My Textbook says that:
Reflection of sound waves for displacement from a rigid boundary (e.g. closed end of an organ pipe) is
analogous to reflection of a string wave from rigid boundary; reflection accompanied by an inversion i.e.
an abrupt phase change of
$\pi$. This is consistent with the requirement of displacement amplitude to
remain zero at the rigid end, since a medium particle at the rigid end can not vibrate. As the excess pressure and displacement corresponding to the same sound wave vary by $\pi/2$ in term of phase, a displacement minima at the rigid end will be a point of pressure maxima. This implies that the reflected pressure wave from the rigid boundary will have same phase as the incident wave, i.e., a compression pulse is reflected as a compression pulse and a rarefaction pulse is reflected as a rarefaction pulse.
I did not understand the last statement. If the phase change after reflection from a rigid boundary is $\pi$, then shouldn't a rarefaction pulse be reflected as a compression pulse and a compression pulse as a rarefaction pulse, afterall the phase difference between a particle at rarefaction and compression is $\pi$. Where am I wrong? I think I have trouble understand the $\pi$ phase change in the case of sound wave. The same thing was easy to understand in the case of wave on a string. Please clear this confusion!
Best Answer
The answer you seek is actually in the passage.
There are two ways of describing a sound wave: as a variation in pressure wave and as a mean displacement of particles wave as illustrated below.
. . . the excess pressure and displacement corresponding to the same sound wave vary by π/2 in term of phase . . .
. . . a displacement minima at the rigid end will be a point of pressure maxima. . .
So at a rigid boundary there is zero mean displacement whereas the variation in pressure is a maximum.
Which leads on to the statement,
This implies that the reflected pressure wave from the rigid boundary will have same phase as the incident wave, i.e., a compression pulse is reflected as a compression pulse and a rarefaction pulse is reflected as a rarefaction pulse.
If the phase change after reflection from a rigid boundary is π, . . .
This is for the displacement wave not the pressure wave.
Whereas for the pressure wave which is $\pi/2$ out of phase with the displacement wave there is no phase change at a rigid boundary.