Tunamis on a deep open sea and on a sloping beach -- a mathematical theory

TL;DR

This paper develops a mathematical theory describing how shallow water surface waves, or tunamis, experience infinite propagation speeds at certain points on a sloping beach, leading to extreme wave interactions.

Contribution

It introduces a novel mathematical framework for understanding the behavior of tunamis on sloping beaches, highlighting the infinite speed phenomena at specific surface points.

Findings

Wave speed becomes infinite at the surface point where the tangent matches the bottom slope.

Infinite wave speeds occur just before crest and after trough, causing intense wave interactions.

The theory explains the mathematical structure of tunamis on sloping beaches.

Abstract

Approaching a sloping beach, shallow water surface waves of Airy get suddenly $ +\infty$ or $ -\infty$ propagation speed at the point of surface $x = x_0$, say, where the tangent $\varGamma_x$ of the surface $y = \varGamma$ "coincide" with that $b_x$ of the water-bottom $y = b(x)$, losing the cruising sound speed of propagation so high on a deep open sea. That is, the tunamis gain instantaneously a $ +\infty$ propagation speed just before the crest as $(\varGamma_x - b_x)(x) \to +0$, $x \to x_0\!-\!0$ , and a $ -\infty$ propagation speed just after the trough as $(\varGamma_x - b_x)(x) \to -0$, $x \to x_0\!-\!0$. We would have thus a big crush between the crest rushing forward and the trough rushing backward. This is a mathematical structure of tunamis "on" a sloping beach, in particular.