= Solution
Use the <Fourier transform> convention
$$
\widehat\eta(k,t)=\int_{-\infty}^{\infty}\eta(x,t)e^{-ikx}\,dx,
\qquad
\eta(x,t)=\frac1{2\pi}\int_{-\infty}^{\infty}\widehat\eta(k,t)e^{ikx}\,dk.
$$
The <deep-water capillary-wave dispersion relation> makes each Fourier mode satisfy
$$
\partial_t^2\widehat\eta+S^2|k|^3\widehat\eta=0.
$$
The transformed initial data are
$$
\widehat\eta(k,0)=0,
\qquad
\partial_t\widehat\eta(k,0)
=-W\int_{-\epsilon}^{\epsilon}e^{-ikx}\,dx
=-\frac{2W\sin(k\epsilon)}k.
$$
Therefore
$$
\widehat\eta(k,t)
=-\frac{2W\sin(k\epsilon)}{kS|k|^{3/2}}
\sin(S|k|^{3/2}t).
$$
The integrand is even, so Fourier inversion gives the <surface response to a localized impulsive velocity>
$$
\boxed{
\eta(x,t)=-\frac{2W}{\pi S}
\int_0^\infty
\frac{\sin(k\epsilon)\sin(Sk^{3/2}t)\cos(kx)}{k^{5/2}}\,dk}.
$$
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