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For wavenumber selection by a moving observer of deep-water waves, an observer with nonzero velocity $V$ follows $x=Vt$ up to a constant initial displacement. Splitting the cosine into exponentials gives phase functions ${\psi }_{\pm }(k)=kV\pm \sqrt{g|k|}$. Their stationary points satisfyso the selected wavenumbers areThe two signs arise in the conjugate branches needed for a real wave. Equivalently the observer selects wave packets whose group velocity equals $V$. Stationary-phase contributions decay as $t^{-1/2}$, whereas smooth nonstationary contributions decay more rapidly, so these are the dominant wavenumbers when the initial spectrum is nonzero there.

For real even data, one may make this explicit. Set $k_{\ast }=g/(4V^2)$. The stationary curvature has magnitude $2|V{|}^3/g$, yieldingwhen its leading coefficient is nonzero. If that spectral value vanishes, the leading stationary term vanishes too. At $V=0$ there is no finite nonzero stationary wavenumber selected by this condition.