Traveling Sound Waves — Question 1

Question 1

Pressure follows the gradient of the displacement field

A rightward plane sound wave is $\xi=a\cos(kx-\omega t)$ with $a=2.0\,\mu\mathrm m$, $f=500\,\mathrm{Hz}$, $c=340\,\mathrm{m/s}$ and $\rho_0=1.20\,\mathrm{kg/m^3}$. Use $p=-B\xi_x$, $u=\xi_t$, and $B=\rho_0c^2$.

1. Derive the pressure and particle-velocity fields, including their phases.

2. Calculate both amplitudes and the pressure when displacement is at its positive maximum.

Question 1 - Solution

Pressure follows the gradient of the displacement field

(a) Method and calculation

Write $\Phi=kx-\omega t$, with $\omega=2\pi f$ and $k=\omega/c$. Differentiating gives$$p=Bka\sin\Phi,\qquad u=\omega a\sin\Phi.$$$$\frac{p}{u}=\frac{Bk}{\omega}=\rho_0c\equiv Z.$$Pressure and velocity are in phase; each is a quarter-cycle from displacement.

(b) Interpretation and further calculation

The characteristic impedance is $Z=408\,\mathrm{Pa\,s/m}$. Thus$$u_m=2\pi(500)(2\times10^{-6})=6.283\times10^{-3}\,\mathrm{m/s},$$$$p_m=Zu_m=2.5635\,\mathrm{Pa}.$$At positive displacement maximum, $\Phi=2m\pi$, so $p=u=0$. Maximum displacement is not maximum compression.

Physical check

The pressure amplitude has units $Z u_m=\mathrm{Pa}$. A positive pressure crest travels with a positive particle velocity for a rightward plane wave.