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In acoustics the pressure wave has a $\pi/2$ phase difference with the displacement wave. But I do not understand how this leads to a different position of nodes of pressure inside a tube with respects to the nodes of displacement.

In interference, what is important is the phase difference so if I have two waves interfering creating a standing wave, if both of the pressure waves have a $\pi/2$ phase difference with respect to the corrisponding displacemente waves, then the phase difference between the two pressure waves would be the same of the one between the two displacement waves.

So apparently there is no difference in the interference of pressure or displacemente waves, hence I do not see the reason of the different location of nodes, for istances in the followin situation

Consider two speakers at the same frequency one in front of the other that interfere creating a standing wave: the nodes of pressure would be located exactly where the displacement nodes are, because the phase difference is zero at the midpoint between the speakers both for displacement and pressure waves, then at space intervals of $\lambda/4$ from midpoint the nodes of pressure (and displacement too) would be located.

So why the nodes of a pressure standing wave should have a different location with respect to the ones of the corresponding displacement standing wave?


To clarify the question : in the picture there is a possible situation described with the two speakers, which emits sound in phase at the same frequency. I'm sure that the displacement waves will have an antinode at the midpoint, but what about the pressure wave. Is $A$ or $B$ the correct diagram of the corresponding pressure wave? enter image description here

On the one side there is a pase difference of $\pi/2$ with the displacement that makes me think about $B$ (node in the center instead of an antinode).

On the other side if both of the waves coming from the two different speakers have the same phase difference with the corresponding pressure wave, then at the midpoint there should still be an antinode (as for displacement) because the phase difference of pressure waves calculated at midpoint should be $$\Phi_1-\Phi_2=k(\frac{L}{2})-\frac{\pi}{2}-(k(\frac{L}{2})-\frac{\pi}{2})=0$$ Which means constructive interference at the midpoint ($L$ is the distance between the speakers and $k=\frac{2\pi}{\lambda}$).

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    $\begingroup$ I'm not sure I understand the question. In the context of, say, a Ruben's tube it's the average pressure (and therefore the net gas flow) that varies differently than the amplitude of the displacement field. So you're not talking about a different macroscopic result rather than a different microscopic wave. $\endgroup$ – dmckee Jul 3 '16 at 21:44
  • $\begingroup$ @dmckee Thanks for the answer! I added some explanations in the question, in particular is $A$ or $B$ correct in the diagram? $\endgroup$ – Sørën Jul 4 '16 at 15:58
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    $\begingroup$ Speakers work by displacing the air next to them. Therefore your displacement diagram is wrong. There is an antinode at both speaker positions. $\endgroup$ – Robin Jul 7 '16 at 14:49
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There is a difference between pressure and displacement: pressure is "absolute", displacement has a direction. With speakers on the left and right, pointing to each other, the pressure wave from the right will be the mirror image of the one from the left, the displacement wave is not only mirrored but also changes sign (mirrored by the x-axis). Taking that phase change into account, it should make more sense.

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