Can we distinguish if a general wave or disturbance $f$ (not necessarily electromagnetic) which satisfies the wave equation
\begin{equation}
\frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}
= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \equiv \nabla^2 f
\end{equation}
is transverse or longitudinal?