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I'm currently learning about separation of variables as applied to situations where the boundary conditions are not homogeneous. I'm having trouble deciding how to handle one of the boundary conditions, namely if I should split up the solution into subproblems.

Consider a circular sector filled with water, governed by Laplace's equation for the velocity potential $\phi(r,\theta,z,t)$:

$$\frac{∂^2\phi}{∂z^2}+\frac{1}{r^2}\frac{∂^2\phi}{∂θ^2}+\frac{1}{r}\frac{∂}{∂r}(r\frac{∂\phi}{∂r}) = 0$$

The circular sector of radius $R$ has surface of water at $z=0$, a flat floor at $z = -H$, and flat walls at $\theta = 0$ and $\theta = \frac{2\pi}{3}$.

The boundary conditions are: $$\frac{∂\phi}{∂z}(r,\theta,z=-H,t) = 0$$ $$\frac{∂\phi}{∂r}(r = R,\theta, z, t) = 0$$ $$\frac{∂\phi}{∂\theta} (r,\theta=0, z, t) = \frac{∂\phi}{∂\theta} (r,\theta=\frac{2\pi}{3}, z, t)=0$$ $$\frac{∂^2\phi}{∂t^2} (r,\theta, z=0, t) + g\frac{∂\phi}{∂z} (r,\theta, z=0, t) =0 $$

The last boundary condition (for the surface of the water) includes a time derivative and g, gravitational acceleration. I've never encountered such a boundary condition before during study of separation of variables and eigenfunction expansion, so I'm wondering how to interpret this boundary condition and how it affects a eigenfunction expansion problem for $\phi$. Any insight is thoroughly appreciated!

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Let $\phi(r,\theta,z,t) = P(r)\Phi(\theta)Z(z)T(t)$. The equation has no derivatives in $t$ so you'll only have 3 equations after separation of variables:

\begin{cases} \Phi'' + \mu^2 \Phi = 0 \\ Z'' - \lambda^2 Z = 0 \\ r^2P'' + rP' + (\lambda^2r^2 - \mu^2)Z = 0 \end{cases}

Solving these with the first 3 boundary conditions gives you this solution

$$ \phi(r,\theta,z,t) = T_{0,0}(t) + \sum_{n=0,m=1}^\infty T_{n,m}(t)\cosh\big(\lambda_m(H+z)\big)\cos\left(\frac{3n}{2}\theta\right)J_{3n/2}\big(\lambda_m r\big) $$

such that $\lambda_mR$ are the zeroes of $J_{3n/2}'(x)$.

The last boundary condition gives

\begin{align} T_{0,0}''(t) &= 0 \\ T_{n,m}''(t) + g\lambda_m\tanh(\lambda_mH) T_{n,m}(t) &= 0 \end{align}

These determine the general form of $T(t)$. Two initial conditions at $t=0$ are needed to finish the problem.

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