Sound Waves — Question 1

Question 1

An echo clock includes both legs of the acoustic journey

A colocated loudspeaker and microphone face a flat wall. A pulse echo arrives $0.240\,\mathrm{s}$ after emission. The still-air speed is $340\,\mathrm{m/s}$. The apparatus requires $0.010\,\mathrm{s}$ after emission before it can receive.

1. Find the wall distance and the shortest measurable distance.

2. If the wall recedes at $u=2.0\,\mathrm{m/s}$, find its distance at emission for the same measured echo time. Treat reflection as instantaneous.

Question 1 - Solution

An echo clock includes both legs of the acoustic journey

(a) Method and calculation

For a stationary wall, the travel length is $2d$. Hence$$2d=ct,\qquad d=\frac{340(0.240)}2=40.8\,\mathrm m.$$The receiving delay sets $d_{\min}=c(0.010)/2=1.70\,\mathrm m$. This is an instrumental limit, not a change in propagation speed.

(b) Interpretation and further calculation

Let the outbound time be $t_1$. The meeting condition is $ct_1=d_0+ut_1$. The reflection point is $ct_1$ from the apparatus, so the return time is also $t_1$. Thus$$t=\frac{2d_0}{c-u},\qquad d_0=\frac{(340-2)(0.240)}2=40.56\,\mathrm m.$$At reflection the wall is at $40.80\,\mathrm m$; its emission position is closer.

Physical check

The moving result tends to $ct/2$ as $u\to0$. Echo ranging measures a propagation path, not automatically the wall position at the observation time.