Instability of Two Interacting Quasi-Monochromatic Waves in Shallow Water

China Ocern Finginepring, Vol.21, No.3, pp. 451- 460C 2007 China Ocean Press, ISSN 0890-5487Instability of Two Interacting Quasi-MonochromaticWaves in Shallow Water*TANC Shu-pian (汤叔梗)°.l，WEI Li-xin (书立新)卢and XU Yuan (徐援)“且School of Sciences，llebei University of Science and Tehnology ,Shijiazhuang 050018, China”The Surey Bureoau of Hydrology ard Wuter Resures of louer Reaches ofChangjiang Riter, CWRC，Nanjing 210011, ChinaColage of Sciences，Hohai Unirersity, Nanjing 210098, China(Received 6 November 2006; acrepled 25 April 2007)ABSTRACTThe nonlinear inleractiuns of waves with a double- peaked power spetum have been studied in shallow water. 'Thestarting poinl is dhe prololypical eqpuation for nonlinear uridirectional wavcs in shallow water, i.e. the Korteweg de: Vriesecquation. By mears o[ a muliple- scale technique two defocusing coupled Nonlinear Schridinger equations are derived. IIis found analyically lhal plare wave solutions of such a system are unstable: for small px:turbations, showing that the cxis-tence of a new energy exchange mechanism which can influence the behavior of occan wavcs in shallw water.Key words: intbility; ocean wue; Koreweg de lries equntion; coupled nonlinear Schridinger equation; plane uanxesolution1. IntroductionThe propagation of multiple wave-train systems in shalow water has historically received less at-tention than the propagation of a single wave train. Nevertheless, experimental studies carried out byThompson ( 1980) in representative sites near the coasts of the United States reveal that in 65% of the .analyzed datu, ocean wave spectra show two or more separaled peaks in the frequency dornain (Thomp-son，1980; Egashira, 1985; Smith and Vinccnt, 1992). In this framework, experimental work in thelaboralory has been performed (Smith and Vincent, 1992) . They propagated iregular wave trains withtwo distinct spectral peaks in a wavc lune with 1: 30 slope for different valucs of the peak frequeneyand significant wave height. From a physical point of view, this condition mimics the interaction of twowave regimes, a“swell” and a“sea” ，propagating in the same direction toward shore in shallow wa-ter. Thcir major observation was in a decay of the higher frequency peak along the flumc. They hypoth-esizxd three possible explanations for such experimental results: (i) resonant inleractions amongwaves, (i) botton friction that acls differently for the two dominant wave peaks, ( ii) breaking of theshorter waves enhanced by the presence of the longer waves. More recently, using a higher orderBoussinesq model, Chen et al. ( 1997) have shown .中国煤化Interacions, without* The paper was financialy suponted by he Nalinal Natural SoYH. C N MH Gu No. 40476062) andFoundation of Hebei Universily of Science and Technology1 Cnorresponding author. E-mail: tangshupian@ sina. com452TANG Shu-pian et al. / Chin (kewn Enpineering. 21(3). 451 - 460invoking bottom friction or wave breaking，is sufficient w account for the decay of the high frequencypeak. Even though these numerical simulations of the Boussinesq equation can qualitatively reproducethe experimental results, data on the occurence of double peaked spectra in the North Sea have beenanalyoed (Soares， 1991) ，indicating that they appeur to have similar probability of occurence than inthe North Atlantic, as previously reported, the basic physical mechanisms of interaction of wave trainswith double peaked spectra in shallow water are far from being completely understood.A great deal of progress in understanding wave propagation in shallow water has been achieved by .investigating the Korteveg -de Vries (KdV) equation, which can be considered as the basic weakly non-linear model for unidirectional shallow water waves. The analytical properties of the KdV equation ( itis integrable by the Inversc Scatlering Trnsform ( Ablowitz and Scgur, 1981)) have improved basicknowledge of the nonlinear dynamics of water waves (Mei, 1993; Osbome and Petti, 1994 ; Osbormeet al.，1991; Osborne et al. 1998). A one-dimensional and a horizontally two dimensional randornwave transformnation models based on a spectral wave equation with a probabilistic bore-type energy dis-sipation term， and examines the validity of the wave models by comparing the predictions with the ob-servations. The model solves the spatial evolution of complex Fourier amplitudes from which energyspectra are calculated ( Mase and Kitano, 2000). In particular after the seminal work by Zabusky andKruskal ( 1965), extensive studies have been carried out on the evolution of a sine wave in shallow wa-ter (Mei, 1993). However, less attention has been paid to the evolution of two nonochromatic waveswith different wave numbers. In this paper discussed is a fundamental instability that occurs betweentwo quasi-monochromatic interacting wave trains. Furthermore, the focus herein is not to model oceanwaves but to study leading order eleels using he sinplest weakly nonlinear and dispersive model in .shallow water. The subject is therefore an investigation of the basic nonlinear interaction that may lakeplace between two separated narow banded wave spectra in shallow water. For this aimn, in suitableassumptions, and with a multiple-scale technique, we derive a system of two Coupled NonlinearSchridinger (CNIS) equations from the KdV equation. Plane wave solutions of such a systerm are thensludied analyically by a stlandard linear stabilty analysis, resulting in the presence of an instahility re-gion.2. The Plane Wave Solution of CNLS Equations'The KdV equation can be formally derived from the Euler equations for water waves ( Mei, 1993;Whitham，1973) under the assumption that waves are small ( but finite amplitude) and long whenwomnpared with the water depth at rest. In a frame of reference moving with the velocity co=V gh,where h is the water dcpth and g is gravity acceleration, the KdV equation in non-dinensional formreads:中国煤化工ζ+ μ5ζx + Aζx(1).YH.CNMHGlere ζ= ζ(x,t) is the free surface elevation, x and t anc spaut anu uue vatiawnes; p and λ are thenonlinear and dispersive small parameters: μ = 3a/(2h) and λ= (h/l)2/6, with a characteristicTANC Shu-pian u nl. / China Ocean Engineering .21(3).451 - 460453wave -amplitude and l a characteristic wavelength. We are interested in invesligating the interaction oftwo waves, cenlered al non-dimnensional wave nrumbers k; and kz, propagating in the positive x direc-tion. We consider the case of narow banded speetra, i.e. △h;/h;2.01.00.5.5Fig.1. Growlh-rate a a function of non-dimcnsional Fig. 2. Growth-ralte 出a furctlion of non- dimensionalwave mmber 1k2 and perturbation K for A= Bwave. numberand 的and perturbation K for=0.2 and h= 1. Axes are in non-dimension-A= B=0.4 and k=1. Axes are in mourdi-al units.mensional units.4. The ExperimentThe experiments were canied out in a towing tank with 260 m in length and 10.5 m in width,and waves were generated by a horizontally double-hinged fap-type wave maker. The depth of the tankis 10 m for the first 80 m section closest lo the wave maker, and 5 m elsewhere. A vertical bottomjump connects the two tank parts. A sloping beach is located at the far end of the tank opposite thewave maker. The wave elevation was measured by wave staffs simultaneously at different locationsdownstreamn the tank as shown in Fig. 3.WavemakerBeach、235 678!0--- *9.3 408(120160200x (m)Fig. 3. Sketch of the wave tiank with theI中国煤化工The time series measured by probe 1 at 9.3 m is used^TYHCNMHG_ion of the numer-cal simulations. Probe2 is at 40 m and is0.5 m in front ot probe 5. r'robes 0 and 8 are 0.4 m abeadof and behind probe 7 at 160 m. Probe 9 is 1.5 m from the tank wall and serves to assess the impor-TANG Su-pian e al. / Chiru 0ovn kingivrering , 21(3), 451 -460457tance of trarlsverse modulations .Two experiments with bichromatic wave ( test 60 and test 61) obtained by mixing two qusi-monochromatic waves were carried out according to the following specifications:Wave periodsWave heightsTestT|(s)T2(s) .H(m)H2(m)501.902.100.16510.95.1.050.04Further details of the experiment have been reported in Stansberg ( 1998) .5. Comparison of Simulation and ExperimentExperimental test 60 is periodic with period 39.9 s. A computational donain of length 279.3 s,corresponding to 7 periods is adopted after skipping the first 19.7 s of startup. The non-dimensionaldepthis kh= 10 for the first 80 m and kgh =5 for the rest of the tank, however we herein presentsimulations using kqh = 10 for the entire tank. Simulations with hgh = 5 revealed only insignificantmodifications for large fetch, thus as far as this comparison between experiment and simulations is con-cemned, the effect of the jump at 80 m is not important.The time series measured at wave staff 1 is used for initialization. Here the transient effecls ofstartup do not occur in the computational domain. At successive wave staffs, the transient effects ofstartup propagate into the computational domain, but are not accounted for in the numerical simulation .Results are presented from the last period in the computational domain which remains unaffected bytransient effects of startup for the duration of the simulation. Measurements and initialization at staff Iis shown in Fig. 4.Experiment and all wave theories).3-Experiment(Linear,NLS,CNLS)CNLS0.2.2.1--0.1 t-0.1-0.2-0.3245 250 255 260 265 270~ 275 280-0.34245250 2: 55260 265 270 275 2801(8)t(s)Fig. 4. Test 60, wave staffI ial 9.3 m, usexdFig. 5. Test 60, staff2 at 40 m.for initialization.The NLS equation accounts for a nonlinear increase中国煤化工ds good ugreenentwith the observed phase velocity in the experiment. TheDYHCNMHGCunt for he nonlin-ear increase in group-velocity. The NLS equation does account Ior tne nonnear Increasc in arnplitudeof the group, but does not capture the asymmetric forward leaning evolution as seen in the experi-458TANC Shu-piun el al. / China Cevun kingineiug. 21(3). 451 - 4600.3-Expcriment0.ExperimentCNLS ACNLS0.24息ofvwIMN息oMWA0.1-0.1).2 t-0.2-0.3 L2452502:55260265270275280-0.3- 245 250 255 260 265 270 275 280t(s)t(8)Fig. 6. Test 60, staff4 at 80 m.Fg.7. Test 60. sa[5 at 120 m. .0.30.2).1息0wwww.Mwwhl息wllww-0.3C245 250 255 260 265 270 275 280-0.3- 245 250~ 255 260 265 270 275 280t()Fig. 8. Tesl 60, slaff7 all 160 m.Fig. 9. Test60, statf l0 al 200 m.0.080.08 [:NLS0.04MMMM巨0Mw-0.04-0.04 t-0.0840445450460-0.08455ig.10. Test 61. staff 1 at 9.3 m, used for inialization.Fig. 11. Test 61, stall2 al 40 m.CNLS simulation results al slaffs2, 4, 5, 7 and 10 are shown in Figs. 5~ 9. The CNLS equa-tion accounts for nonlinear increase in both the phase and group velocities in gvod agreenent with theexperinental observations. The nonlinear increase in amplitude of the group and the asymmetric for-ward-leaning evolution of the group seen in the experimenls 0 ale rantied。中国煤化工。Experiment 61 was done for shorter wavelength and Iffectively muchlonger tank. The waves are periodie with period 19.95 s.TYCNMH G;of length 199.5s corresponding to 10 periods, after skipping the first 270.5 s of lhe time serics . The non-dimernsionalTANC Shu-pim et al. / China (rerun Finginering, 21(3), 451 - 460459depth is kgh =40 for the first 80 m and k,h = 20 for the rest of the tank. These depths aure effectivelyinfinite as far as experiment 61 is concemed.The time series measured at wave staff 1 is used for initialization as shown in Fig. 10. CNLS sim-ulation results at staffs2, 4 and 5 are shown in Figs. 11~ 13. Hig. 11 shows that the qualitalive fea-tures of group spltting are well captured. Fig. 13 reveals thal the simulation rcsults of the CNLS equa-tion become unreliable for large fetch.0.08Experiment ---- CNLS .. Experiment .CNLS0.04 |0.04-0.04-0.08440145450455- 460-0.0840445455460t(s)t(S)Fig. 12. 'Test61, stalf4 at 80 m.Fig. 13. Test61, salf5 at 120 m.The experinents and simulations were done for long-crestod waves. It is necessary to pcrforn ex-periments and simulations for short-crested waves to assess if forecasting of individual wave crests canbe done in a realistie shot-crested sea. However, for long-crested sea states ，we anticipate that themodel is capable of acurately forecasting individual wave cresls over a fetch of several tens of wavc-lengths .6. Concluding RemarksResults can be summarized as follows: in shallow water the dynamics of two wave trains can beunstable; as expected the gruwh ratc and the size of the instabilily region depend on the nonlinearity o[the system.The derived CNLS systerm represents a very crude simplification of the real problem: efct relat-ed lo a non- constant water depth could be considered; directionalily could also be simply included byapplying the multiple scale method to the Kadomtsev-Petviashvili equation ( Ablowitz and Segur, 1981)(an extension of the KdV equation that includes the dynamics of transverse perturbations); higher or-der effects could also be investigaled.We believe that the prcsent work offers new perspectives for understanding the dynamics of dou-ble-peaked spectra in shallow water. Accurate comparison with experiments and numerical simulationsare definitely needed in order to validate the obtained theoreticul rAs a final remnark we would中国煤化工like to stress that we have sturted the derivation of the CNLSAthe Inverse Scat-tering theory furmishes a unique method for investigating allHCNMHG. NM.M.nerefore interest-ing to interpret the unstable solutions of the CNLS equations in temms of the Inverse Scallering modes460TANG Shut-pion pr ru. / (hin (xrn kingineering ，21(3). 451 - 460for the KdV cquation.ReferencesAblowitz. M. J. and Sogur, H.，1981. Solions and Imerse Sonttering Trunsfom. SIAM, Philadelphia, P. A.Chren. Y., Cuza, R. T. and l'lgar, S.. 1997. Modeling spectra of breaking surface waves in shallow water, J. (en-phys. Res.. 102(CII): 25035 ~ 25046.Egashira，K.，Fukuda. S.，Kishira, Y. and Nishinura. T.，1985. Field olbservation of wave: transformation due toreef, Proc. 32rul Jupanese Couslal Eng. Conf., 90~ 94.Forest, M. G.. McLaughlin, D. w.. Munki, D. J. and Wright, 0. C.. 2000. Nonfoxusing Iusubilily in CGoupled,Integrable Norlinear Schrodinger pdes, J. Norlinear Sci.. 10, 291~ 331.Grimshaw, R.， Pelinovsky, D.. Pelinovsky, E. wnd Talipova, T.，2001 . Wave group dynrics in weakly nudlineurlong- wave models, Physica D, 159(1-2): 35~ 57.Hashimoto, H. and Ono, H., 1972. Norlinear modulation of gravity wave, J. Phys. Soc. Jupan, 33. 805-811.Manakov, S. V.， 1974. On the theory of lwo dimensional stalionary self-focusing of electromagnetic waves, Sov.Phys.-JETP, 38, 248 ~ 253.Mase, H. and Kitano, T.. 2000. Spectrum- based prediction model for random wave transformation over arbitrary bottomtopography. (osal Iinginering Joumal, 42(1): 11~ I51.Mei, C. C., 1993. 7he Applied Dynamics of Ocean Surface Waves. John Wiley & Sons.Mernyuk，C. R.. 1987. \onlinear pulse propagation in birefringent optical fibers, IEEE' J. Qucnwum FNlectron., 23(2): 174~ 176.Menyuk. C. R, 1989. Pulse propeagation in an llplicallyl birefringent Kerr mocdium, IEEE J. Qunum Hiectron.. 25(12): 2674 ~ 2682.Okarmura, M.，1984. Instability of weakly nonlinear standing gravity waves, J. Phys. Soc. Japan, 53(11): 3788~3796.Oslbome, A. R., Segre, E., Bfelta, G. and Cavaleni, I., 1991. Soliton basis statcs in shallow- water ocean surfacewaves，Phys. Rer， Lell., 67(5): 592 ~ 595.Osbome，A. R. and Peti, M..1994. Laboralory- generaled, shallow waler surface waves: Analysis using the periodic ,inverse scattering transformn, Phys. of Fluids, 6(5): 1727 ~ 1744.Osborne, A. R., Serio, M., Bergamasco, L. and Cavaleri, L.. 1998. Solions, cnoidal waves and nonlinear interac-tions in shallow- water ocean surface waves, Physica D, 123(1-4): 64~ 81.Pierce, R. D. and Knobloch, E, 1994. On the modulational slability of traveling and standing water waves, Phys. ofF'uids, 6(3): 1177~ 1190.Roskes, C. J, 1976. Nonlinear multiphase deep- water wavetrains, Phys of bluids, 19(8): 1253~ 1254.Smith, J M. and Vincent, C. L.， 1992. Shoaling and decay o[ twn wave trains on a beach, J. Walenvay, Port ,Conast. & 0e. kingrg.. 118(5): 517~ 533.Soares, C. G, 1991. On the occurence of double peaked wave sperctra, Oeean king., 18(1-2): 167- 171.Sansherg，C. T，1998. On the nonlinear behavior of ocean wave groups, Proc. 3rd Intermnational Symposium on OceanWare Measurement and Analysi- WAVES'97, ASCE, 2, 1227~ 1241.'Thompson, E.下，1980. knergy spectra in shallow U. S. coastal uxaters, 'Tech. Paper No.82-2, Coast. Eng. Res.Ctr.，U.S. Army Corps of Engrs., Fort Belvoir, Va.'Tracy, E. R., Harson, J. w.. Usborme, A. R. and Bergamasco, I., 1988. On the nonlinear Schrsdinger equationlimit of the Korteweg-de Vries equation, Physica D, 32(1): 83~ 106.Whitham, B. 1973. Linear and nonlinear waves, Wiley and Sons ./abusky, N. J. and Kruskal, M. D.. 1965. Interaction of solitons in a collisionless plasma and the recurence of initialstates, Phys. Rer. Llt., 15(6): 240~ 243.中国煤化工/akharov，V. E. and Kuznetsov，E. A., 1986. Multi-scale expanzegrable by the inversescattering lransform, Physica D.18(1-3): 455 ~ 463..YHCNMHG