# Distinguishing transverse and longitudinal from wave equation

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

• What do you mean by "in general"? Do you want the identification to be based entirely on the nature of the wave function? Commented Mar 6, 2016 at 6:15
• @SchrodingersCat Yes. Maybe I myself am not very sure at this point, but is there a way to extract information about the mode of propagation by looking at the wave equation? If not, how else could we distinguish them? Commented Mar 6, 2016 at 7:05
• I think that you have to consider a wave equation in dimension>1 since the notions of longitudinal and transverse usually refer to vectors. Commented Mar 6, 2016 at 10:43
• @Urgje yes, the full equation is actually three dimensional. If there is three dimensional, how would I distinguish it then? Commented Mar 6, 2016 at 20:41
• No, because the wave equation applies to both longitudinal and transverse waves. Physical context should tell you what $f$ is supposed to be. Commented Mar 7, 2016 at 1:47

Transverse waves obey $$\nabla\times f≠0$$ and $$\nabla f=0$$.
Longitudinal waves obey $$\nabla\times f=0$$ and $$\nabla f≠0$$.
• This implies that $f$ is a vector. That might not be the case. Commented Apr 8, 2020 at 15:34