Im trying to solve the following problem
\begin{aligned} \psi_xx+\psi_zz &= 0, -h(x,t) < z < \eta(x,t) \\ \psi_z + h_x \psi_x + h_t &= 0, z = -h(x,t) \\ \psi_z - \eta_x \psi_x - \eta_t &= 0, z = \eta(x,t) \\ \psi_t + \frac{1}{2} \nabla \cdot \nabla \psi + gz &= 0, z = \eta(x,t) \\ \psi_x (-L,z,t) &= 0 \\ \psi_x (L,z,t) &= 0 \\ \end{aligned}
First I have tried to linearize the problem, assuming that $ \psi $ and $ \eta $ is small and that h(x,t) is given by $ h(x,t) = h_0 + \xi (x,t) $. I have then
\begin{aligned} \nabla^2\psi &= 0 \\ \psi_z - \eta_t &= 0, z = 0 \\ \psi_z + \xi_t &= 0, z = -h (x,t) \\ \psi_t + g \eta &= 0 , z = 0 \end{aligned}
Then I want to try and solve the problem using eigenfunction expansion \begin{aligned} \psi(x,z,t) &= \sum_{n=0}^{\infty} \psi_n (z,t) X_n(x) \\ \xi (x,t) &= \sum_{n=0}^{\infty} \xi_n (t) X_n(x) \\ \end{aligned}
The $ X_n (x) $ I have found to be
\begin{aligned} X_n(x) &= \cos(k_n(x + L)),n = 0,1,2... \\ k_n &= \frac{n \pi}{2L} \\ \end{aligned}
Now im not sure how to proceed. I think I must do an eigenfunction expansion of $ \psi_n $ but im not sure exactly how to start. The extra dimension confuses me.
Thanks for any help that can be given. The problem comes from water waves.