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I came up with a doubt about standing waves and path difference in general. Consider these two different cases as examples.

  • If I have a rope fixed at one end and I make the free end oscillating, I can get the formation of a standing wave

$$\xi(x,t)=\xi_{incident}+\xi_{reflected}=A[\mathrm{sin}(kx+\omega t)+\mathrm{sin}(kx-\omega t)]=2A\mathrm{sin}(kx )\mathrm{cos}(\omega t)\tag{1}$$

  • If I have two coherent sources in front of a wall, depending on the path difference of the two waves I get a path difference interference, and there would be different points along the wall where the intensity has a maximum, independent of time. That would happen if

$$k \Delta r=2n \pi$$ Where $\Delta r$ is the path difference.

The difference between the two situations is that in the first case I get points where there is an oscillation in time, while in the second one, as stated, the points "do not oscillate": where there is a minimum, that minimum stays there in time and the same happens for maxima.


Now if I have two speakers one in front of the other, emitting with the same frequency and with no difference in phase, what do I get? A standing wave or a path difference interference?

The situations cannot be the same in the two cases, for the reason stated above, and both of them are possible in this case. But I do not see the conditions for which I do get one thing instead of the other.

On the one hand there is the interference of two identical waves travelling in opposite directions (exactly like in $(1)$), on the other hand I saw problems where it is explicitly said that in this case there is a path interference.

How can I distinguish between the two phenomena?

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closed as unclear what you're asking by ACuriousMind, user36790, knzhou, John Rennie, honeste_vivere Jun 26 '16 at 17:38

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    $\begingroup$ You say intensity has a maximum, independently from time but that isn't true. At the maxima the intensity oscillates from zero to its maximum value at the frequency of the light wave i.e. about 500THz. You can see the rope oscillating because its frequency is nearer 1Hz, but that's the only difference. $\endgroup$ – John Rennie Jun 25 '16 at 6:16
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    $\begingroup$ Worth a look? physics.stackexchange.com/q/69587 $\endgroup$ – Farcher Jun 25 '16 at 7:14
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In the first case I get points where there is an oscillation in time, while in the second one, as stated, the points "do not oscillate": where there is a minimum, that minimum stays there in time and the same happens for maxima.

This is not true. There is identical time dependence, of frequency $\omega$, in both situations; as I said in my answer to your other question, the two approaches are identical.

You're claiming there's no time dependence of the resulting wave in path length difference interference, but that's just plainly untrue. Consider double slit interference with red light: we see a red pattern on the screen. That pattern can only be red because the light on the screen is oscillating with the same frequency we put in.

It's true that the locations of the intensity maxima, when you average the power over a full oscillation, do not change. But that's true for standing wave interference too.

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