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For a smooth solution, the first equation is $u_x+(uv{)}_y=0$. Multiplying it by $v$ and adding $u$ times the second equation givesThus the appropriate shallow water equations conservation variables and flux areUse this conservative extension to define a weak solution across $y=F(x)$. A jump contributes $-F^{\prime }[W]\delta (y-F(x))$ to $W_x$ and $[H]\delta (y-F(x))$ to $H_y$. Cancellation of the singular parts gives the Rankine-Hugoniot conditionsEquivalently this follows by integrating each conservation law over a thin space-time control region crossing the discontinuity.

The conservative extension is essential here. Piecewise smoothness by itself does not imply any jump law. Moreover the smooth second equation can also be written $v_x+(u+v^2/2{)}_y=0$; imposing that different conservation law distributionally would give a different shock condition. Weak conservation laws depend on conserved variables. The requested jump law specifically selects mass $u$ and momentum $uv$, and this is the extension used below.