Wavenumber selection by a moving observer of deep-water waves ($k_{\ast }=\pm g/(4V^2)$)

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.