Asymptotic, Convergent, and Exact Truncating Series Solutions of the Linear Shallow Water Equations for Channels with Power Law Geometry

Abstract

The present study was originally motivated by some intriguing exact solutions for waves propagating in nonuniform media. In particular, for special depth profiles reflected waves did not appear and ray theory became exact. Herein, geometrical optics is employed to obtain asymptotic series for waves of general shapes in nonuniform narrow channels, within the framework of linear shallow water theory. While being kept simple, the series incorporate higher order contributions that may describe the evolution of waves with high accuracy. The higher orders also contain a secondary wave system. For selected classes of geometries and wave shapes explicit solutions are calculated and compared to numerical solutions. Apart from the vicinity of shorelines, say, higher order expansions generally may provide very accurate approximations to the full solutions. The asymptotic series are analyzed for different wave shapes and are found to be convergent for cases where the basic wave profiles have compact support (finite length). A number of new, closed form, exact solutions are also found. The asymptotic expansion is put into a context by employing it for the transmission of waves from a uniform channel section into a nonuniform one. Additional results and side topics are presented in a supplement.