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Why is it that when a transverse wave is reflected from a 'rigid' surface, it undergoes a phase change of $\pi$ radians, whereas when a longitudinal wave is reflected from a rigid surface, it does not show any change of phase? For example, if a wave pulse in the form of a crest is sent down a stretched string whose other end is attached to a wall, it gets reflected as a trough. But if a wave pulse is sent down an air column closed at one end, a compression returns as a compression and a rarefaction returns as a rarefaction.

Update: I have an explanation (provided by Pygmalion) for what happens at the molecular level during reflection of a sound wave from a rigid boundary. The particles at the boundary are unable to vibrate. Thus a reflected wave is generated which interferes with the oncoming wave to produce zero displacement at the rigid boundary. I think this is true for transverse waves as well. Thus in both cases, there is a phase change of $\pi$ in the displacement of the particle reflected at the boundary. But I still don’t understand why there is no change of phase in the pressure variation. Can anyone explain this properly?

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For a standing wave produced by reflection of a longitudinal wave from a rigid surface there is always a displacement node and a pressure antinode at the rigid surface ie pressure variation is maximum but displacement variation is minimum. Now I'm really confused. You're right, if the reflected wave is in phase with the original wave then there should be a crest of the standing wave at the rigid surface. But does that mean displacement or pressure variation is maximum? – Amu Apr 16 '12 at 14:34
This might help you. – Vijay Murthy Apr 16 '12 at 14:55
Thanks, Vijay Murthy! That link helped clarify things. – Amu Apr 16 '12 at 16:16
Amu, would you like me to merge your other account into this one? – David Z Apr 16 '12 at 17:46
@DavidZaslavsky Yes please! Thank you :) – Amu Apr 17 '12 at 7:14

Great question!

You might have learned that the amplitude of compression and the amplitude of particle displacements are not synonymous. In fact, the maximum amplitude of pressure and the maximum amplitude of particle displacements are out of phase for $\pi/2$. And twice $\pi/2$ (one for original, and one for the reflected wave) accounts for the missing $\pi$ in the phase change of particle displacement.

Imagine, that rarefaction travels towards the wall, which is on the right side. On the moment the wave strikes the wall, maximum displacement is left of rarefaction, that is $\pi/2$ behind it. The same is true for the reflected wave, that is, maximum displacement is again left of the rarefaction, only the direction of the wave is opposite, so maximum displacement amplitude is $\pi/2$ in front of rarefaction.

Thus, the phase of particle displacement changes phase for $\pi$, while the phase of pressure does not change at all at rigid surface.

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I'm sorry, I didn't really understand what you're saying. Isn't maximum displacement in a sound wave at the rarefaction? Rarefaction is the point of minimum pressure and maximum displacement, and compression is the point of maximum pressure and minimum displacement, right? Correct me if I'm wrong. – Amu Apr 16 '12 at 16:10
No, wrong. Maximum displacement is $\pi/2$ away of rarefaction. See – Pygmalion Apr 16 '12 at 16:24
Watch this: or You will see, that when particle is in its equilibrium position, you have either rarefaction or compression (maximum pressure amplitude). Not really intuitive, but real. – Pygmalion Apr 16 '12 at 16:29
The animation certainly suggests that when a particle is in its equilibrium position, you have either rarefaction or compression, which I now realize makes perfect sense as compression and rarefaction are both extremes of pressure, and they're accompanied by zero displacement. Thanks for correcting me! :) – Amu Apr 16 '12 at 17:00
Thanks for up-voting. By the way, the logic is the same for the compression. Imagine that you produce compression or rarefaction by pulling or pushing a membrane. In both cases will the maximum displacement trail maximum pressure change for $\pi/2$ going right and lead maximum pressure change for $\pi/2$ returning left. However, I cannot find any simulation comparable to those shown above... – Pygmalion Apr 16 '12 at 17:09

Here is another possible way of explanation:

Reflection of the wave is similar process as crushing two waves, one from the left and one from the right, which meet exactly at the surface. Now, if you wish that particle at the surface has zero displacement, then the wave on the right must be point-symmetrical through that particle to the wave on the left.

If left wave pulls particle at the surface up, the right wave must pull it down. If left wave pulls particle at the surface down, the right wave must pull it up.

Obviously, trough and crest match.

If left wave pulls particle at the surface left, the right wave must pull it right.

However, pulling left from the left and right from the right both corresponds to rarefaction.

If left wave pushes particle at the surface right, the right wave must push it left.

However, pushing right from the left and pushing left from the right both corresponds to compression.

I really love this problem (as mentioned above) but this explanation is furthest my mind is able to go...

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