Parabolic approximations for water waves in conformal coordinate systems

A general formulation of small and large-angle parabolic approximations in conformally mapped coordinate systems is introduced. The technique is applied to the study of two particular cases involving a polar coordinate system. Comparisons to data and full solutions of the governing Helmholtz equation are given. For the case of waves between diverging breakwaters, we find that distinct differences exist between the lowest order parabolic approximation and an analytic solution in the Kirchoff approximation in polar coordinates. The errors are only partially alleviated in the next higher order approximation. For the case of waves in a circular channel bend, we find a similar level of disagreement between lowest-order parabolic approximations and full solutions. The higher order approximation produces results which are reasonably accurate in this case. In both cases, we also investigate the effects of wave nonlinearity, and investigate the growth of Mach stems at the outer wall of circular channel bends.