Asymptotic solution for the problem of sound propagation in a shallow sea with the bathymetry described by a parametric quadratic function

A wide class of three-dimensional sound propagation problems in shallow water where the bathymetry can be locally approximated by a parametric quadratic function is considered. Relevant examples of such bathymetry functions include underwater canyons and ridges. Asymptotic solution for this class of problems is derived under the adiabaticity assumption. The solution is based on mode parabolic equation theory and group-theoretical disentanglement formulas for operator exponentials. Two examples are considered (a ridge and a bottom with quadratic slope), and respective wavefront geometry is studied.