used. E x a m p l e s of these d a t a are g i v e n by H o g b e n a n d L u m b [10], W a l d e n [27] a n d Lee et al. [15]. I n the n e x t p a r a g r a p h s the a t t e n t i o n w i l l be f o c u s e d t o t h e m o d e l l i n g o f t h e s h o r t - t e r m e n v i r o n m e n t a l c o n d i t i o n s s a t i s f y i n g realistic wave s p e c t r a a n d d i s t r i b u t i o n f u n c t i o n s o f the wave heights.

3 Choice of the wave spectrum

For ship response c a l c u l a t i o n s i n i r r e g u l a r waves a s u i t a b l e wave s p e c t r u m m o d e l has t o be chosen. I n t h e design o f offshore s t r u c t u r e s , w h i c h are l o c a t i o n -b o u n d , measured spectra m a y -be availa-ble. U s u a l l y t h i s is n o t t h e case f o r vessels w h i c h are s a i l i n g at m o r e or less r a n d o m locations o n the ocean. For t h i s purpose, t h e o r e t i c a l s p e c t r a l models have been developed f r o m w h i c h the m o d i f i e d P i e r s o n - M o s k o w i t z s p e c t r u m a n d the J O N S W A P s p e c t r u m are the m o s t c o m m o n l y used spectra. A l t h o u g h these s p e c t r a are f r e q u e n t l y used f o r the c a l c u l a t i o n of response spectra f o r a l l p r o b a b l e c o m b i n a t i o n s o f s i g n i f i c a n t wave h e i g h t a n d p e r i o d , some l i m i t a t i o n s on t h e i r a p p l i c a b i l i t y exist.

T h e m o d i f i e d P - M s p e c t r u m can be w r i t t e n as

T h e m o d e l is developed f o r f u l l y developed sea states, w h i c h occur o n l y d u r i n g l i m i t e d periods. F u r t h e r m o r e i t appears t h a t t h e a s s u m p t i o n o f n a r r o w -bandedness does n o t h o l d a n y m o r e since t h e f o u r t h m o m e n t o f the sp e c t r u m does not exist unless some corrections are a p p l i e d [23]. C o m p a r i s o n w i t h field d a t a measured on t h e open ocean showed t h a t t h i s s p e c t r u m u n i f o r m l y overestimates t h e measurements [11].

T h e J O N S W A P s p e c t r u m was developed by Hasselman et al. [9] a n d is a c t u a l l y the P - M f o r m u l a t i o n m u l t i p l i e d w i t h a peak-enhancement f a c t o r . Five free p a r a m e t e r s d e t e r m i n e the final shape o f t h e s p e c t r u m . These p a r a m e t e r s have t o be d e t e r m i n e d o n t h e base o f e m p e r i c a l f u n c t i o n s o f t h e n o n - d i m e n s i o n a l peak f r e q u e n c y . A l t h o u g h the n u m b e r o f p a r a m e t e r s is u s u a l l y r e d u c e d t o t w o (a peakedness p a r a m e t e r a n d t h e peak f r e q u e n c y ) , these t w o p a r a m e t e r s s t i l l have t o be d e t e r m i n e d b y t r i a l a n d e r r o r since t h e m o m e n t s of the s p e c t r u m can n o t be d e t e r m i n e d a n a l y t i c a l l y . T h i s s p e c t r a l shape is o n l y v a l i d f o r f e t c h -l i m i t e d d e v e -l o p i n g sea states.

H u a n g et al. [11] developed t h e WaUops s p e c t r u m , a u n i f i e d t w o - p a r a m e t e r s p e c t r u m w h i c h represents the sea states u n d e r a l l stages o f wave development a n d decay. T h i s s p e c t r u m o n l y depends on i n t e r n a l p a r a m e t e r s w h i c h can be d e r i v e d a n a l y t i c a l l y based o n an i n p u t peak frecjuency a n d variance of the wave surface e l e v a t i o n . F u r t h e r m o r e the d e r i v a t i o n has been based o n fluid-dynamical considerations. T h e s p e c t r u m has the a n a l y t i c a l f o r m u l a t i o n o f a P - M s p e c t r u m and c o m p a r i s o n w i t h field d a t a show t h a t t h e results are c o m p a r a b l e w i t h the J O N S W A P m o d e l . W h e n the e n e r g y c o n t a i n i n g waves are a p p r o a c h i n g b r e a k i n g , t h e W a l l o p s s p e c t r u m reduces e x a c t l y t o the P M m o d e l a n d t o t h e J O N -S W A P m o d e l f o r w i c h the peak- enhancement f u n c t i o n is t h e n equal t o 1. T h e W a l l o p s s p e c t r u m is i n d e p e n d e n t o f t h e w i n d speed b u t depends o n l y o n the peak p e r i o d a n d the s i g n i f i c a n t wave slope. A n i m p o r t a n t f e a t u r e of t h i s s p e c t r a l f o r m u l a t i o n is the variable b a n d w i d t h , w h i c h has i t s i n f u e n c e on the d i s t r i b u t i o n of wave heights a n d periods and t h u s o f t h e responses t o o .

T h e i m p o r t a n c e of a correct choice o f the wave s p e c t r u m m a y be questioned especially w i t h r e g a r d t o the l o n g - t e r m p r e d i c t i o n s of responses [7], b u t the short- t e r m response statistics are c e r t a i n l y effected b y the s p e c t r a l shape. I t is not i m m e d i a t e l y clear h o w a s p e c t r u m w i t h variable b a n d w i d t h effects the l o n g t e r m statistics. Due t o t h e lack of measurements a choice of a m a t h e m a t i -cal s p e c t r u m has t o be made. A l t h o u g h the I T T C r e c o m m e n d e d t h e m o d i f i e d P - M s p e c t r u m , some questions arise w h e n t h i s s p e c t r a l shape is regarded more a c c u r a t e l y a n d w h e n i t is c o m p a r e d w i t h field data. T h e J O N S W A P s p e c t r u m also has i t s l i m i t a t i o n s since i t is o n l y applicable t o a few cases i n the scatter d i a g r a m , representing t h e c o m b i n a t i o n s of s i g n i f i c a n t wave h e i g h t and charac-t e r i s charac-t i c wave p e r i o d . T h e W a h o p s s p e c charac-t r u m does n o charac-t show charac-these h m i charac-t a charac-t i o n s w h i l e i t is easy t o apply, t h e m o m e n t s can be d e t e r m i n e d a n a l y t i c a l l y . I n t h e f u r t h e r s t u d y , t h i s s p e c t r a l m o d e l w i l l be used. T h e t h e o r e t i c a l f o r m u l a t i o n o f the s p e c t r u m is g i v e n by

UJ ' (3)

i n w h i c h iOp is t h e f r e q u e n c y o f the waves at the s p e c t r a l peak a n d /? a n d m are f u n c t i o n s of the i n t e r n a l p a r a m e t e r , t h e s i g n i f i c a n t slope ? of the wave field w h i c h is defined as

? =

or t h e r o o t of the m e a n squared value o f t h e surface e l e v a t i o n d i v i d e d b y t h e wave l e n g t h o f the waves at t h e s p e c t r a l peak. T h i s characteristic wave l e n g t h is d e t e r m i n e d by means o f t h e dispersion r e l a t i o n . T h e coefficients /? a n d m are respectively c a l c u l a t e d u s i n g the f o l l o w i n g t w o f u n c t i o n s of m l o g ( V 2 7rv)2 l o g 2 a n d (5) ( 2 7 r < r ) 2 ? 7 i ( ' " - i ) / 4 1 (6) 4 ( m - 5 ) / 4 p ( - m - l ^

where r(.) is the g a m m a f u n c t i o n . A f t e r some algebra t h e s p e c t r a l m o m e n t s can be d e r i v e d , w h i c h are g i v e n b y

777,0

^ ^ 2 4 ( m - 5 ) / 4 1

F r o m field observations the pealc f r e q u e n c y a n d s i g n i f i c a n t wave h e i g h t are k n o w n . FoUowing f r o m s h o r t t e r m wave statistics t h e r e l a t i o n between t h e v a r i -ance o f a Gaussian d i s t r i b u t e d sea and the s i g n i f i c a n t wave h e i g h t is k n o w n , so the variance a n d t h e r e s u l t i n g s i g n i f i c a n t slope can be d e t e r m i n e d . Some examples o f a W a l l o p s s p e c t r u m a n d a n I T T C s p e c t r u m w i t h t h e same variance a n d peak p e r i o d are s h o w n i n figure 1.

D e t e r m i n a t i o n o f the variance o n the basis of a k n o w n s i g n i f i c a n t wave h e i g h t is s h o w n i n t h e n e x t p a r a g r a p h .

WAVE S P E C T R A I T T C - s p e c t r u m v e r s u s W a l l o p s - s p e c t r u m S [ m 2 s ] 0.5 1 1.5 a a.5 o m e g a [ r a d / s ] Pealc p e r i o d Tp = 8 [ s ] V a r i a n c e m O = 2.25 [ m 2 ] WAVE S P E C T R A I T T C - s p e c t r u m v e r s u s W a l l o p s - s p e c t r u m S [ m 2 s ] 0 0.5 1 1.5 2 2.5 3 o m e g a [ r a d / s ] P e a k p e r i o d T p = 1 6 . 4 [ s ] V a r i a n c e m O = 0 . 7 6 6 [ m 2 ]