Series solutions for shallow water waves

Abstract

Introduction. The main purpose of this note is to convert the second-order terms given by Laitone [1960, 1962] for finite amplitude cnoidal waves from their reference depth based upon the water depth below the wave trough ho to the more appropriate expression in terms of the mean or still water depth d. An additional purpose is to answer some of the questions raised by Chappelear [1962] concerning the differences he found between his numerical calculations and Laitone's [1960] second-order series expansions for cnoidal waves. The cnoidal waves under consideration are the finite amplitude progressive waves in constant-depth shallow water that have a periodic wave profile η which is represented mathematically by the Jacobian elliptic functions. The nonlinear shallow water theory used by both Chappelear [1962] and Laitone [1960, 1962] is based upon the systematic asymptotic expansion method introduced by Friedrichs [1948]. He generalized the classical nonlinear shallow water first approximation by a suitable stretching of the vertical dimensions which are considered to be everywhere physically smaller than the wavelength. As shown by many comparisons, e.g. see Laitone [1962], this shallow water expansion method is satisfactory for wavelengths greater than 5 times the water depth.