= Wavenumber selection by a moving observer of deep-water waves
{title2=$k_*=\pm g/(4V^2)$}
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.
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