Stationary-phase wake of an impulsive capillary wave
= Stationary-phase wake of an impulsive capillary wave
Along $x=Vt>0$, if $\epsilon V^2/S^2\ll1$, the stationary wavenumber $k_0=4V^2/(9S^2)$ obeys $k_0\epsilon\ll1$. The <one-dimensional stationary-phase formula> then gives
$$
\eta(x,t)\sim
\frac{W\epsilon S t^2}{x^{5/2}}
\frac9{2\sqrt\pi}
\sin\left(\frac{4x^3}{27S^2t^2}-\frac\pi4\right).
$$
Only the integrated initial velocity $-2W\epsilon$ survives in this long-wave far field.