Reflectionless wave propagation on shallow water with variable bathymetry and current

TL;DR

This paper investigates conditions under which long waves can propagate without reflection in shallow water with varying depth, width, and current, providing exact solutions relevant for marine engineering and coastal safety.

Contribution

It presents the first analytical conditions and exact solutions for reflectionless wave propagation in variable shallow water environments.

Findings

Existence of global and bounded exact solutions for reflectionless waves

Reflectionless propagation occurs for specific spatial variations of depth, width, and current

Potential applications in mitigating wave impact on ships and coastal structures

Abstract

In the linear approximation, we study a one-dimensional problem of the reflectionless wave propagation on a surface of a shallow duct with the spatially varying water depth, duct width, and current. We show that both global and bounded exact solutions describing reflectionless propagation in opposite directions of long waves of arbitrary shape exist for the particular variations of these parameters. A general analysis of the problem is illustrated by a few solutions constructed for the specific cases of spatial profiles of the flow velocity. The results obtained can be of interest to mitigate the possible impact of waves on ships, marine engineering constructions, and human activity in the coastal zones.