Wave propagation in density stratified fluids

The fundamental properties of internal gravity are investigated with respect to the effect that the basic physical quantities of fluid velocity and buoyancy frequency have upon the propagation of these waves. Among other things, it is shown that when the wavenumber tends to infinity the wave phase speeds are then found to depend only upon the local behavior of the mean flow near an overall maximum or minimum of velocity. In shallow stratified fluids internal solitary waves are described to first order in wave amplitude, by the Korteweg-de Vries equation. This theory is extended to second order in wave amplitude. The second order correction to the wave profile and the phase speed and the first order correction to the wavelength are all determined. Strong interactions between weakly nonlinear long internal gravity wave modes are studied. Strong interactions occur when the linear long wave phase speeds are nearly equal although the waves belong to different modes. In shallow stratified fluids it is shown that this situation is described by two coupled Korteweg-de Vries equations.