具有小密度差的两层流体中运动点源的二阶内波解

在具有自由面的两层流体中，运动点源生成的Kelvin船波存在两种模式，即表面波模式和内波模式。当上、下层流体密度比趋于1时，由内波模式计算的界面波幅趋于无穷大，这与实验事实相违背。为克服此困难，在自由面和界面作小波幅运动的假设，引入一个小密度差参数。研究了运动点源在无粘、不可压且具有小密度差的两层有限深流体中生成的高阶波动。首先利用摄动方法推导了各阶小参数满足的边值问题；其次，给出了小密度差情形下的可解性条件。证明了在密度比趋于1的极限情形，不存在导致界面波幅无穷大的内波模式；最后，利用Phillips的非线性共振相互作用理论，构造了具有自由面的两层有限深流体中Kelvin船波系的二阶一致有效波动解，并证明了该解在深水情形下退化为Newman关于均匀流体中自由面的二阶波动解。

A Second-order Solution on Internal Waves Generated by a Moving Source in Two Layers Fluid with Small Density Difference

Kelvin ship-waves possess two kinds of wave modes in a two-layer fluid system with a free sur face, one is surface-wave and the other is internal-wave. Contradicting to experimental results, the interfacial amplitude calculated by the internal-wave mode becomes unlimited when the density ratio betwe the two layer fluids approaches one. To overcome this difficulty, upon the assumption that the free surface and interface be in small amplitude motion, a parameter of density difference was introduced. Higher order waves generated by a moving source in a two-layer inviscid, incompressible fluid of finite depth with small density difference were studied. Using perturbation method, we first derived a set of nonlinear boundary conditions at the free surface and the interface for various orders of small parameter. Then, the boundary-value problems for successive approximations were derived and the solvability condition for small density difference was obtained. It was presented that for the limit case of the density ratio closing to one there exists no internal-wave mode, which results in the infinite amplitude at the interface. Last, in terms of Phillip's theony of nonlinear resonant interaction, a second-order uniformly valid Kelvin ship-waves in two-layer fluid of finite depth with free surface was constructed. Furthermore, it is demonstmted that the solution will degenerate exactly to Newman's one for single-layer fluid if the density ratio is set to one.