Experiments and Numerics of Bichromatic Wave Groups

Transcription

1 ^<^)\- ( O Experiments and Numerics of Bichromatic Wave Groups J. Westhuis* 1 E. van Groesen* R. Huijsmans* Astract In this paper we report on extensive experiments on nonlinear wave groups that evolve in a hydrodynamic laoratory over iong distances from the generation of simple ichromatic waves. The experiments show large deformations of the wave group, with large increase of wave heights, aove a critical value for the quotient of wave amplitude and frequency difference. This parameter and its critical value are identified. A very efficient and accurate numerical code ased on the full nonlinear surface wave equations that has een developed for this purpose, reconstructs the experiments and enales to investigate the evolution over much longer distances than in the laoratory. 1 Introduction Nonlinear effects can dominate the evolution of free surface waves to a considerale extent as previous and recent investigations have shown. From increasing experimental data, numerical and theoretical tools the understanding is growing consideraly. In this paper we will contriute y providing some results of a large series of experiments performed in a wave tank at MARIN and y descriing a computer code ased on the full surface wave equations and showing results of simulations; a parameter that determines the complexity of the phenomena will e identified. The instaility of wave trains has een a topic of research since Benjamin & Feir (1967) showed that the travelling wave solution of the non-linear water wave prolem is unstale to modulational perturations of its envelope. Many authors have studied these instailities and confirmed the B-F instaility growth-rates and the recurrent character of evolution (e.g. Stiassnie & Kroszynski (1982), Lo & Mei (1985)). When Yuen & Lake (1975) showed that the Benjamin-Feir instaility is also produced y the Nonlinear-Schrödinger (NLS) equation, the long term ehaviour of the initial instaility has een sought in terms of disintegration of the modulated wavetrain into solitary wave envelopes (see e.g. Chu & Mei (1971); Hasimoto & Ono (1972)). In this study we investigate the long time ehaviour of unstale ichromatic waves. An experimental study y Stanserg (1997) in which the spatial evolution of a ichromatic wave group envelope is reported, motivated the research of which some results will e presented in this presentation. Experiments performed at the water asins of MARIN Westhuis & Huijsmans (1999) confirm the reported ichromatic evolution (Sec. 2), and the large series of experiments makes it possile to verify the critical value of a characteristic parameter that determines the 'complexity' of the resulting wave profiles. The waves in the laoratory are rather short crested (with wave length approximately equal to the depth), and comparale to experiments performed y Stanserg, ut complementary to those reported y Shemer et al. (1998). who studied desintegration for relatively long crested i-chromatic waves. "Department of Mathematical Sciences, University of Twente, The Netherlands 'Maritime Research Institute Netherlands (MARIN) Wageningen, The Netherlands. 1

2 x=40m x=20 m x=60 m x=160 m x=180m z=0 x=0 Figure 1: Scaled sketch of the experiment setup performed at the MARIN high speed asin (actual depth h=3.6 [mj). The wave elevation was measured at the center line of the tank at x = 0, 10, 40, 60,..., 180 [mj and at x = 80 [m] and x = 120 [m] at \ of the tank width. Often, numerical simulations are reported ased on simplified descriptions of the full wave prolem; KdV-type of equations are often used for that aim, such as recently e.g. y Pelinovsky et al. (1999). However, the wavelengths of interest in this paper are comparale to the depth which makes the classical Boussinesq assumption an invalid approximation of the dispersion. An improved model with correct dispersion has een descried in Van Groesen (1998), Van Groesen et al. (1999) and exploited for theoretical investigation. Numerical procedures to deal with the full nonlinear prolem often use Boundary Element Methods. Instead, we will descrie a Finite Element/Finite Difference method (an extension of the principal idea of Greaves et al. (1997) to a numerical wave tank) that is more efficient for the specific calculations in wave tanks. The numerical code includes various types of wave makers and a numerical asoring each and is suited to perform very long time/length simulations. By comparison with experiments we show that the code produces very accurate results (Sec. 3, 4). Besides that, simulations are performed on a length scale of approximately 5 times the maximum laoratory scale, thus exposing dynamics not found in laoratory experiments. As in the reported cases for unstale wave trains, we also found recurrence phenomena for some particular cases of ichromatic wave evolution. For moderate amplitude cases a 'simple' periodicity in the spatial evolution of the spectrum may e oserved. However, with increasing initial amplitudes or narrowing frequencies, more complicated (periodic) structures may e oserved. We visualize an experimentally relation etween characteristic spatial quantities related to the recurrence of the main sideand modes. Oservations of the various simulations/experiments indicate that the staility of the evolving envelopes depends on the frequency-difference as well as on the amplitude. The experiments of Stanserg (1997), Westhuis & Huijsmans (1999) and the supporting calculations presented in the first sections, will e taken as a characteristic example to distinguish various stages in the unstale evolution. To otain a (partial) explanation, we use the NLS (Non linear Schrodinger) - type of envelope equation. 2 Measurements The measurements on ichromatic wave groups were performed at the High Speed Basin at MARIN. A schematic sketch of this facility can e seen in Fig. 1. In order to measure the spatial evolution of the wave group, resistance type wave proes were positioned at x=0 [m], x=10 [m], x=40 [m], x=60 [m], x=80 [m], x=100 [m], x=120 [m], x=140 [m], x=160 [m] and x=180[m] at the centreline of the tank. At x=80 [m] and x=120 [m], additional proes were positioned at 1/4 of the tank readth to investigate possile 3D effects. The proe at x=0 [m] is mounted on the wave generator and thus measures the run-up against the wave oard. The time interval during which the signals were measured is 700 s. More details on Isr-01 JW & EvG Page: 2 of 16

3 AT = 0.4 AT = 0.2 AT = 0.15 AT = 0.1 AT = 0.05 q = 0.03 q = 0.04 q = 0.05 g = 0.06 q = = = 0.09 q = 0.10 E S E S E S E S E S Tale 1: Overview of experiments (E) and numerical simulations (S) of i-chromatic wave groups. The signal sent to the wavemaker is 2q(cos(^-J + cos( MJ; T\ = 2 AT [sj; T2 = 2 + AT [sj; h = 5 [m] q is the flap stroke amplitude related y linear theory to the desired wave amplitude q. The cross in a cell (x) indicates that the experiment c.q. numerical simulation with the corresponding values for T\, T2 and A was performed and () indicates that the experiment was performed ut that either wave reaking was oserved in the wavetank resp. the numerical scheme halted ecause of wave reaking. the measurement setup, measurement data and data analysis can e found in Westhuis & Huijsmans (1999) The steering signal Eq. (1) sent to the wavemaker is strictly ichromatic S(t) = ^icosfwii) + A 2 cos(u 2 t), (1) and no second order wavemaker theory was used. Using linear theory (e.g. Schaffer (1996)) the value of Ai is related to the desired value of the wave amplitude A{. In Tale 1 the experiments performed for different cominations of periods T\ and T 2 and the amplitudes Ai = A2 = q are summarized. The performed measurements are in the columns identified y (E) and cells marked with an (x). Remark that in all cases the wavelength of the carrier wave of the wave group is constant and satisfies (L aj 6.25 fa 1.25/i). All measurements are found to e periodic in the modulation period T mo a = ^. Examination of the additional signals at tank readth at x = 80 [m] and x = 120 [m] show no signs of standing waves across the readth of the tank. Based on linear theory the envelope of the periodic signals should e the same at all the measurement locations. Fig. 2 however, clearly shows that linear theory is not valid for some of the generated ichromatic wave groups. Fig. 2 (a) and () show the wave elevation signal measured at x = 10 [m] resp. x = 180 [m]. As would e expected from linear theory in this case no significant change in envelope can e oserved. This was actually oserved for all experiments performed with AT = 0.4. However, as Fig. 2 (c)-(f) clearly show, the envelope significantly changes with increasing distance for higher values of q when AT = The experiments and numerical simulations for q = 0.08 [m] and AT = 0.2 [s] correspond to the experiment performed y Stanserg (1997). Similar to the measurements presented y Stanserg, asymmetric wave group evolution was found. Figures of these measurements and corresponding simulations are presented at the end of the next section. 3 Numerical Simulation Method The Huris 2D code, developed y MARIN and Twente University, is used to simulate the aove descried experiments. The code is an extension of the comination of a Finite Element/Finite Difference method as proposed y Greaves et al. (1997). Isr-01 JW & EvG Page: 3 of 16

4 x=10 [AT«0.4,q=0.06] x=180 AT=0.4, q=0.06] C) x=10[at=0.15],q=0.04j x=180[at«0.15],q=0.04] x-10[at=0.15.q=0.06] x=180[at=0.15,q=0.06] Figure 2: Measurements on different ichromatic wave groups. (a,c,e) show signals measured at x = 10 [mj from the wave maker; (,d,e) show signals at x = 180 [m] from the wave maker. Figures (a)-() (with AT = 0.4^ show no significant change in wave group envelope at increasing distance from the wave maker. However, figure (c)-(f) clearly show the change in envelope with increasing distance for higher values of q when AT = Isr-01 JW k EvG Page: 4 of 16

5 If a fluid is assumed incompressile and inviscid and the flow is assumed irrotational, the velocity field (u) of the fluid can e defined using a potential, that satisfies Laplace's equation u = V$ (2) A$ = 0 (3) On a material oundary the following condition for the potential holds were n is the outward direction normal to the oundary and v n is the normal velocity of the oundary. At the free surface z ~ j)(x, t), which is assumed to e a function of the horizontal spatial variale x = (x,y) and of time t, the Bernoulli pressure of the fluid must equal the atmospheric pressure, which is assumed to e constant and equal to zero. _ = -_ V* 2 -pi, (5) Here g is the gravitational acceleration in the negative z direction. This condition is called the dynamic oundary condition. Control over the grid points is otained using an Aritrary Lagrangian-Eulerian method y adding a grid correction vector. The grid correction vector must e constructed perpendicular to the surface normal, and its length (in the two dimensional case) is parameterized y a as in Eq. (6). Vr.or = (T (6) For every grid point the parametrization is such that a = 0 corresponds to Lagrangian movement of the grid point, and a = 1 corresponds to only vertical movement of the grid point. In the case of the wavemaker, the correction vectors are dynamically determined in such a way that the horizontal velocity of the grid points close to the wavemaker are identical to the horizontal velocity of the grid point on the intersection etween the wavemaker and the free surface. The horizontal velocities of the grid points further downstream from the wavemaker are smoothly adapted to the desired grid strategy for the rest of the domain. The position of the grid point is denoted y the vector a; gr id and the potential at this grid point is denoted as $ gr id- The kinematic equation expressing a velocity identity is thus given as Dx «=V* + vcor (7) Dt From Eq. (5) and (7) the evolution of the position of the grid point is given y, the dynamic equation for the evolution of the potential along the trajectory of a grid point is given y Eq. (9). fif- = d ( $ gri d + (V$ grid +u cor )-V$ grid (8) = f - V$ grid + V C0T j V$ gri d - 9 gri d 2 (9) When equations (3), (4), (9) and (7) are supplemented with appropriate initial conditions, a closed set of equations is otained. The Huris 2D code solves these equations y a 5 step 4'th order Runga-Kutte integrator for the time dependent oundary quantities (4), (9) and (7) and a linear Finite Element method Isr-01 JW & EvG Page: 5 of 16

6 for otaining the solution of the Laplace prolem on every stage. The used Finite Element mesh is refined near the free surface and derivatives of the potential are otained y second order Finite Difference methods. For more details aout the method see Westhuis et al. (1998). The code has een verified for many situations and performs well and efficient. In another pulication Van Groesen & Westhuis (2000) we will show results of long distance calculations of ocean wave groups aove horizontal and sloping ottom. For the applications in this paper the performance of the typical wave tank attriutes, i.e. the wave generation and asoring each, will now e riefly descried. An artificial each is constructed using a comination of pressure damping, grid stretching and a Sommerfeld condition (Eq. 10) on the outflow oundary, where c a is set at the critical wave speed yfgh. + C- *,=0 (10) Fig. 3 shows a time trace of the wave elevation for a characteristic computation (h=5 m, A ~ 5m, A= H ~ 0.2m). The continuous line shows the elevation from a computation where the numerical each starts at x=400m, the dotted line shows the signal at the same position when the each is positioned after 1000m. Clearly, no distinguishale effects of the numerical each can e oserved which has also een demonstrated in Westhuis (2000). In order to exclude that the oserved effects in the numerical simulation of the ichromatic wave evolution are essentially related to the specific way of wave generation, three different kind of generators were examined. 1. Moving flap: a moving flap hinged at 2.67 [m] elow the still water level, 2. Linearized flap: the fluxes of the real flap are generated on a static vertical wall 3. Linear solution; the flux generated at the fixed vertical wall is derived from the expression for linear water waves. Although the signals differ from each other (which is to e expected), this difference consists of relatively small phase differences, the characteristic deformation of the wave group envelope is still oserved (see Fig. 4) and power spectra are identical. For further calculations we used the linearized flap, ecause it is more easy to perform stale computations than with the moving geometry and it is more similar to a physical wave tank than the imposed linear solution. Using this method many numerical simulations have een performed regarding i-chromatic wave groups (see Tale 1). Most convincing argument that the numerical simulations are accurate representations of actual physics is given in Fig. 5 were simulations and measurements are directly verified. Although some deviations can e oserved and it seems that the phase velocity of the numerical waves is slightly lower than that of the measured waves, overall agreement is excellent. From these oservations regarding the each and wave generation and from qualitative and quantitative comparisons of the numerical results with experiments, we conclude that the oserved phenomena are only due to the nonlinearity of the equations and are not a consequence of numerical wave generation or asortion. They also suggest that calculations over a greater length than the laoratory dimension (and thus can not e experimentally verified) still hold physical significance. 4 Description of Experimental and Numerical Oservations We now descrie in some more detail the phenomena that can e oserved in the experiments. Since the numerical simulations are reliale, for a etter understanding of the phenomena Isr-01 JW & EvG Page: 6 of 16

7 0.3 Elevation at x=400m 400 m + each 1000m + each Wsl Figure 3: Comparison etween two numerical simulations. The wave elevation at the position where the numerical each starts (approximately 60 wavelengths from the wave generator) and a computation where the numerical each is placed 100 wavelengths away from the previous position. The small differences are negligile compared to the deformation of the initially symmetric wave group. water elevation at x=120 m - real geometry exact lin.influx lin. geometry t[s] 125 Figure 4: Comparison of three numerical simulations: the wave elevation of the unstale ichromatic wave generated y a moving hinged flap (dashed line), a hinged flap with linearized geometry (full line) and with the exact linear flux solution of the ichromatic twat/e prescried (dotted line). Isr-01 JW & EvG Page: 7 of 16

8 dots: calculations line: experiments x-40 x x x= Figure 5: Comparison of the measured (Sec. 2) and numerically simulated (Sec. 3) wave signals of the ichromatic wave (h 5[m] with 7\ = 1.9 [sj, T 2 = 2.1 [sj and Ai A 2 = 0.08 fm]) at increasing distances from the wave maker x = 40 [mj,..., x = 180 [m] Isr-01 JW & EvG Page: 8 of 16

9 we exploit the numerical simulations that have een produced for evolutions over a distance of 1200 m, while the laoratory experiments are over a distance of 200 m. To descrie the evolution, first oserve that the wave maker signal Eq. (1) would lead in a linear theory to a surface elevation given y Vtin(x,t) = 2qcos(Akx - Aut)cos(kx - üt) (11) where k, Q are the averaged wave numer and frequency and Ak, Aa; half the differences. For notational convenience we have also introduced q = A\ = A 2. The modulation period is T it J mod A W The evolution along the wave tank of the signal Eq. (11) shows various characteristic stages that we will now identify. The description mainly deals with changes in the envelope. All plots are for the typical example corresponding to the experiments used in the previous sections: oth waves of amplitude 0.08 [m], on depth of 5 [m], with periods of the waves T\ = 1.9 [s] and T 2 = 2.1 s. Stage 0 is the evolution along a short distance, during which the wave seems to develop as the linear wave Eq. (11) (see Fig. 6, x=10). In the consecutive stage 1 the envelope steepens at the front, without noticeale increase of amplitude (see Fig. 6, x=40). During this steepening process, the symmetry in the envelope is lost within each modulation period. Then, in stage 2, fast changes in the envelope show an increase of the amplitude at the front, and a corresponding decrease at the rear. The precise development depends very much on the values of the parameters (wave height and frequency difference). For small wave heights, large frequency differences, the envelope remains rather symmetric and the increase of amplitude is small. For larger wave height, smaller frequency differences, the changes are more dramatic: large amplitude increase in the front, with large changes in envelope at the rear. Visual oservation shows a process of 'splitting off' a separate wave, see Fig. 6, Fig. 7 and Fig. 8. The envelope plotted in Fig. 7 is otained y using Hilert transform methods and applying a low-pass filter, thus filtering out the carrier wave frequencies. Fig. 8 is a density plot of the extracted wave envelope in a moving frame of reference where the frame velocity is set to the experimentally determined velocity of the T [s] and A\ A 2 = 0.08[m] was found, main wave group. These stages can e distinguished in the experiments and the calculations, and the spatial intervals of the successive stages are roughly stage 0 = [0,40], stage 1=[40, 120], and stage 2=[120,160]. After stage 2, the numerical calculations however show another stage 3, during which an 'interaction' process etween the large-amplitude waves and the splitted waves seems to take part. Although a time-periodic T mo d ehaviour is still present, now clearly there is an interaction of the larger (faster) wave from one period with the smaller (slower) wave of the previous time interval. During this stage, the waves that splitted during Stage 2, ehave in a soliton-type of way, as is clearly visile in Fig. 7 and Fig 8. Other experiments showed that details of the long-time evolution depend rather critical on the precise values of the parameters. A different way to descrie the changes during the evolution of the waves is y investigating the spectral properties; we report on some results as follows. As can e oserved in Fig. 9 the spectrum of the time signal of the wave elevation at a fixed position, changes significantly with increasing distance from the wavemaker. Although some regularity may e oserved, the growth and decay of several sideand modes is remarkale. In situations where the initial frequencies are well separated (T l,t 2 ) = (1.8,2.2) no spatial variations have een oserved. The first side and modes in Fig. 9 (2a>i o>2, 2^2 OJI) are excited y the nonlinear resonance etween the difference frequency (as a result of quadratic nonlinear effects) and the original frequencies. In Fig. 9 for larger amplitudes even significant energies at the frequencies Zu>i 2u 2 and Au\ ZÜJ 2 are clearly visile. We were not ale to do calculations with higher initial amplitudes ecause reaking would occur within 900 seconds and the numerical simulation halts. This was also confirmed y experiments showing spilling and reaking of the wave for q > In Fig. 10 (a)-(d) we have tried to categorize some of the results related tot the spatial structure of the primary sideand modes. The results for two sets of ichromatic experiments {{T U T 2 ) = (1.9,2.1) and (T U T 2 ) = (1.925,2.075)) have een used. In (a) the initial spatial Isr-01 JW k EvG Page: 9 of 16

10 u.o 0.2 d 0.1 wy i. x=40 A AA A A AA^A vypvv * I A n M \N\h\\ M (l x=200 II r AAAAM Figure 6: Time signals at susequent positions x=10,40,80,120,160,200m, showing the evolution from stage 0 at x=10, to stage 1 at x=40 m, to stage 2 at x=80, 120m, and stage S at x=160, 200 m. Isr-01 JW & EvG Page: 10 of 16

11 O O Y Figure 7: Plots of signals of the numerical simulation (T\ = 2.1 fsj, T 2 = 1.9 [sj q = 0.08 [mj, h = 5 [mj) at x= m, showing the splitting and merging process etween 200 and 450, resemling soliton-interactions. Oserve the long interaction region, x= during which the two waves have merged (and slown down). Isr-01 JW & EvG Page: 11 of 16

12 t[s] Figure 8: Density plot of the spatial-temporal structure of the envelope of the wave group in Fig. 7. Results from a numerical simulation in a frame of reference moving with the velocity of the largest (most right) waves. The splitting and merging of small slow waves (faint) with large and faster waves indicate the 'typical' soliton-type ehaviour during stage 3. During the interaction region x= the two waves are merged.(entrance effects are visile in the left lower-corner of the plot.) growth rate of the primary sideand modes is plotted against the wave numer of their closest original mode. Plots () and (c) show the amplitude and the wave numer of the spatial variations of these sideand oscillations. In (d) the results of () and (c) are comined to give a relation etween the wave steepness of the individual components of the ichromatic wave and the steepness of the spatial oscillations of their corresponding sideands. An interesting oservation is that can e made from (c) is that the spatial wavenumer associated with the (2,1) frequencies are increasing functions of ka while the (1,2) spatial wavenumers decrease with increasing ka. It is also remarkale to notice that the side and frequencies that lay well out of the B-F instaility intervals (situations in Fig. 9 with q=0.04, q=0.06) show significant growth rates, which indicates that a different kind of analysis is needed when examining the instaility of these wave groups. Remark that the presented numerical results are otained y simulating a 1200 m wavetank (h=5m) which uncovers dynamics that can not e otained in physical laoratories. Also remark that the numerical method to solve the fully nonlinear equations is not in any way a spectral method so periodicity of the modes can not e a numerical artefact in that sense. The oserved spectral structure as oserved in Fig. 9 indicates that a many-mode model (e.g. Krasitskii (1994)) can also e employed to simulate this wave group ehaviour. However, it seems that the mere simulation using the presented method as well as the many-mode methods will not present any physical insight y themselves. A different approach in trying to understand and predict the spectral evolution of the unstale wave groups is to examine the envelope evolution of these wave groups. In the next sections this approach will e further outlined. Isr-01 JW k EvG Page: 12 of 16

13 q=0.04 q=0.07 q-0.06 q=0.08 Figure 9: Numerical simulations on a spatial domain of 1000 [mj. Evolution of the spectrum of the ichromatic wave for different amplitudes. The periods of the ichromatic modes are (T U T 2 ) = (1.9,2.1) [s], q = 0.08 [mj and the depth of the tank is 5 [mj. Clearly the periodic structure of the nonlinear spatial evolution is visile for smaller amplitudes. However, for larger amplitudes this simple periodicity does no longer hold due to the involvement of more modes. Isr-01 JW & EvG Page: 13 of 16

14 xlo" if oe 73-Q» s ale al SJ 1? e- dt-0.2(2,-1) O dt-0.2 (-1,2) -x- dt-0.15(2,1) K dt-0.15 (-1,2) I ' ka Figure 10: The initial growth rate (a), the amplitude (), the wavelength (c) and the 'wave steepness' (d) of the spatial oscillations of the two largest sideand modes. These values are plotted against the wavenumer of their corresponding principal mode in the ichromatic wave. Data are plotted from two sets of numerical simulations. The first set of ichromatic waves has angular frequencies (rad/s) of {u\,u) 2 ) = (3.307,2.2992), the second set has frequencies (wi,w2) = (3.263,3.028). The data are calculated and plotted for the modes (m,n) = mui + ^2,(771,71) e {(2,-1), (-1,2)} Isr-01 JW & EvG Page: 14 of 16

15 5 Conclusion and Discussion We showed some results aout deformations of ichromatic wave groups. Experiments and accurate numerical simulations ased on the full nonlinear surface wave equations dealt with rather short crested waves, implying that dispersion and nonlinearity are in the self-focussing regime. Our results confirm previously reported findings aout large deformations of envelopes, and, more importantly, give an indication of the values of relevant parameters for which these large deformations appear. The relevant characteristic parameter, found from theoretical analysis that will e pulished (see also Westhuis et at. (2000)), is the quotient of initial amplitude and frequency difference, -^. A detailed investigation of the governing phase-amplitude equations, in particular the nonlinear dispersion relation with the extension of the Fornerg-Whitham term (see e.g. Chu & Mei (1970), leads to this parameter. Another useful interpretation that can serve as a phenomenological explanation, is possile y looking at the energy so i lr mod ^ a 'soliton-type' of solution that resemle the large amplitude waves oserved in the experimental and simulated evolutions. The requirement that this soliton is (practically) confined within one (enforced) modulation period T m0 d, is met if the critical value is exceeded. In that case, the amount of energy generated at the wave maker exceeds the minimum value of the energy of a soliton in the modulation period Since r mo d is inversely proportional to the frequency difference Aw, this illuminates its appearance in the characteristics parameter ^_, which is related to the generated energy in one period. Except the emphasis on the changes in envelope, from the simulations we derived in this paper detailed information aout spectral properties and related growth rates. Actually, there is still no easy way to relate spectral information to the changes in envelope. We showed that during the evolution remarkale log time evolution of the spectra of the wave groups have een oserved. In the spatial range that validation with experiments is possile, good mutual agreement etween measurements and simulations was found (Fig. 5). During the evolution of ichromatic wave groups, energy is transferred to sideand modes in a quasi periodic manner. Although in first instance a relation was suspected etween the B-F instaility and the resulting growth rates of side and nodes, no direct relation has een found. When the spatial structure per mode could e considered periodic, an experimental dependence etween the spatial wavenumer and the wavenumer of the principal mode was found (Fig. 10). The phenomena may proaly e reproduced y numerical simulation using many node-interactions. An alternative approach is possile y a low dimensional mode-analysis of the envelope, as will e descried in a forthcoming paper. References BENJAMIN, T. h FEIR, J The desintegration of wave trains in deep water, part 1. theory. J. Fluid Mech. 27, 417. CHU, V. h MEI, C 1970 On slowly varying stokes waves. J. Fluid Mech. 41, 873. CHU, V. &; MEI, C The nonlinear evolution of stokes waves in deep water. J. Fluid Mech. 47, 337. GREAVES, D., BORTHWICK, A., Wu, G. &; TAYLOR, R. E A moving oundary finite element method for fully nonlinear wave simulations. Journal of Ship Research 41 (3), VAN GROESEN, E Wave groups in uni-directional surface wave models. J. Eng. Math. 34, Isr-01 JW & EvG Page: 15 of 16

16 VAN GROESEN, E., ANDONOWATI & SOEWONO, E Nonlinear effects in Dichromatic surface waves. Proc. Estonian Acad, Sci. Phys. Math. 48, VAN GROESEN, E. & WESTHUIS, J Simulations of nonlinear wave groups in coastal regions, in preparation. HASIMOTO, H. & ONO, H Nonlinear modulation of gravity waves. J. Phys. Soc. Japan 33, 805. KRASITSKII, V On reduced equations in the hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1. Lo, E. h MEI, C A numerical study of water-wave modulation ased on a higher-order nonlinear schrodinger equation. J. Fluid Mech. 150, 395. PELINOVSKY, E., TALIPOVA, T., KIT, E. & ETIAN, O Nonlinear wave paket evolution in shallow water. In Int. Symp. On Prog. In Coastal Engineering and Oceanography, pp Seoul, Korea. SCHAFFER, H. A Second-Order wavemaker theory for irregular waves. Ocean Engng. 23, SHEMER, L., KIT, E., JIAO, H. & ETIAN, O Experiments on nonlinear wave groups in intermediate water depth. Journal of Waterway, Port, Coastal and Ocean Engineering nov/dec, STANSBERG, C On the nonlinear ehavior of ocean wave groups, proc. Waves '97 2, STIASSNIE, M. & KROSZYNSKI, U. I Long-time evolution of an unstale water-wave train. J. Fluid Mech. 116, WESTHUIS, J Approximate analytic solutions and numerical wave tank results for the reflection coefficients of a class of numerical eaches. In Proc. of the 10th ISOPE conference. WESTHUIS, J., ANDONOWATI & VAN GROESEN, E Efficient numerical calculations on fully nonlinear water waves in large two dimensional domains. Sumitted. WESTHUIS, J., VAN GROESEN, E. & HUIJSMANS, R. H Long time evolution of unstale Dichromatic waves. In 15th IWWW&FB (ed. T. Miloh). Caesarea, Israël. WESTHUIS, J. & HUIJSMANS, R Unstale ichromatic wavegroups: Experimental results. Tech. Rep MARIN. YUEN, H. & LAKE, B Nonlinear deep water waves: theory and experiment. Phys. Fluids 18, 956. Isr-01 JW & EvG Page: 16 of 16