Suppose the fluid occupies the semi-infinite channel $0\leq y\leq $ an $ x\geq X(y, t)$. The upper and lower walls are rigid and stationary. The left-hand boundary executes small oscillations so that$$ X(y, t) = ε sin(ωt) cos(\frac{πy}{H}) $$with ε $≪$ 1.
Derive the linearised boundary conditions
$$\frac{∂φ}{∂y} = 0 $$ on y = 0 $$\frac{∂φ}{∂y} = 0 $$ on y = H $${\frac{∂φ}{∂x} = εωcos(ωt)cos(\frac{πy}{H})}$$ on x = 0
Find the φ that describes small oscillations of the fluid. Show that there are two qualitatively different regimes for $ 0 < ω < ω_c$ and$ ω > ω_c$, and determine the critical frequency $ω_c$. What condition must φ satisfy as x → +∞?
I have derived the boundary condition but how to find the φ and the critical frequency?
Let $${φ= f(x)cos(ωt)cos(\frac{πy}{H})}$$
and plug back, $f'(0)=εω$ and the Laplace's equation, $f''(0)=(\frac{\pi}{H})^2f(0)$ and what about the critical frequency.