Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 36B (Waves)

Linearizing the shallow water equations about depth $h$ at rest gives wave speed $c=\sqrt{gh}$. Replacing $h$ by $c$ in the nonlinear equations yields $c_t+u{c}_x+(c/2)u_x=0$ and $u_t+u{u}_x+2c{c}_x=0$. Direct addition and subtraction giveThus the Riemann invariants $J_{\pm }=u\pm 2(c-c_0)$ are constant on the characteristics $C_{\pm }$ with speeds $u\pm c$. In this problem $c_0=\sqrt{g{h}_0}$.

Before a signal reaches the wall, the portion to the right of the leading characteristic $x=c_{0}t$ is unaffected. Its initial depth and velocity persist, giving $\boxed{u=0,c=c_0(c_{0}t\le x\le a)}$ for $0\le t\le a/c_0$.

The release produces a rarefaction wave expanding into the dry region. Every incoming $C_{-}$ characteristic connects to the undisturbed pool, where $J_{-}=0$, so throughout the simple wave $u=2(c-c_0)$. The outgoing $C_{+}$ rays emanate from $(0,0)$ and have $x/t=u+c=3c-2c_0$. Solving givesThe depth is $h=c^2/g$. At the right edge these formulas match the stationary pool; at the left edge $c=0$ and the fluid front travels at $-2c_0$. Beyond it the bed is dry, so a fluid velocity need not be assigned. Values at $t=0$ are understood as one-sided limits of the fan.

The first wall interaction occurs at $t_0=a/c_0$. The leftmost reflected signal follows a $C_{-}$ ray through $(a,t_0)$. Until a point has encountered this signal it remains in the original fan. In that fan its speed is $u-c=x/(3t)-4c_0/3$. ThusMatching at $(a,t_0)$ gives the reflected frontConsequently the same rarefaction formula remains valid for $-2c_{0}t\le x\le x_f(t)$ when $t\ge t_0$. Between $x_f(t)$ and the wall, information from the wall changes $J_{-}$ and the simple-wave relation no longer holds. In particular continuing the old formula to $x=a$ would give a negative velocity for $t>t_0$, violating the no-penetration condition $u(a,t)=0$. The reflected region must accommodate both characteristic families; it is not another untouched part of the original fan.