# On the absence of trapped water waves near a cliffed cape

## TL;DR

This paper proves that in certain semi-infinite water domains with cliffed capes, there are no trapped water wave modes, indicating the absence of localized oscillations near such coastal geometries.

## Contribution

The study demonstrates the non-existence of trapped water wave solutions in domains with a sector-shaped free surface exceeding a vertex angle of π.

## Key findings

- No trapped mode solutions exist for the considered domain geometry.
- The spectral analysis shows no eigenvalues embedded in the continuous spectrum.
- Results apply to water layers of constant depth with cliffed capes.

## Abstract

The water wave problem is considered for a class of semi-infinite domains each having its shore shaped as a cliffed cape. In particular, the free surface of a water domain is supposed to be an infinite sector whose vertex angle is greater than $\pi$, whereas the water layer lying under the free surface is of constant depth. Under these assumptions, it is shown that there are no trapped mode solutions of the problem for all values of a non-dimensional spectral parameter; in other words, no point eigenvalues are embedded in the problem's continuous spectrum.