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For a localized terrain source in uniform flow, the stationary phase method applied to $kx+m(k)z$ gives phase $\Phi =(N/U)\sqrt{x^2+z^2}$ on the downstream branch. Permanent phase lines are circular arcs, whereas an individual monochromatic component has straight phase lines. Reflected arcs use an image source; transmitted phase is continuous across the velocity interface.

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.