Note on the equations of Long waves over an uneven bottom

Abstract

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.