= Solution
Measure <pressure> relative to atmospheric <pressure>. At leading order the stress-free upper surface gives $p(H)=0$, $\tau(H)=0$. Integrating the normal balance first yields
$$
p(x,z)=\rho g\cos\alpha[H(x,t)-z].
$$
At fixed $z$, $p_x=\rho g\cos\alpha H_x$. Therefore $\tau_z=\rho g(\cos\alpha H_x-\sin\alpha)$, independent of $z$. Integrating from $z$ to $H$ gives
$$
\boxed{\tau(x,z)=\rho g(\sin\alpha-\cos\alpha H_x)(H-z).}
$$
This is linear in $z$, is zero at the surface, and reaches its basal value at $z=0$. In particular the surface slope modifies the shear driving; a flat upper surface relative to the horizontal eliminates it in this approximation.
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