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

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.