Group velocity and nonlinear dispersive wave propagation

Abstract

By the use of a Hamiltonian formulation, a basic group velocity is defined as the derivative of frequency with respect to wavenumber keeping action density constant, and is shown to represent an incremental action velocity in the general nonlinear case. The stability treatment of Whitham and Lighthill is extended to several dimensions. The water-wave analysis of Whitham (1967 a ) is extended to two space dimensions, and is shown to predict oblique­-mode instabilities for kh < 1.36. A treatment of Lighthill’s (1965) solution in the one­-dimensional elliptic case resolves the problem of the energy distribution in the solution past the critical time. A note on diffraction effects on quasilinear solutions of the Whitham type is presented.