 Solution

In the inviscid incompressible irrotational model, write the total velocity as $(U+{\varphi }_x,{\varphi }_y)$. The perturbation potential obeys Laplace's equation in $-h<y<\eta (x,t)$. The complete boundary conditions, neglecting surface tension and taking the air pressure constant, areThe first surface condition expresses that the free surface is material; the second is the Bernoulli equation with atmospheric pressure and the constant background kinetic energy absorbed into the potential's time gauge.

At first order, evaluation is on $y=0$ and products of perturbations vanish. HenceFor a mode $e^{i(\omega t-kx)}$, the bottom condition selects $\varphi =C\cosh [k(y+h)]e^{i(\omega t-kx)}$. Let $\Omega =\omega -kU$. The linear surface conditions areEliminating $C$ gives the dispersion relationIt is the intrinsic gravity-wave frequency squared, with the uniform current supplying the Doppler shift. For either sign of real $k$, $k\tanh (kh)$ is nonnegative.