 Energy of a deep-water standing gravity wave

For elevation $h_0\sin (\omega t)\sin (kx)$, depth-integrated kinetic energy per surface area is $\rho h_0^2{\omega }^2{\cos }^2(\omega t)/(4k)$. Wavelength-averaged excess gravitational energy is $\rho g{h}_0^2{\sin }^2(\omega t)/4$. Conservation gives ${\omega }^2=gk$ in the inviscid gravity-only model.