Weakly-Nonlinear, Long Internal Gravity Waves in Stratified Fluids of Finite Depth

Abstract

This paper presents an analytical investigation of the propagation of a weakly-nonlinear, long internal gravity wave in a stratified medium of finite total depth. The governing equation is derived and shown to reduce to the KdV equation in the shallow-water limit and to the Benjamin/Ono equation in the deep-water limit. The equation is also shown to possess four conserved quantities, just as was the case for the deep-water waves considered by Ono. The equation suggests the existence of a steady-state waveform described by two parameters which degenerates into the one-parameter, steady-state waveforms discussed by Benjamin. A numerical approach using Fornberg's pseudospectral method is used to examine the solution. The results demonstrate the existence of a solitary wavelike steady-state solution with a solitonlike behavior. The effect of finite water depth on the waveform and the wave speed of the steady-state solitary waves is discussed.