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Background:

https://en.wikipedia.org/wiki/Longitudinal_wave

'Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave. ... The other main type of wave is the transverse wave, in which the displacements of the medium are at right angles to the direction of propagation.'

My question was essentially asked before. See:

Distinguishing transverse and longitudinal from wave equation

but was never answered. Maybe the answer is that it can not be done--so then the answer to my question would be yes.

My question is: Can the same 1D wave equation be used for transverse waves on strings and longitudinal waves from infinite planar sources?

A Google search for 'longitudinal vs transverse wave equation' did not produce an answer that I could find.

I would appreciate any additional info along with the yes or no.

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  • $\begingroup$ Probably wasn't answered b/c it's unclear what to do. I suggest you write down the wave equations for each case and see how they are related. $\endgroup$ – JEB Jul 28 '18 at 19:24
  • $\begingroup$ If I was forced to answer I would say they are the same and would write down the same equation for each--but that seems fundamentally strange to me. $\endgroup$ – user45664 Jul 28 '18 at 19:29
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Both equations and solutions for transverse and longitudinal waves are the same. The only difference is the direction of the displacement.

There is nothing strange about it, because, at high level, both types of waves behave identically. Let's for simplicity, consider a $1$D wave, which can be described by the standard wave formula:

$A(x,t)=A_{max} cos(kx-\omega t)$.

Looking at this formula, we can say - without specifying the direction of the displacement $A$ - that it describes a sine wave with amplitude $A_{max}$, wave number $k$ and angular frequency $\omega$, moving at a constant velocity $v=\frac \omega k$ in the $x$ direction and that, at any point $x$, the displacement changes as a sine wave function.

This description perfectly fits both transverse and longitudinal waves. So we need to differentiate between them only, when we get to a lower level and look at specific physical mechanisms associated with each wave type.

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