Long Wave Dynamics along a Convex Bottom

TL;DR

This paper investigates long wave transformations over a convex bottom profile using shallow water equations, revealing unique traveling wave solutions, wave reflection behaviors, and conditions leading to increased runup heights.

Contribution

It establishes the existence and uniqueness of traveling wave solutions in convex bottom geometries and analyzes wave reflection and runup phenomena in this context.

Findings

Traveling wave solutions exist and are unique in convex bottom profiles.

Reflected waves are inverted and form a zone of weak current.

Runup height can be significantly larger on convex profiles than on linear slopes.

Abstract

Long linear wave transformation in the basin of varying depth is studied for a case of a convex bottom profile in the framework of one-dimensional shallow water equation. The existence of travelling wave solutions in this geometry and the uniqueness of this wave class is established through construction of a 1:1 transformation of the general 1D wave equation to the analogous wave equation with constant coefficients. The general solution of the Cauchy problem consists of two travelling waves propagating in opposite directions. It is found that generally a zone of a weak current is formed between these two waves. Waves are reflected from the coastline so that their profile is inverted with respect to the calm water surface. Long wave runup on a beach with this profile is studied for sine pulse, KdV soliton and N-wave. Shown is that in certain cases the runup height along the convex profile is considerably larger than for beaches with a linear slope. The analysis of wave reflection from the bottom containing a shallow coastal area of constant depth and a section with the convex profile shows that a transmitted wave always has a sign-variable shape.