Traveling Sound Waves — Question 3

Question 3

A phase-resolved measurement can determine propagation direction

At a point in a uniform fluid, measurements give $p(t)=0.60\cos\omega t\,\mathrm{Pa}$ and $u(t)=-1.50\times10^{-3}\cos\omega t\,\mathrm{m/s}$, where positive velocity is rightward. The fluid impedance is $Z=400\,\mathrm{Pa\,s/m}$. Assume only two counterpropagating plane waves at this frequency.

1. Resolve the measured pressure into rightward and leftward components.

2. If the velocity sign alone is reversed, repeat the inference. Why would a pressure-only measurement not resolve the direction?

Question 3 - Solution

A phase-resolved measurement can determine propagation direction

(a) Method and calculation

Let $p_+$ and $p_-$ denote the signed rightward and leftward pressure signals at the point. Then$$p=p_++p_-,\qquad Zu=p_+-p_-,$$$$p_+=\tfrac12(p+Zu)=0,\qquad p_-=\tfrac12(p-Zu)=0.60\cos\omega t.$$The measured field is purely leftward under the two-wave assumption.

(b) Interpretation and further calculation

Reversing the measured velocity produces $Zu=+0.60\cos\omega t$, so $p_+=0.60\cos\omega t$ and $p_-=0$. Both cases have identical pressure histories. Direction is encoded in the pressure-velocity phase relation, not in pressure amplitude alone.

Physical check

For a single progressive wave $|p/u|=Z$. Zero net velocity with nonzero pressure would instead require both components.