= Solution
{c}
Seek nonzero <time-harmonic waves> of the form $\phi=\operatorname{Re}[F(y)e^{i(kx-\omega t)}]$ and $\eta=\operatorname{Re}[A e^{i(kx-\omega t)}]$, taking $k>0$. <Laplace's equation> reduces to $F''-k^2F=0$. The impermeable-bed condition $F'(-h)=0$ gives $F(y)=C\cosh(k(y+h))$.
The linearized kinematic and dynamic <linearized free-surface boundary conditions> at zero become
$$
Ck\sinh(kh)=-i\omega A,\qquad
-i\omega C\cosh(kh)+gA=0.
$$
Eliminating $A$ and the nonzero amplitude $C$ gives
$$
\boxed{\omega^2=gk\tanh(kh).}
$$
Conversely, 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.
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