Effect of continuous variation of the density of a liquid on the waves generated by moving surface pressures

Abstract

This study investigates the plane linear problem of steady-state internal waves in an ideal incompressible liquid with nonuniform density. The waves are generated by surface pressures applied in a bounded region which moves at constant velocity. It is assumed that the density in the unperturbed state varies continuously with depth, remaining constant in the upper and lower layers and varying according to an exponential law in the middle layer. The problem may be regarded, in particular, as a hydrodynamic model for the study of internal waves produced by a cyclone moving over the surface of the ocean. Analogous investigations for a homogeneous liquid were carried out in [1–3]; internal waves for a liquid with the above-mentioned law of density variation but with stationary pressure changes which are periodic with respect to time were studied in [4]. Problems analogous to the one considered here, both for exponential variation of density in the entire layer and for the case of a nonuniform layer near the surface, were investigated in [5, 6]. An analysis of non-linear waves of the steady-state type with arbitrary distribution of vorticity and density with respect to depth was carried out in [7, 8].