Let $\Phi(x,y,z,t)$ be the velocity potential and $\eta(x,y,t)$ be the free surface elevation induced by an impulsive surface source $\delta_x\delta_y\delta'_t$ with support on the boundary $z=0$ at the space-time point $x=y=t=0$. The linear problem is governed by the following equations, where $c$ is the speed of sound in water and $g$ is the acceleration due to gravity. The computational domain is $ \mathcal{D} =\lbrace(x,y,t): -\infty<x,y,t<\infty\rbrace, -h\leq z\leq 0$.
\begin{eqnarray} \nabla^2\Phi-\dfrac{1}{c^2}\dfrac{\partial^2\Phi}{\partial t^2}&=0&\, -h\leq z< 0,(x,y,t)\in\mathcal{D}\\ g\dfrac{\partial\Phi}{\partial z}+\dfrac{\partial^2\Phi}{\partial t^2}&=0&z=0,(x,y,t)\in\mathcal{D}\setminus (0,0,0)\\ \eta+\dfrac{1}{g}\dfrac{\partial\Phi}{\partial t}&=0&z=0,(x,y,t)\in\mathcal{D}\setminus (0,0,0)\\ \dfrac{\partial\Phi}{\partial z}&=0&z=-h,(x,y,t)\in\mathcal{D}\\ \dfrac{\partial\Phi}{\partial z}-\delta_x\delta_y\delta'_t&=0&z=0,(x,y,t)=(0,0,0) \end{eqnarray}
By direct substitution and the properties of the Bessel function the following integral transform satisfies all these equations for any constant $\mathcal{C}$
\begin{equation} \Phi(r,z,t)=\mathcal{C}\int\limits_{-\infty}^{\infty}\int\limits_{0}^{\infty}e^{i \omega t}\Gamma(z;\mu)J_0(rk)k\omega\,\,d k d\omega,\;-h\leq z\leq 0,\infty<x,y,t<\infty \end{equation}
where $r^2=x^2+y^2$ and the the depth profile function $\Gamma(z;\mu)$ can take the following two equivalent forms
\begin{eqnarray} \Gamma(z;\mu)&=&\dfrac{\cosh(\mu(h+z))}{\mu\sinh(\mu h)}\\ \Gamma(z;\mu)&=&\dfrac{g\cosh(\mu(h+z))}{\omega^2\cosh(\mu h)} \label{eq:Depth_Profile} \end{eqnarray}
where
\begin{equation} \mu^2={k^2-\dfrac{\omega^2}{c^2}} \end{equation}
and \begin{equation} \omega^2=g\mu\tanh(h\mu) \end{equation}