Traveling Sound Waves — Question 2

Question 2

Reversing propagation reverses the pressure-velocity sign

A leftward pressure wave is $p=P\cos(kx+\omega t)$ with $P=0.80\,\mathrm{Pa}$, $f=200\,\mathrm{Hz}$ and $Z=400\,\mathrm{Pa\,s/m}$. Use the linear momentum equation $\rho_0u_t=-p_x$ and $\omega=ck$.

1. Derive $u(x,t)$, taking no constant background flow.

2. Find a zero-mean displacement field and its amplitude. What is the sign of particle velocity at a positive pressure crest?

Question 2 - Solution

Reversing propagation reverses the pressure-velocity sign

(a) Method and calculation

With $\Psi=kx+\omega t$, the momentum equation gives$$\rho_0u_t=kP\sin\Psi,\qquad u=-\frac{kP}{\rho_0\omega}\cos\Psi=-\frac PZ\cos\Psi.$$The negative sign is required by leftward propagation, even though pressure is a scalar.

(b) Interpretation and further calculation

Integrate $u=\xi_t$ without a static displacement offset:$$\xi=-\frac{P}{Z\omega}\sin\Psi,\qquad a=\frac{0.80}{400(2\pi)(200)}=1.592\,\mu\mathrm m.$$At $p=+P$, $u=-P/Z=-2.0\times10^{-3}\,\mathrm{m/s}$. Particle motion is leftward at this compression crest, consistent with propagation.

Physical check

The amplitudes still satisfy $p_m=Z u_m$; it is the signed field relation that changes.