= Solution
The fixed upper and lower walls have zero normal <velocity>. Since $v=\phi_y$, their <no-penetration boundary conditions> are
$$
\boxed{\phi_y=0\quad\text{at }y=0,H.}
$$
For the moving end wall, use its level-set equation $F(x,y,t)=x-X(y,t)=0$. The <kinematic boundary condition> $DF/Dt=0$ gives the exact impermeability relation
$$
u=X_t+vX_y\quad\text{at }x=X(y,t).
$$
Both $v$ and $X_y$ are first order, so their product is second order. A <Taylor expansion> from $x=X$ to $x=0$ changes the first-order perturbation <velocity> only at second order as well. Thus
$$
\boxed{\phi_x(0,y,t)=X_t(y,t)=\epsilon\omega\cos(\omega t)\cos\frac{\pi y}H.}
$$
At infinity, select a bounded response with \b[no waves incident from $x=+\infty$]. A propagating <acoustic waveguide mode> must satisfy the outgoing <radiation condition for a waveguide mode>, while a nonpropagating mode must decay. For a time dependence $e^{-i\omega t}$, a propagating axial factor has the form $e^{iqx}$, $q>0$, and obeys $\Phi_x-iq\Phi\to0$. An evanescent factor is $e^{-\alpha x}$, $\alpha>0$.
Requiring $\phi\to0$ for all frequencies would incorrectly exclude the radiated traveling waves. In this lossless channel they retain finite <amplitude> arbitrarily far downstream.
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