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I know that for transverse waves, the particles themselves have a different speed to the wave itself but is this also true for longitudinal waves? It seems intuitively that since the displacement is parallel to the direction of the wave, particles that are displaced positively (or in same direction as the direction of energy transfer) should have same speed right?

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  • $\begingroup$ If you’re talking about light, all the particles (photons) Move at the same speed of light. $\endgroup$ Jun 2 '20 at 14:31
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The answer is no. I'll give you two qualitative reasons.
First of all, take for example a monochromatic wave, it has fixed speed, while displaced particles are accelerated back and forth.
But even mean speed of particles and speed of the wave are unrelated things because of this second point:
consider a series of particles of mass $m$ (separated at a distance $a$ while at rest) connected by springs (with characteristic coefficient $k$, at rest while their extension equals $a$).
The speed of the particles depends only on two of those three parameters ($m$ and $k$), while wave speed also depends on the distance $a$ between particles:
so I can change the last one keeping the others fixed, changing wave speed while keeping the dynamics of single particles unchanged

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In longitudinal waves (in matter), the particles bounce back and forth. Their motion is often modeled as being sinusoidal with a varying speed.

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