Deep-water gravity wave

A small-amplitude deep-water wave with surface elevation $\eta =A{e}^{i(kx-\omega t)}$ has dispersion relation ${\omega }^2=gk$ and a velocity potential that decays as $e^{kz}$ below the surface.

A deep-water gravity wave has dispersion $\omega (k)=\sqrt{g|k|}$. An observer moving with nonzero speed $V$ samples Fourier phases $kV\pm \omega (k)$. The stationary phase method selects the points where their derivatives vanish, which gives the displayed wavenumbers. Their group velocity matches the observer's velocity. For real even initial displacement with a nonzero Fourier coefficient there, the resulting signal has leading size $t^{-1/2}$ and phase $gt/(4|V|)-\pi /4$. If the initial coefficient vanishes, that leading term is absent; a stationary observer does not select a finite nonzero wavenumber by this rule.

For two wave periods emitted together at one location, deep-water group velocity is $gT/(4\pi )$. A distant observer sees the shorter period $T_2<T_1$ later, with $\Delta t=(4\pi D/g)(1/T_2-1/T_1)$. This yields the displayed range estimate. Different emission times, a moving storm, finite-depth dispersion, currents and refraction are possible confounders; the propagation calculation alone does not establish a unique source range without these assumptions.

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.

For an impulsive distant source of deep-water gravity waves, a component of period $T$ travels at group velocity $gT/(4\pi )$. Its arrival at distance $L$ obeys the displayed law, so inverse period increases linearly with arrival time. Two arrivals determine both $L$ and $t_0$. Finite source duration, currents, refraction and passage through sea ice can invalidate the simple inversion.

For small kinematic viscosity $\nu$, the mean viscous dissipation per unit horizontal area is $2\rho \nu g{k}^2|A{|}^2$. Since the mean wave energy is $\rho g|A{|}^2/2$, the amplitude obeys