Sound Waves — Question 3

Question 3

An acoustic packet outruns the motion of its material particles

A $400\,\mathrm{Hz}$ tone is emitted for $0.050\,\mathrm s$ into still air at $c=340\,\mathrm{m/s}$. Inside the packet, particle displacement amplitude is $a=3.0\,\mu\mathrm m$. Neglect turn-on and turn-off transients and acoustic streaming.

1. Find the number of cycles, wavelength and packet length.

2. Find maximum particle speed and compare it with packet speed. Does a tagged particle travel the packet length?

Question 3 - Solution

An acoustic packet outruns the motion of its material particles

(a) Method and calculation

The duration counts temporal cycles, while each cycle occupies one wavelength:$$N=f\Delta t=20,\quad\lambda=\frac cf=0.850\,\mathrm m,\quad\ell=c\Delta t=N\lambda=17.0\,\mathrm m.$$

(b) Interpretation and further calculation

For $\xi=a\cos\omega t$,$$u=-\omega a\sin\omega t,\quad u_{\max}=2\pi(400)(3\times10^{-6})=7.54\times10^{-3}\,\mathrm{m/s}.$$$$\frac{u_{\max}}c=2.22\times10^{-5}.$$A tagged particle oscillates over a peak-to-peak distance $2a=6\,\mu\mathrm m$. Over the ideal twenty complete cycles its net displacement is zero; the packet advances through successive particles.

Physical check

Information and energy can move over macroscopic distances while material displacement remains microscopic. The no-drift conclusion is within the stated linear model.