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

We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear flow. The far-field modal and amplitude equations for the ring waves are presented in dimensional form. We re-derive them from the formulation for plane waves tangent to the ring wave, which opens a way to obtaining important characteristics of the ring waves (e.g. group speed) and to constructing more general `hybrid solutions' consisting of a part of a ring wave and two tangent plane waves. The modal equations constitute a new spectral problem, and are analysed for a number of examples of surface ring waves in a homogeneous fluid and internal ring waves in a stratified fluid. The detailed analysis is developed for the case of a two-layered fluid with a linear shear current where we study their wavefronts and two-dimensional modal structure. Comparisons are made between the modal functions of the surface waves in a homogeneous and two-layered fluids, as well as the interfacial waves described exactly and in the rigid-lid approximation. We also analyse the wavefronts of surface and interfacial waves for a family of power-law upper-layer currents, which can be used to model wind generated currents, river inflows and exchange flows in straits. A global and local measure of the deformation of wavefronts are introduced and evaluated.