Past exam of the applied partial differential equations course of the University of Oxford / 2023 / 3 / b

Apply the Rankine-Hugoniot conditions to the conserved variables $(h,hu)$ and flux $(hu,h{u}^2+h^2/2)$. They giveFor a genuine shock, $h_1\ne 1$. Eliminating $u_1$ yields $s^2=h_1(h_1+1)/2$. Taking the positive branch requested,When $h_1=1$, the jump conditions instead force $u_1=0$ and describe a constant state; the labelled interface can then move at any speed and is not a shock. The nontrivial formulas describe a shallow-water shock into stationary unit-depth water.