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

Approaching a sloping beach, shallow water surface waves of Airy get suddenly $+\mathrm{\infty }$ or $-\mathrm{\infty }$ propagation speed at the point of surface $x=x_0$, say, where the tangent ${\Gamma }_x$ of the surface $y=\Gamma$ “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 $+\mathrm{\infty }$ propagation speed just before the crest as $({\Gamma }_x-b_x)(x)\to +0$, $x\to x_0\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}0$ , and a $-\mathrm{\infty }$ propagation speed just after the trough as $({\Gamma }_x-b_x)(x)\to -0$, $x\to x_0\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}0$. 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.