Traveling Sound Waves — Question 8

Question 8

A rigid end doubles pressure while stopping normal motion

A rightward incident pressure wave is $p_i=P\cos(kx-\omega t)$ in $x<0$. A rigid wall at $x=0$ imposes total normal particle velocity $u(0,t)=0$. The medium has impedance $Z$.

1. Find the reflected wave and the combined pressure and velocity fields.

2. Locate the nearest pressure node and velocity antinode to the wall. Is $p=Zu$ valid for the combined field?

Question 8 - Solution

A rigid end doubles pressure while stopping normal motion

(a) Method and calculation

The reflected pressure must be $p_r=P\cos(kx+\omega t)$ because its velocity has the opposite impedance sign. Adding gives$$p=2P\cos kx\cos\omega t,$$$$u=\frac PZ[\cos(kx-\omega t)-\cos(kx+\omega t)]=\frac{2P}{Z}\sin kx\sin\omega t.$$At $x=0$, pressure amplitude is $2P$ and velocity is zero.

(b) Interpretation and further calculation

The nearest pressure node is at $x=-\pi/(2k)=-\lambda/4$, also a velocity antinode. The relation $p=Zu$ describes one rightward progressive component, not this standing superposition. Indeed, the wall has nonzero pressure even though its normal velocity vanishes.

Physical check

At the wall the instantaneous power per area $pu$ is zero, consistent with a stationary rigid boundary doing no work.