Note on the equations of Long waves over an uneven bottom
Note on the equations of Long waves over an uneven bottom
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关于不平坦底部长波方程的注记
DOI：
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发表时间：
1966
期刊：
影响因子：
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通讯作者：
B. Méhauté
中科院分区：
文献类型：
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作者：
C. Mei;B. Méhauté
本文系统地导出了不均匀海底浅水长波方程。在假定深度小于水平尺度的基础上，根据波幅的大小，提出了三种近似形式：（a）当波幅与深度为同一数量级时，重新推导了Airy方程作为一级近似;（B）当波幅与深度的立方相当时，两个长度是无量纲的关于一个共同的水平尺度，得到两个高阶非线性方程，其中包括经典椭圆余弦波作为一个特殊的解决方案，为水平底部。这些方程可以转化为一组一阶拟线性双曲方程，其特征曲线在x-t平面上与底部剖面直接相关。为了便于数值计算，然后将它们写为沿特性沿着的微分方程。在第三个区域（c）的非常小的波幅，适当的线性化方程。
Equations for long waves in shallow water are derived systematically for an uneven bottom. With the basic assumption that the depth is small in comparison with a horizontal length scale, three regimes of approximation are presented according to the magnitude of the wave amplitude, (a) When the amplitude is of the same order of magnitude as the depth, the Airy equations are rederived as the first approximation, (b) When the amplitude is comparable to the cube of the depth, both lengths being nondimensionalized with respect to a common horizontal scale, two high-order nonlinear equations are obtained which include the classical cnoidal wave as a special solution for a horizontal bottom. These equations may be transformed to a set of first-order quasi-linear hyperbolic equations with the characteristic curves in the x-t plane directly related to the bottom profile. To facilitate numerical computations they are then written as differential equations along the characteristics. In the third regime (c) of extremely small wave amplitudes the appropriate linearized equations are given.