Energy of a deep-water standing gravity wave
= 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 $\rho gh_0^2\sin^2(\omega t)/4$. Conservation gives $\omega^2=gk$ in the inviscid gravity-only model.