有限深水中骑行波的显式Hamilton描述

小尺度波（扰动波）迭加在大尺度波（未受扰动波）上形成的波动一般之为“骑行波”。研究了有限可变深度的理想不可压缩流体中的骑行波的显式Hamliltn表示，考虑了自由面上流体与空气之间的表面张力。采用自由面高度和自由面上速度势构成的Hamilton正则变量表示骑行波的动能密度，并在未受扰动波的自由面上作一阶展开。运用复变函数论方法处理了二维流动。先用保角变换将物理平面上的流动区域变换到复势平面上的无限长带形区域，然后在复势平面上用Fourier变换解出Laplace方程，最后经Fourier逆变换求出了扰动波速度热所满足的积分方程。作为特例考虑了平坦底部的流动，导出了动能密度的显式表达式。这里给出的积分方程可以替代相当难解的Hamilton正则方程。通过求解积分方程可得出agrange密度的显式表达式。本文提出的方法约研究骑行波的Hamilton描述以及波的相互作用问题提供了新的途径，有助于了解海面的小尺度波的精细结构。

An Explicit Hamiltonian Formulation of Riding Waves in Shallow Water of Finite Depth

The wave motion formed by perturbation waves of small scale superposed on long scale unperturbed waves is generally termed 'riding waves'. The explicit Hamilton formulation of riding waves in i-deal incompressible fluid of finite variable depth was studied, with the surface tension between fluid and air on the free surface was taken into account. The pair of Hamilton canonical variables formed by the height of the free surface and the velocity potential on free surface was used to express the density of kinetic energy of riding waves, and was expanded to the first order on the free surface of the unperturbed waves. Two-dimensional flow was dealt with using the method of functions of complex variables. Firstly, the flow region on the physical plane was transformed by conformal mapping to the infinite strip on the plane of complex potential; then the Laplace equation on the potential plane was solved by Fourier transform; and finally an integral equation satisfied by the velocity potential of the perturbatoin waves by way of inverse Fourier transform. The flow of even bottom was considered as a special case, explicit expression of the density of kinetic energy was derived. The integral equation given here may replace the Hamil-tonian cononical equations, which are rather difficult to solve. The explicit expression of Lagrangian density can be obtained by solving the integral equation. The method presented in this paper provide a new path to study the Hamiltonian formulation of riding waves and the problem of waves interactions, and it is useful to the understanding of the fine structure of small scale waves on the ocean surface.