Deep-water capillary-wave dispersion relation (${\omega }^2=(T/\rho )|k{|}^3$)

For a capillary wave on deep water with surface tension $T$ and density $\rho$, the dispersion relation isIts group velocity is $(3S/2)|k{|}^{1/2}$ in the direction of the wavevector.

If an initially flat one-dimensional surface receives velocity $-W$ on $|x|<ϵ$ and zero velocity elsewhere, its linear capillary-wave response is

Along $x=Vt>0$, if $ϵV^2/S^2\ll 1$, the stationary wavenumber $k_0=4V^2/(9S^2)$ obeys $k_0ϵ\ll 1$. The one-dimensional stationary-phase formula then givesOnly the integrated initial velocity $-2Wϵ$ survives in this long-wave far field.