To facilitate the coordinate transformation, denote the new spatial coordinates with primes, i.e., \((x',y',z',\tau)\). Start by writing the partial derivatives in the old \((x,y,z,t)\) coordinates in terms of the new \((x',y',z',\tau)\) coordinates:
\begin{align*}
\frac{\partial}{\partial x} &= \frac{\partial}{\partial x'}\frac{\partial x'}{\partial x} + \frac{\partial}{\partial y'}\frac{\partial y'}{\partial x} + \frac{\partial}{\partial z'}\frac{\partial z'}{\partial x} + \frac{\partial}{\partial \tau}\frac{\partial \tau}{\partial x} = \frac{\partial}{\partial x'}\\
\frac{\partial}{\partial y} &= \frac{\partial}{\partial x'}\frac{\partial x'}{\partial y} + \frac{\partial}{\partial y'}\frac{\partial y'}{\partial y} + \frac{\partial}{\partial z'}\frac{\partial z'}{\partial y} + \frac{\partial}{\partial \tau}\frac{\partial \tau}{\partial y} = \frac{\partial}{\partial y'}\\
\frac{\partial}{\partial z} &= \frac{\partial}{\partial x'}\frac{\partial x'}{\partial z} + \frac{\partial}{\partial y'}\frac{\partial y'}{\partial z} + \frac{\partial}{\partial z'}\frac{\partial z'}{\partial z} + \frac{\partial}{\partial \tau}\frac{\partial \tau}{\partial z} =\frac{\partial}{\partial z'}-\frac{1}{c_0} \frac{\partial}{\partial \tau}\\
\frac{\partial}{\partial t} &= \frac{\partial}{\partial x'}\frac{\partial x'}{\partial t} + \frac{\partial}{\partial y'}\frac{\partial y'}{\partial t} + \frac{\partial}{\partial z'}\frac{\partial z'}{\partial t} + \frac{\partial}{\partial \tau}\frac{\partial \tau}{\partial t} = \frac{\partial}{\partial \tau}
\end{align*}
Using these relations, note that
\begin{align*}
\frac{\partial^2}{\partial x^2} &= \frac{\partial^2}{\partial x'^2}\\
\frac{\partial^2}{\partial y^2} &= \frac{\partial^2}{\partial y'^2}\\
\frac{\partial^2}{\partial z^2} &= \frac{\partial^2}{\partial z'^2} - \frac{2}{c_0}\frac{\partial }{\partial z'}\frac{\partial}{\partial\tau} + \frac{1}{c_0^2}\frac{\partial^2}{\partial \tau^2}\\
\frac{\partial^2}{\partial t^2} &= \frac{\partial^2}{\partial \tau^2}
\end{align*}
Thus equation (\ref{waver}) becomes
\begin{align*}
\bigg(\frac{\partial^2}{\partial x'^2} + \frac{\partial^2}{\partial y'^2} +\frac{\partial^2}{\partial z'^2}-\frac{2}{c_0}\frac{\partial^2}{\partial z'\partial \tau} + \frac{1}{c_0^2}\frac{\partial^2}{\partial \tau^2}\bigg)p - \frac{1}{c_0^2}\frac{\partial^2p}{\partial\tau^2}&=0\,.
\end{align*}
Canceling common terms and rearranging gives
\begin{equation*}
\nabla^2 p = \frac{2}{c_0}\frac{\partial^2 p}{\partial z'\partial\tau}\,.
\end{equation*}
Finally, recalling that \(z'\) is but a placeholder, and that in fact \(z' = z\), gives the desired equation (\ref{wave}),
\begin{equation*}
\frac{\partial^2 p}{\partial \tau \partial z} = \frac{c_0}{2}\nabla^2 p\,.
\end{equation*}