ALTERNATIVE MODEL FOR NONLINEAR WATER WAVES OVER ARBITRARY DEPTHS

95Journal of Hydrodynamics, Ser. B,2 (2003),95- 100China Ocean Press，Beijing 一Printed in ChinaALTERNATIVE MODEL FOR NONLINEAR WATER WAVES OVER ARBI-TRARY DEPTHS'Huang HuShanghai University, Shanghai Institute of Applied Mechematics and Mechanics, Shanghai 200072，Chi-naLaboratory of Nonlinear Mechanics， Institute of Mechanics， Chinese Academy of Science， Beijing100080，China(Received June 1, 2001)ABSTRACT: To account for effects of nonlinearty on the cause relatively large variations of the wave proper-wave propagation characteristics. by using Green 's secondi- ties particularly after significant propagation dis-dentity a nonlinear consistent equation for water waves propa-tance. Surface wave scattering by uneven bottoms,gating over arbitrary depths is derived by introducing a func-generally assumed to be horizontal, or mildly var-tion as approximation to the exact velocity protential functionying”，or rapidly changeable with some specificfor the nonlinear governing equations， which can be simpli -fied to teh linear uniform mild-slope equation given by Zhangscaling orders-4， has been studied by many rand Edge-1] recently. In shallow water the equation reduces searchers in the past fifeen years, which to a cer-to a nonlinear equation of Boussinesq-type. In deep water the tain extent can predict some typical bottom topog-nonlinear dispersion relation for Stokes expansion is found.raphy，such as rippled bed and sand bar. Due to avariety of bottoms these models may fail to simu-KEY WORDS: nonlinearity， variable depth, mild-slope e-late completely the various mechanisms that inducequation，Boussinesq- type equationthe transformation of waves. In effect recent stud-ies[5-7] showed that some linear models are ineffec-1. INTRODUCTIONtive in predicting some resonance peaks associatedThe mathematical difficultes inherent in deter-closely with bathymetry variation. In other wordsmining the evolution of an incident wave train andthese models break down at some frequencies, sim-the associated phenomena result mainly from twoilar to the classic mild-slope equation’ s breakingnonlinear boundary conditions at the itially un- down as described by Kirbyl2].known free surface, a kinematic condition and a dy-It can be seen clearly that the major questionnamic condition. Thus two assumptions, the lin- facing the modeler is how to represent nonlinearityearization and the small-amplitude waves, are often in a consistent wave model on frequency. In theadopted to obtain the linear analytical solution. Al- present study, starting directly from the governingthough linear wave theory serves as the starting equations for water waves a nonlinear equation withpoint for any weakly nonlinear model, it does not no restriction on the variation of water depth hasgive us much of a route to describe the complexity been developed by applying Green' s identity.of wave motion in the coastal region.In shallow water, waves are strongly affected 2. THE CONSISTENT EQUATIONby depth variations, yet water depth near a site ofWe sunnnse that inromnressible, homogeneous中国煤化工interest is often not well known and may vary rap- fluidover a bed of varyingidly in time and space. Slight depth variations can quiesYHC N M H G y denoting horizontalProject supported by the National Natural Science Foundation of China (Grant No: 10272072)，the Open Foundationof the State Key Laboratory of Estuarine and Coastal Research (SKLEC), the Open Foundation of the State Key Laboratoryof Nonlined?存熱据ics (LNM), and Shanghai Key Subjeet Program.)6cartesian coordinates.' The vertical coordinate, Z，2 non propagating modes(7)is measured positively upwards with the undis-turbed free surface atz = 0. The governing equa-We then use Green's second identity of Smith andtions for the velocity potential 中(x，y，z，l) and .SprinksL1o] to extract the propagating component ofthe free-surface elevation ζ(x,y,l) are given as fol-lows:af.a中,0f-5°φ2- φV°φ+°∞=0,-h≤z≤5ζ(1)[[要-az2]dz=「r'Jz°8z_-r(8)刁2 φaφ_Lr日。1.laφ子一at+gg+a+2Vφ.V+2和和"or[(中)2 +1 vφ|°]=0,z= 5(x,y,t) (2)j_[jv°φ.十虫共7dz=-「f驶p。习z(9)aφ黑+Vh. Vφ=0，z=h(3)The integrals are mainpulated to obtainwhere又=x’v) and g is the gravitational ac-. (k°f+ f°V°φ+2fVφ●Vf+ pfV'f)celeration constant. The free-surface elevation ζ fol-lows as when φ has been determined:dx= 1{”中十wφφ+raφ+1[就i+2Vφ.又+gat2ζ==一1φ[(Vφ)2 + (独)2]}，g Al2-0z1 aφ 2r亚。艺x工2》+1口01)--atz=ζ(4)[fVh●(fVφ+φvf)]--h(10)Adopting Radder and Dingeman's choice[8] for see-king a trial functionφ(x，y, z t; ζ,中) as an ap- Applying Leibniz's rule, we getproximation to the exact function 中(x，y, z, t)，which is motivated by a modification to linear wave1”中+ wφ十日中。theory as proposed by Heg', we use thesolu- J_ (帐kfs +v°)dz =g'Ztion for linear periodic waves applied at the freesurfacez= ζ :1a中。ar/a中。2V中.V+2x(十φ(x,y,z,t;5,$)=chk(h+)+(x,y.t)=伟chk(h + 5ζ)V中。|2]}2-:+二v●(CC,Vφ)-(5)where the real functionk is given as the positive so-[fVh●▼f]--h(11)lution of the equationwhere C and C。are function of the local depthh + ζw = gkthk(h+ 5)中国煤化工MHCNMHGwhere wo is the frequency. The solutionto Eqs. (1)- c_ wCr1+- 2k(h+5)(12):.发，Cg=(3) may be expressed assh2k(h+ξ)-φ。(x,y,z,t) =. f(h,k,ζ,z)$(x,y,t)+and)7of、_ afL af、(13a)∞vh-kvk+gVζin which, m = 2(Vφ- koφVζ),n= k"(1-σ° )Vh+[σ+q(1-σ )]Vk- k°σVζ. The dimensionlessV'f =fv2h+ fvk+ a上、parameters of a;(i = 1,..,9) are given in Appen-hVkξV s+ h‘dix.If the terms conceming ζ in F and the nonlinear(Vh●Vh)+af(vk. vk)+d5.term N are ignored, Eq. (13) is the same as the u-okeniform mild-slope equation derived recently byZhang and Edge]. Tsay et al.11] also derived a(Vζ●▼5) +-2[3f(Vh.vk)+ fnonlinear equation similar to Eq. (13) when slope-JhakJhaζof water bottom is mild.(Vh. v5)+a”f(vk●vζ)](13b) 3.THE GEOMETRIC-OPTICS APPROXIMA-kζTIONThe nonlinear Eq. (15) can be linearized asThe detailed expressions for the derivatives of f aregiven in Appendix: Therefore from Eqs. (11) andφ_V●(CC。Vφ) + (o - k°CC。)φ-(4) for中。we derive a nonlinear consistent equationat2of the velocity potential φ for surface waves over ar-bitrary depths in the following form:gFφ= 0(17)P-口. (CC2V$) + (wζ-k°CCg$)-whereVik+at(Vh●Vh)k十gFφ+N=0.(14) F =a1V*h+a2为2Vk .Vk+,Vh●Vka:(18)k3F= a1V"h+a2V"kg+asV°ζ+ a;(Vh●In order to clarify the relation between Eq. (17)and the geometric optics-approximation, we substi-Vh)k+asVk●Vk+ a6(Vζ●Vζ)k十tute the wave- like structureφ(x,y,t) = Re {ei5-)}(19)又ζ.▼k27-+ as(Vh●Vζ)k+ ag1(15)into Eq. (17). Here the amplitudeb(x, y) and theahase functon S(x，y) are real functions of x andy. From the real part of the resulting equation theN= <φ+ψvφ.v+1中是(中)2+eikonal equation can be obtained, while from the i-alT22dzJzHzmaginary part a transport equation for the wave en-ergy is obtained, i. e.| Vφ。|2]>.-;= 2顿°2中- ko中当)+Htat(VS)2= k+V(CC2).▼b.V2b.gF(20)CC。‘bCCgk'rφ +m●{n(ks$°-φ5>+ ko●中国煤化工.MHCNMHG(21)[koφφ_ 5→φ-φ、vζ-心<骤>]+From Eq. (20) it follows that for the geometric op-)t~Attics- approximation (VS)2 = k2 to be valid, the fol-lowing condition should be met口(箭方数据”. v(fVφ+φVf)|=:) (16)98V(CCg)。Vb .V2bBased on the Boussinesq assumptions， i. e.，O(e)CC。《k”，b< k2，=O(μ2)<1, Eq. (13) readsE