= Solution
For a strictly hyperbolic two-component <system of conservation laws> with ordered speeds $\lambda_-<\lambda_+$, a compressive family-$j$ shock has its family characteristics entering from both sides: $\lambda_j(U_R)<s<\lambda_j(U_L)$. The other family crosses the interface with the same sign of relative speed on both sides. These are the <Lax shock admissibility conditions> and give three incoming characteristic directions for two conservation laws.
In the present problem,
$$
\lambda_\pm(U_R)=\pm1,\qquad\lambda_\pm(U_L)=u_1\pm\sqrt{h_1}.
$$
Since $u_1=s-s/h_1$, we always have $u_1-\sqrt{h_1}<s$. Hence a positive-speed shock of this form cannot be a family-minus shock. For a family-plus shock the remaining strict inequalities are
$$
1<s<u_1+\sqrt{h_1}.
$$
The first is equivalent to $h_1(h_1+1)>2$, or $h_1>1$. The second is equivalent to $s<h_1^{3/2}$, whose square reduces to $h_1+1<2h_1^2$, again equivalent to $h_1>1$ for positive $h_1$. The minus speed on the right is $-1<s$ automatically. Thus all causal inequalities hold exactly when
$$
\boxed{h_1>H,\qquad H=1}.
$$
Depth increases behind the right-moving shock. For $0<h_1<1$, the algebraic jump solution has the wrong characteristic orientation and is an inadmissible expansion shock; equality is the no-jump limit.
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