= Solution
Let the bed be $z=0$ and the free surface $z=h(x,t)$. In a long, shallow layer, vertical acceleration is smaller than gravity, so the vertical <Euler equation> gives <hydrostatic pressure> $p=p_{\rm atm}+\rho g(h-z)$. <Incompressibility> makes the vertical speed smaller than the horizontal speed by the <aspect ratio>; <irrotational flow>[irrotationality] then makes the leading horizontal <velocity> depth-independent. The horizontal <Euler equation> becomes
$$
\boxed{u_t+uu_x+gh_x=0.}
$$
Integrating $u_x+w_z=0$ across the depth, using bed impermeability $w(0)=0$ and the surface condition $w(h)=h_t+uh_x$, yields
$$
\boxed{h_t+(hu)_x=0.}
$$
Combining these gives the conservative momentum equation $(hu)_t+(hu^2+gh^2/2)_x=0$. Integral mass and <momentum balance for a material fluid element>[momentum balance] across a stationary thin jump therefore require
$$
\boxed{[hu]_-^+=0,\qquad[hu^2+gh^2/2]_-^+=0.}
$$
These conservation laws still apply across a jump where the smooth hydrostatic/<irrotational> derivation does not resolve the internal turbulent structure.
Put $A=h_-$, $B=h_+$ and $q=Au_-=Bu_+>0$, orienting the flow from minus to plus. For a nontrivial jump the momentum condition gives $q^2=gAB(A+B)/2$. Its change in mechanical-<energy> <flux> is
$$
\begin{aligned}
[\rho hu(u^2/2+gh)]_-^+
&=\rho q\left[\frac{q^2}{2}\left(\frac1{B^2}-\frac1{A^2}\right)+g(B-A)\right]\\
&=-\frac{\rho gq(B-A)^3}{4AB}
=\boxed{\frac{\rho gu_-(h_--h_+)^3}{4h_+}}.
\end{aligned}
$$
A passive jump cannot create mechanical <energy>, so the change is nonpositive. Since $q>0$, $\boxed{h_+>h_-}$ for a genuine jump; equality describes no jump. The upstream flow is supercritical and the downstream flow subcritical, since $Fr_-^2=B(A+B)/(2A^2)>1$ and $Fr_+^2=A(A+B)/(2B^2)<1$.
The gas-dynamic analogue is a compressive <normal shock wave>[normal shock]: <density> and <pressure> increase and flow changes from <supersonic> to <subsonic flow>[subsonic]. <Entropy> increases; directed mechanical <energy> is irreversibly converted into <internal energy>. Total gas <energy>, including <internal energy>, is conserved across an <adiabatic process>[adiabatic] <shock>. This distinguishes that total-<energy> balance from the reduced shallow-water mechanical-<energy> loss.
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