Sound Waves — Question 10

Question 10

A stationary boundary preserves the acoustic clock

A monochromatic sound source drives a stationary boundary between air ($c_a=340\,\mathrm{m/s}$) and water ($c_w=1500\,\mathrm{m/s}$) at $f=500\,\mathrm{Hz}$. Assume a transmitted wave; amplitude transmission is outside this question.

1. Find period and wavelength on both sides.

2. Find the number of cycles contained in a $3.0\,\mathrm m$ segment on each side. Explain why the longer water wavelength does not imply a slower oscillation.

Question 10 - Solution

A stationary boundary preserves the acoustic clock

(a) Method and calculation

The common periodic boundary forces the same temporal frequency on the transmitted field:$$T_a=T_w=1/f=2.00\,\mathrm{ms}.$$$$\lambda_a=c_a/f=0.680\,\mathrm m,\qquad\lambda_w=c_w/f=3.00\,\mathrm m.$$

(b) Interpretation and further calculation

The spatial cycle counts are$$N_a=3.0/0.680=4.412,\qquad N_w=3.0/3.00=1.00.$$Both points still execute $500$ cycles each second. A water phase front travels farther during each common period. This continuity of frequency does not require equal amplitudes, impedances or transmitted powers.

Physical check

The ratio $\lambda_w/\lambda_a=c_w/c_a$ follows directly from fixed frequency. These statements rely on a stationary, time-independent interface.