Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

Abstract

We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear current. The far-field modal equations for the ring waves are formulated in dimensional form. They are derived both directly from the dimensional set of Euler equations written in the cylindrical coordinate system, and from the dimensional formulation for plane waves tangent to the ring. A number of examples for both surface waves in a homogeneous fluid and internal waves in a stratified fluid are considered to illustrate the theory. The detailed analysis is developed for the case of a two-layer fluid with a linear shear current where we construct the necessary singular solutions of the equation defining the modification of the speed of the long surface and interfacial ring waves in different directions and study their wavefronts and two-dimensional modal structure. Comparisons are made between the surface waves in a homogeneous and two-layered fluids, as well as the interfacial waves described exactly and in the rigid-lid approximation.