= Solution
Fourier modes of the harmonic potential decay as $e^{|k|y}$; the two surface conditions give $\eta_{tt}+|k|^3\eta=0$, hence $\omega^2=|k|^3$. The initial transform is $F(k)=-i\sqrt\pi,k e^{-k^2/4}/2$. At $x=Vt$, the contributing stationary wavenumber is $k_0=4V^2/9$, giving
$$
\eta(Vt,t)\sim\frac{4V^{5/2}}{27\sqrt t}e^{-4V^4/81}\sin\left(\frac{4V^3t}{27}-\frac\pi4\right).
$$
Different observer speeds select different group-velocity wavenumbers; fast observers see exponentially weaker, higher-frequency ripples.
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