Generation of random waves in time-dependent extended mild-slope. equations using a source function method

Transcription

1 Generation of random waves in time-dependent etended mild-slope equations usin a source function method Gunwoo Kim a, Chanhoon Lee b*, Kyun-Duck Suh c a School of Civil, Urban, and Geosystem Enineerin, Seoul National University, San 56-1, Shinlim-don, Gwanak-u, Seoul , South Korea b Department of Civil and Environmental Enineerin, Sejon University, 98 Kunja-don, Kwanjin-u, Seoul , South Korea c School of Civil, Urban, and Geosystem Enineerin & Enineerin Research Institute, Seoul National University, San 56-1, Shinlim-don, Gwanak-u, Seoul , South Korea * Correspondin author. Fa: addresses: maossi1@snu.ac.kr (G. Kim), clee@sejon.ac.kr (C. Lee), kdsuh@snu.ac.kr (K.-D. Suh).

2 Abstract We develop techniques of numerical wave eneration in the time-dependent etended mild-slope equations of Suh et al. (1997) and Lee et al. (003) for random waves usin a source function method. Numerical results for both reular and irreular waves in one and two horizontal dimensions show that the wave heihts and the frequency spectra are properly reproduced. The waves that pass throuh the wave eneration reion do not cause any numerical disturbances, showin usefulness of the source function method in avoidin re-reflection problems at the offshore boundary. Keywords: Source function method; Etended mild-slope equation; Numerical eneration of wave; Random waves; Numerical eperiment 1. Introduction The mild-slope equation, which was derived first by Berkhoff (197), is able to predict the transformation of waves due to refraction, diffraction, shoalin, and reflection from deep to shallow water. Nishimura et al. (1983) and Copeland (1985) developed hyperbolic equations for reular waves. Smith and Sprinks (1975), Radder and

3 Dinemans (1985), and Kubo et al. (199) developed time-dependent mild-slope equations that are able to predict the transformation of random waves. Recently, efforts have been made to improve the mild-slope equation by includin hiher-order bottom effect terms proportional to the bottom curvature and to the square of bottom slope, which were nelected in the derivation of the mild-slope equation. The resultin equations are called the etended mild-slope equations or modified mild-slope equations. These equations may be rearded as a complete form of the linear refractiondiffraction equation because it is derived by interatin the Laplace equation from bottom to mean water surface without nelectin any terms. In an elliptic form, Massel (1993) and Chamberlain and Porter (1995) etended the model of Berkhoff. In a hyperbolic form, Suh et al. (1997) etended the two models of Smith and Sprinks, and Radder and Dinemans, Lee et al. (1998) etended the model of Copeland, and Lee et al. (003) etended the model of Kubo et al. To predict the wave field in a concerned reion usin time-dependent wave equations, waves should be enerated at the offshore boundary and propaate into the model domain. Waves reflected from the model domain should pass throuh the offshore boundary without any numerical distortions. Otherwise, the waves re-reflected at the

4 offshore boundary may influence the numerical solution in the model domain. In order to avoid the re-reflection problem at the wave eneration boundary, internal wave eneration techniques have been widely used. Larsen and Dancy (1983) and Madsen and Larsen (1987) employed a line source method to enerate waves for the Pererine s (1967) Boussinesq equations and Copeland s mild-slope equations, respectively, by usin a mass transport approach. Lee and Suh (1998) and Lee et al. (001) corrected the line source method by usin the enery transport approach for the Radder and Dinemans mild-slope equations and the Nwou s (1993) Boussinesq equations, respectively. In the mass and enery transport approaches, the phase and enery velocities are used as the velocities of water mass and wave envelope, respectively, at the wave eneration line. Recently, Wei et al. (1999) developed a source function method for eneratin waves in the Boussinesq equations of Pererine and Nwou. The method employs a source term added to the overnin equations, either in the form of a mass source in the continuity equation or an applied pressure forcin in the momentum equations. They used the Green function method to solve the overnin equation includin a spatially distributed source term. The line source method enerates waves at a sinle point alon the wave propaation

5 direction, while the source function method enerates waves at a source band consistin of several rids alon the wave propaation direction. In the line source method, waves are assumed to be enerated at either side of the source point and to propaate across the source point at the speed of enery velocity. In the source function method, waves are assumed to be enerated at the source band and to propaate continuously. Therefore, both methods can enerate waves successfully without interferin the waves crossin the wave eneration line or band. The line source method yielded a solution with sliht numerical noises in some wave equations such as Radder and Dinemans mild-slope equations (Lee and Suh, 1998) and Nwou s Boussinesq equations (Lee et al., 001), while such noises were not found in the Copeland s mild-slope equations (Lee and Suh, 1998). However, the source function method yielded a smooth solution without any numerical noise in the Boussinesq equations of Nwou (Wei et al., 1999). In this study, to enerate waves without any noise in the model domain, source functions are developed for the time-dependent etended mild-slope equations of Suh et al. (1997) and Lee et al. (003). And, numerical eperiments were conducted to verify the source function method. In section, a source function is derived for each model. In section 3, numerical eperiments are conducted for eneratin desired incident waves, includin monochromatic and random waves in one and two horizontal dimensions. In

6 section 4, summaries and discussions are presented.. Derivation of source function The time-dependent etended mild-slope equations of Suh et al. (1997) and Lee et al. (003) are reduced to the Helmholtz equation in a constant-depth reion. In this section, we first derive a source function for the Helmholtz equation which is then used to obtain source functions for Suh et al. and Lee et al. s equations..1 Helmholtz equation with source function The Helmholtz equation is the overnin equation for linear waves propaatin on a flat bottom. The Helmholtz equation with a source function is iven by: F k F f (1) where F is the wave property such as surface elevation or velocity potential, k is the wave number, f is the source function, and is the horizontal radient operator. For waves propaatin obliquely with respect to the -ais, we may have the followin

7 epressions for F and f : ep i k y F(, y) F () ep i k y y f (, y) f (3) y where k y is the wave number in the y -direction and is the wave phase. Substitutin Eqs. () and (3) into Eq. (1) yields an ordinary differential equation for F with respect to : F k F f (4) where k is the wave number in the -direction. We can find a non-homoeneous solution for Eq. (4) by followin the method of Wei et al. (1999). A detailed procedure is iven in the Appendi A. The source function can adopt a smooth Gaussian shape: f Dep s (5) where D and are parameters associated with the manitude and width, respectively, of the source function and s is the location of the center of the source reion.

8 Introducin the Green s function method to the differential equation ives the wave property correspondin to the source function: F ( ) ADI ep ik s (6) where A and I are iven by i A (7) k k I ep (8) 4 The parameter D can be determined by the taret wave property and the parameter. Althouh a lare value of is preferred since the source reion can become narrower, too narrow a source reion may cause a poor representation of the source function in the discretized equation. We follow Wei et al. s recommendation to take the width of the source function as half of the wavelenth. If we nelect the source function where the value of the Gaussian function ( ep ( ) ) is smaller than ep s, the width parameter is determined as 80 /( L), where the value of is in the rane of 0.3 to 0.5 and L is the wavelenth correspondin to the carrier frequency.

9 . Source function for Suh et al. s model The time dependent mild-slope equations of Suh et al. (1997) are iven by CC t 0 t k CC R h R h 0 1 (9) (10) where is the surface elevation, is the velocity potential at mean water level, and C and C are the phase speed and the roup velocity, respectively, is the ravitational acceleration, h is the water depth, and the over bar indicates the variables associated with the carrier anular frequency. If R R 0, the above equations are reduced to 1 the model of Radder and Dinemans (1985). f S We consider waves on a flat bottom. Addin the source function ( D ep ) to the riht-hand side of Eq. (9), where the subscript S S s implies the model of Suh et al., and eliminatin in favor of ive the followin equation:

10 t CC k CC f S. (11) For a component wave whose anular frequency is, the y -directional and temporal harmonic terms of (, y, t) and f S y, t, can be separated as ep i k y y t (1) f S f ep S i k y t y (13) Substitutin Eq. (1) and (13) into Eq. (11) yields the one-dimensional Helmholtz equation in terms of : k CC f S (14) where the riht-hand-side term can be rearded as the source function f in the Helmholtz Eq. (4), and the wave number k is the wave number correspondin to the anular frequency in this model, which can be obtained by the dispersion relation of the model (Lee et al., 004) as

11 k k C 1 1 C (15) Followin the procedures from Eq. (4) to (8), the velocity potential function ( ) correspondin to the source function can be obtained as ( AD I S ) ep ik s (16) CC where A and I are iven in Eqs. (7) and (8). On the other hand, the taret velocity potential function ( ) is iven as s (17) a i ep ik where a is the wave amplitude. Thus, the parameter D S in Eq. (16) can be obtained by equatin ( ) in Eqs. (16) and (17) as D S a CC i. (18) AI Thus, the source function f S for Suh et al. s model is obtained as

12 t y k i AI a CC i f y s S ep ep. (19).3 Source function for Lee et al. s model The equation of Lee et al. (003) are iven by 0 ) ( ) ( 1 1 t h u h u CC k i t CC i h u h u CC k CC (0) where is related to the water surface elevation by it ep. If 0 1 u u, the above equation is reduced to the model of Kubo et al. (199). We consider waves on a flat bottom. Addin the source function ep ) ( s L L D f to the riht-hand side of Eq. (0), where the subscript L implies the model of Lee et al., ives the follow equation: L f t CC k i t CC i CC k C C. (1)

13 For a component wave whose anular frequency is, we have the followin relation: i t () where. The y -directional and temporal harmonic terms of (, y, t) and f L, y, t can be separated as ep i k y t (, y, t) y (3) f L y, t f ep i k y t,. (4) L y Substitutin Eqs. (3) and (4) into Eq. (1) and usin Eq. () ive the one-dimensional Helmholtz equation in terms of : k CC f L ( CC ) (5) where the riht-hand-side term can be rearded as the source function f in the Helmholtz Eq. (4) and the wave number k is obtained by the dispersion relation of the model (Lee et al., 004) as

14 k k 1 1 (6) C k 1 CC C Followin the procedure from Eq. (4) to (8), the surface elevation function ( ) correspondin to the source function can be obtained as AD I L ( ) ep ik s (7) ( CC ) CC where A and I are iven in Eqs. (7) and (8). On the other hand, the taret water surface elevation function is iven as s a ep ik (8) Thus, the parameter (8) as D L in Eq. (7) can be determined by equatin in Eqs. (7) and D L acc ( CC ) AI (9)

15 Thus, the source function f L for Lee et al. s model is obtained as f L acc ( CC ) ep AI ep i k y t s y. (30) 3. Numerical eperiments In this section, both monochromatic and random waves are tested for the models of Suh et al. (1997) and Lee et al. (003) in horizontally one dimension. And also multidirectional random waves are tested in horizontally two dimensions. For the Suh et al. s model, the time-derivative terms are discretized by a fourth-order Adams-Moulton predictor-corrector method and the spatial derivative terms are discretized by a threepoint symmetric formula. The Lee et al. s model equation is discretized by the Crank- Nicolson method in horizontally one dimension and by the ADI method in horizontally two dimensions. 3.1 Uni-directional monochromatic waves

16 The computational domain consists of an inner domain of 10 L, where L is the wavelenth, and two spone layers with the thickness of.5l at the outside boundaries. The center of the source reion is located at the center of the inner domain (see Fi. 1). The rid spacin is chosen as L 0 and the time step t is chosen so that the Courant number is Cr Cet / 0. 1, where C e is the enery velocity (Lee and Suh, 1998), so that a stable solution is uaranteed. Fi. 1 Firstly, we enerate uni-directional monochromatic waves. Fi. and Fi. 3 show numerical solutions of the surface elevation and wave amplitude in shallow water ( kh 0.05 ) and deep water ( kh ), respectively, at time of t 30T, where T is the wave period. In the fiures, the solutions are normalized with the taret wave amplitude a 0. Both wave amplitudes and wavelenths are almost equal to the taret values. Also, wave enery is dissipated at the spone layer almost perfectly. Fi. Fi. 3

17 Secondly, we test whether the waves pass throuh the wave eneration band without any numerical disturbances. We remove the spone layer at the riht boundary, and reduce the computational domain to 8 L. The relative water depth is kh, and Lee et al. s model is used. Fi. 4 shows a snapshot of water surface elevations at t 30T, 30T T 4, 30T T 4, 30T 3T 4. The waves reflected from the riht boundary do not make any numerical disturbances when passin throuh the wave eneration reion around / L 0, while wave enery is well absorbed in the left spone layer. Fi Uni-directional random waves In eneratin random waves and predictin the propaation of the waves, the frequency of carrier waves must be chosen so as to ive minimal errors in a wide rane of local wave frequencies. As the difference between the component wave frequency and the carrier one becomes larer, the model yields larer errors in both wave phase and enery. The accuracy of the dispersion relation of the model determines the accuracy of the model.

18 Lee et al. (004) found that the dispersion relation of the Suh et al. s model is accurate at hiher frequencies, while the dispersion relation of the Lee et al. s model does not have a particular rane of hih accuracy. Therefore, for the Suh et al. s model, the peak frequency is chosen as the carrier frequency which is relatively lower in whole frequency rane. For the Lee et al. s model, the weiht-averaed frequency is chosen as the carrier frequency which is hiher than the peak frequency. The TMA shallow-water spectrum is used as the taret spectrum (Bouws et al., 1985) : 4 ep f / f p 1 / S( f ) f, h (31) 4 5 f ep 1.5 f / f p k where is a spectral parameter, is the peak enhancement factor, is the spectral width parameter, and the Kitaiordskii shape function, f h incorporates the effect of k, 4 finite water depth , =, and the peak frequency of input wave, f p = Hz is used. The computational domain has an inner domain of 10L ma ( L ma is the maimum wavelenth resultin from the cutoff incident frequency spectrum), and the center of

19 wave eneration band is located at a distance of L ma from the up-wave boundary of the inner domain (see Fi. 5). Two spone layers with the thickness of S.5L are ma placed at the outside boundaries. Fi. 5 To ensure the resolution in space and time as well as the numerical stability, the frequency rane of incident waves is confined to 0.6 to 1.4 Hz. The rid size is chosen as L min 30.3 ( L min is the minimum wavelenth resultin from the cutoff incident frequency spectrum) and the time step is chosen for the maimum Courant number Cr ma Cema t / to be 0.1. Uni-directional random waves are enerated by superposin all the source functions correspondin to all the component waves: f S m, y, t ep m a A m CC m I m ep i ( k y t ) s y m m m (3) f L (, y, t) m a m CC ( CC ) m ep A I m m ep i ( k y t ) s y m m m (33)

20 where m denotes the number of frequency components, a m is the amplitude of a component wave, m is the phase anle which is randomly distributed between 0 and, and A m and I m are iven in Eqs. (7) and (8). The water surface elevations are recorded at a distance of L ma up-wave from the down-wave boundary of the inner domain. The surface elevations are recorded from t 100T p at every 0.05 s, where T p is the peak wave period, and the total number of samples is Fis. 6 and 7 compare simulated wave spectra with the taret in shallow water ( kh 0.05 ) and deep water ( kh ), respectively. The areement between the taret and computed spectrum is ecellent in whole frequency ranes. Fi. 6 Fi Multi-directional random waves Finally, we carry out the test of multi-directional random waves. Lee and Suh (1998) tested the line source method to enerate multi-directional random waves. In this study, we use the same wave conditions as they used. The TMA shallow-water spectrum is used

21 as a taret spectrum, and the conditions of input waves are T 10 s, h 35 m, p , 0. To insure the resolution in space and time and also the numerical stability, the frequency rane of incident waves is confined between Hz and 0.13Hz that covers 85% of the total enery and ives the sinificant wave heiht of H 4.4 m. s The amplitude of each wave component is iven by ai, j S fi G fi, j fi j where the directional spreadin function f, G is obtained by the Fourier series representation for the wrapped normal spreadin function as: N 1 1 n m G f, ep cosn m n1 (34) where N 0 is the number of terms in the series, the mean wave direction m is 0, and the directional spreadin parameter m is 10. These conditions of input waves are the same as the M1 case in Lee and Suh (1998). The representative directions are 14, 5, 0, 5, 14, and these directional components are taken to have equal area of directional spreadin function. The amplitude of each wave component is a S f f / 5 1,,5 i, j i i j.

22 We enerate multi-directional random waves by superposin all the source functions correspondin to all the component waves as: f S m, n y, t ep a CC s ep i ( k y m t m n ), y m, n, m n m Am, ni m, n (35) f L ( CC ) am, n CC m (, y, t) ep y A I m, n, m n m, n m, n s ep i ( k y m t m n ) (36) where m and n denote the components of different frequency and direction, respectively, m,n is the phase anle which is randomly distributed between 0 and, and m n A, and I, are iven in Eqs. (7) and (8). m n The computational domain consists of an inner domain of 3.5L L and the ma 3 ma spone layers with the thickness S.5 L ma at four outside boundaries. The centerline of the source band is located at a distance of 0.5 L from the up-wave boundary of the ma inner domain. Fi. 8 shows a schematic diaram of the computational domain. The water surface elevation ( ) and its slopes in the - and y -directions ( /, / y ) are recorded at a point of (, y L, 1. L ) of the inner domain from 10 Tp at every ma 5 ma 0.3 second until the total number of samples becomes 819.

23 Fi. 8 In order to calculate the directional wave spectrum, we analyzed the data by usin the maimum entropy principle method (MEP). Fi. 9 shows the taret spectrum and the two-dimensional wave spectra calculated from the recorded data. And, Fi. 10 shows the frequency spectra and the directional spreadin function for the peak frequency. All the results show that the areement between the taret spectrum and computed one is ood, and the source function method is very efficient even for eneration of multi-directional random waves. Fi. 9 Fi Conclusions Followin the method of Wei et al. (1999), we have developed a wave eneration method usin a source function for the time-dependent etended mild-slope equations of Suh et al. (1997) and Lee et al. (003). Numerical simulations of uni-directional

24 monochromatic waves showed that the waves were properly enerated for an arbitrary water depth. The wave reflection test showed that the waves passin throuh the source reion did not cause any numerical disturbances. In simulatin random waves, the peak frequency was chosen as a carrier frequency for the Suh et al. s model, and the weiht-averaed frequency for the Lee et al. s model. For uni-directional random waves, the computed wave spectrum was nearly identical to the taret one for an arbitrary relative water depth, and multi-directional random waves were also enerated as desired. The source functions obtained in this study for the etended mild-slope equations of Suh et al. and Lee et al. can also be used for the mild-slope equations of Radder and Dinemans (1985) and Kubo et al. (199), respectively, because the source functions were derived based on the assumption of a constant water depth. In conclusion, we verified that the source function of the Helmholtz equation may be used in eneratin waves in any refraction-diffraction equation models. Appendi A. The method of Green s function In order to obtain the relationship between the source function f (, y, t) and the wave

25 property F (, y, t), a particular solution of the differential equation includin a source function should be obtained. The differential equation after ecludin the component of waves propaatin in the y -direction is iven by F k F f ( ) (4) where the relations between F and F and those between f and f are iven in Eqs. () and (3), respectively. Homoeneous solutions of Eq. (4) correspondin to proressive waves are iven by F h ep( ik ) (37) In order to obtain a particular solution, we seek a Green s function G, which satisfies G k G (38) where and are rearded as the active and fied variables, respectively, and is the Dirac delta function. The wave property and the Green s function should satisfy the

26 radiation boundary conditions iven by as, F ik F G ik G n n n n n n (39) as, F ik F G ik G n n n n n n Interatin Eq. (38) with respect to from 0 to 0, we have G d k G (40) If the Green s function is continuous at, Eq. (40) becomes G (41) The Green s function that satisfies the radiation conditions (39), and the continuity condition (41) at is iven by ik k i G ik k i G G, ep, ep, (4)

27 where A i / k. Multiplyin Eq. (38) by F and interatin with respect to from to ive G F, k G, d F d (43) Interatin by parts and usin the boundary conditions and the definition of the delta function ive F G, f G d, f d G, f d (44) The arbitrary function f can be selected as a smooth Gaussian shape iven by f Dep s (45) where the parameter D is the source function amplitude, and is related to the width of the source function. The source function is neliibly small ecept in the vicinity of s. Therefore, in an interested reion ( s ), the second interal in the rih-hand side of

28 Eq. (44) is neliible, and thus we have F G, f G, ADI ep ik f d d s (46) where I is defined by k ep ik d ep 4 I ep (47) For waves that are enerated at a point s and propaate in the opposite reion ( s ), a similar approach is used to ive F G, f d ADI ep ik (48) s

29 References Berkhoff, J.C.W., 197. Computation of combined refraction-diffraction. Proc. 13th Int. Conf. On Coastal En. Conf., ASCE, Vancouver, Canada, pp Bouws, E., Günther, H., Rosenthal, W., Vincent, C.L., Similarity of the wind wave spectrum in finite depth 1. Spectral form. Journal of Geophysical Research 90(C1), Chamberlain, P.G., Porter, D., The modified mild-slope equation. Journal of Fluid Mechanics 91, Copeland, G.J.M., A practical alternative to the mild-slope wave equation. Coastal Enineerin 9, Kubo, Y., Kotake, Y., Isobe, M., Watanabe, A., 199. Time-dependent mild slope equation for random waves. Proc. 3rd Int. Conf. On Coastal En., ASCE, Venice, Italy, pp Larsen, J., Dancy, H., Open boundaries in short wave simulations a new approach. Coastal Enineerin 7, Lee, C., Cho, Y.-S., Yum, K., 001. Internal eneration of waves for etended Boussinesq equations. Coastal Enineerin 4, Lee, C., Kim, G., Suh, K.D., 003. Etended mild-slope equation for random waves.

30 Coastal Enineerin 48, Lee, C., Kim, G., Suh K.D., 004. Comparison of time-dependent etended mild-slope equations for random waves. Proc. Asian and Pacific Coasts 003. Lee, C., Suh, K.D., Internal eneration of waves for time-dependent mild-slope equations. Coastal Enineerin 34, Madsen, P. A. Larsen, J., An efficient finite-difference approach to the mild-slope equation, Coastal Enineerin 11, Massel, S.R., Etended refraction-diffraction equation for surface waves. Coastal Enineerin 19, Nishimura, H., Maruyama, K., Hirakuchi, H., Wave field analysis by finite difference method. Proc. 30th Japanese Conf. Coastal En., pp (in Japanese). Nwou, O., Alternative form of Boussinesq equations for nearshore wave propaation. Journal of Waterway, Port, Coastal and Ocean Enineerin 119, Pererine, D.H., Lon waves on a beach. Journal of Fluid Mechanics 7, Radder, A.C., Dinemans, M.W., Canonical equations for almost periodic, weakly nonlinear ravity waves. Wave Motion 7, Smith, R., Sprinks, T., Scatterin of surface waves by a conical island. Journal of

31 Fluid Mechanics 7, Suh, K.D., Lee, C., Park, W.S., Time-dependent equations for wave propaation on rapidly varyin toporaphy. Coastal Enineerin 3, Wei, G., Kirby, J.T., Sinha, A., Generation of waves in Boussinesq models usin a source function method. Coastal. Enineerin 36,

32 Captions of fiures Fi. 1. Computational domain for eneratin uni-directional monochromatic waves. Fi.. Normalized water surface elevations and amplitudes of uni-directional monochromatic waves at t 30T in shallow water with kh 0.05 ; solid line = water surface elevation, dashed line = wave amplitude. (a) Suh et al. s model, (b) Lee et al. s model. Fi. 3. Normalized water surface elevations and amplitudes of uni-directional monochromatic waves at t 30T in deep water with kh ; solid line = water surface elevation, dashed line = wave amplitude. (a) Suh et al. s model, (b) Lee et al. s model. Fi. 4. Snapshots of water surface elevations at t 30T, 30T T/4, 30T T/4, 30T 3T/4. Fi. 5. Computational domain for eneratin uni-directional random waves. Fi. 6. Frequency spectra of uni-directional random waves with k p h 0.05 ; solid line = numerical solution, dashed line = taret. (a) Suh et al. s model, (b) Lee et al. s model. Fi. 7. Frequency spectra of uni-directional random waves with k p h ; solid line = numerical solution, dashed line = taret. (a) Suh et al. s model, (b) Lee et al. s model. Fi. 8. Computational domain for eneratin multi-directional random waves.

33 Fi. 9. Computed D directional wave spectra. (a) Taret spectrum, (b) Suh et al. s model, (c) Lee et al. s model. Fi. 10. (a) Frequency spectra and (b) directional spreadin functions (solid line= taret spectrum, dashed line= Suh et al. s model, dash-dotted line= Lee et al. s model).

34 Spone layer Source band Spone layer S 5L 5L S Fi. 1. Computational domain for eneratin uni-directional monochromatic waves.

35 (a) a /L (b) a /L Fi.. Normalized water surface elevations and amplitudes of uni-directional monochromatic waves at t 30T in shallow water with kh 0.05 ; solid line = water surface elevation, dashed line = wave amplitude. (a) Suh et al. s model, (b) Lee et al. s model.

36 (a) a /L (b) a /L Fi. 3. Normalized water surface elevations and amplitudes of uni-directional monochromatic waves at t 30T in deep water with kh ; solid line = water surface elevation, dashed line = wave amplitude. (a) Suh et al. s model, (b) Lee et al. s model.

37 1 / a /L Fi. 4. Snapshots of water surface elevations at 30T 3T/4. t 30T, 30T T/4, 30T T/,

38 Spone layer Source band Wave aue Spone layer S L ma 8L ma L ma S Fi. 5. Computational domain for eneratin uni-directional random waves.

39 Enery density (cm /Hz) (a) Frequency (Hz) Fi. 6. Frequency spectra of uni-directional random waves with k p h 0.05 ; solid line = numerical solution, dashed line = taret. (a) Suh et al. s model, (b) Lee et al. s model.

40 Enery density (cm /Hz) (b) Frequency (Hz) Fi. 6. (Continued).

41 Enery density (cm /Hz) (a) Frequency (Hz) Fi. 7. Frequency spectra of uni-directional random waves with k p h ; solid line = numerical solution, dashed line = taret. (a) Suh et al. s model, (b) Lee et al. s model.

42 Enery density (cm /Hz) (b) Frequency (Hz) Fi. 7. (Continued).

43 .5L ma.5l ma.5l ma L ma Spone layer Wave aue Source band 3L ma.5l ma.5l ma Fi. 8 Computational domain for eneratin multi-directional random waves.

44 Wave direction (de) (a) Frequency (Hz) Fi. 9. Computed D directional wave spectra. (a) Taret spectrum, (b) Suh et al. s model, (c) Lee et al. s model.

45 Wave direction (de) (b) Frequency (Hz) Fi. 9. (Continued).

46 Wave direction (de) ( c ) Frequency (Hz) Fi. 9. (Continued).

47 Enery density (cm /Hz) (a) Frequency (Hz) Fi. 10. (a) Frequency spectra and (b) directional spreadin functions (solid line= taret spectrum, dashed line= Suh et al. s model, dash-dotted line= Lee et al. s model).

48 Direction spreadin function (rad -1 ) (b) Anle ( Fi. 10. (Continued).