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I have an exam question, where they ask me to find the mathematical description of a standing soundwave. They want me to find the displacement wave function $y(x,t)$ and the pressure wave function $p(x,t)$ of a soundwave being reflected at an open end and another being reflected at the closed end of a tube. I know I just have to find the superposition of two waves traveling in the opposite direction.

But what happens with the phase shift for $y(x,t)$ and $p(x,t)$?

I can't find any information in my text book. When a string hits a fixed end/reflected at a node, it experiences a phase shift, because of the force exerted by the wall. But what happens with sound?

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I think this: https://en.wikipedia.org/wiki/Reflection_phase_change gives you some of the information you are looking for, i.e. a sound wave propagating through air in a cavity reflects with no phase change at a solid interface, and with a $\pi$-phase change at the open end of the cavity.

What the article doesn't tell you is whether it is talking about pressure or displacement, so here is how I remember it intuitively:

  • The inside of the open end of the cavity has to have the same pressure as outside. Therefore, you get a $\pi$ phase shift for the pressure (so that it cancels) at the open end. However, the displacement has no such limitation.

  • The closed end of the cavity has a fixed position. Therefore, the displacement has to be 0, which means that it experiences a $\pi$ phase shift at the closed end. However, the pressure has no such limitation.

Also, if I remember well, when you solve the wave equations for sound, you'll see that the nodes (i.e. positions with 0 amplitude) for pressure are the positions where you have maximum amplitude for displacement and vice-versa.

Therefore, the conclusion is this:

  • At an open end, pressure is 0 (or more precisely, the difference of pressure with steady state) but displacement is maximum
  • At a closed end, displacement is 0, but pressure is maximum.

Hope it answers all your questions!

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