# Reflection of sound wave

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!

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.

Considering a wave equation: $$y=A\cos(kx-\omega t+\phi_0) \\ \phi=(kx-\omega t+\phi_0)$$ An abrupt change of $$\pi$$ in the phase $$\phi$$ is like putting $$A\cos(\phi+\pi)$$ instead of $$Acos(\phi)$$ which, comes out to be $$-Acos(\phi)$$

Now, when you look at the graph of one of these sinusoidal functions, you see that the maximas become minimas and those with the value zero stay zero