 Solution

Seek nonzero time-harmonic waves of the form $\varphi =\mathrm{Re}[F(y)e^{i(kx-\omega t)}]$ and $\eta =\mathrm{Re}[A{e}^{i(kx-\omega t)}]$, taking $k>0$. Laplace's equation reduces to $F^{\prime \prime }-k^{2}F=0$. The impermeable-bed condition $F^{\prime }(-h)=0$ gives $F(y)=C\cosh (k(y+h))$.

The linearized kinematic and dynamic linearized free-surface boundary conditions at zero becomeEliminating $A$ and the nonzero amplitude $C$ givesConversely, if this dispersion relation holds, either amplitude can be chosen nonzero and the other set by the second boundary equation, producing a wave satisfying every condition. The expression also holds for negative $k$ because $k\tanh (kh)$ is even.