 36B

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 $c_0=\sqrt{g{h}_0}$.