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This is a basic question but I don't fully understand what is going on.

When we have a pipe in resonance we get standing waves, where displacement antinodes are at pressure nodes and vise versa. So we have a situation where pressure and displacement are 90 degrees out of phase, this means you can differentiate the displacement wave to recover the pressure wave.

Does anyone have a nice explanation of this 90 degree phase difference?

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  • $\begingroup$ What are precisely pressure nodes and displacement nodes? $\endgroup$
    – Lelesquiz
    May 30 '14 at 15:29
  • $\begingroup$ In the pipe, the pressure nodes are the points where there is no pressure variation, that is we have destructive interference. The pressure antinode is where they constructively interfere. A displacement node is where there is no net displacement of the particles. For example at a closed end of a pipe we have displacement node since particles can't move back and forth through a wall, but it is a pressure antinode. $\endgroup$
    – User
    May 30 '14 at 15:32
  • $\begingroup$ You may find your answers here: phys.unsw.edu.au/jw/flutes.v.clarinets.html $\endgroup$
    – Davidmh
    May 30 '14 at 15:35
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See (http://www.berndporr.me.uk/teaching/wave_eq00.pdf).

Formula (12) in particular. It says $$ p = - \gamma_c P_0 \frac{\partial{\xi}}{\partial x} $$.

From this formula you see that pressure and displacement are related by a derivative: if $\xi \propto sin(x)$ then $p \propto cos(x)$. Here's the 90 degrees phase difference.

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