= Solution
Use the <Fourier transform> convention $\widehat W(k,z,t)=\int_{\mathbb R}W(x,z,t)e^{-ikx}dx$, with inverse factor $1/(2\pi)$. The initial profile occupies only the vertical mode $\sin(\pi z/h)$. Its transform is
$$
\widehat W(k,z,0)=\pi hC e^{-\epsilon h|k|}\sin(\pi z/h),\qquad \widehat W_t(k,z,0)=0.
$$
The transformed evolution is a <harmonic oscillator> for each $k$, with
$$
\omega(k)=\frac{\sqrt{\alpha g}\,k}{\sqrt{k^2+\pi^2/h^2+\alpha^2/4}}.
$$
Therefore inversion gives
$$
\boxed{W(x,z,t)=\frac{hC}{2}\sin\!\left(\frac{\pi z}{h}\right)\int_{-\infty}^{\infty}e^{-\epsilon h|k|}\cos(\omega(k)t)e^{ikx}\,dk}.
$$
Using the nonnegative even frequency instead gives the same cosine. At $t=0$, $\int_{\mathbb R}e^{-a|k|}e^{ikx}dk=2a/(a^2+x^2)$ with $a=\epsilon h$, recovering the specified initial profile and its zero time derivative.
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