  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.
