= Solution
At the rigid bed, the <no-penetration boundary condition> gives
$$
\boxed{\phi_z=0\quad\text{at }z=-h}.
$$
The displaced surface is material, so taking the <material derivative> of $z-\eta(x,t)$ gives the exact <kinematic boundary condition for a free-surface graph>:
$$
\boxed{\eta_t+\phi_x\eta_x=\phi_z\quad\text{at }z=\eta(x,t)}.
$$
With upward vertical coordinate $z$, the gravitational potential per unit <mass> is $gz$. The <Unsteady Bernoulli equation>, evaluated on the surface with the prescribed <pressure> $p=B\eta_{xxxx}$, gives
$$
\boxed{\phi_t+\frac12(\phi_x^2+\phi_z^2)
+g\eta+\frac B\rho\eta_{xxxx}=0
\quad\text{at }z=\eta(x,t)}.
$$
Here the spatially uniform time-dependent Bernoulli term has been absorbed into the <velocity potential>. The <pressure> is measured relative to the constant exterior pressure. This dynamic boundary condition uses the specified massless bending-cover model, with no ice inertia or tension.
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