Interference — Question 1

Question 1

A fixed acoustic path difference selects a frequency sequence

Two phase-locked loudspeakers have zero source phase difference. Their paths to a microphone are $3.0$ and $5.0\,\mathrm m$. Each produces the same pressure amplitude at the microphone. Sound speed is $340\,\mathrm{m/s}$.

1. Find the first three positive frequencies giving complete pressure cancellation.

2. At $170\,\mathrm{Hz}$, find the resultant amplitude relative to either contribution. Distinguish a path difference from a whole path length.

Question 1 - Solution

A fixed acoustic path difference selects a frequency sequence

(a) Method and calculation

The relative propagation phase is $\delta=k\Delta r$, with $\Delta r=2.0\,\mathrm m$. Cancellation requires$$\frac{2\pi f\Delta r}{c}=(2m+1)\pi,\qquad f_m=\frac{(2m+1)c}{2\Delta r}.$$For $m=0,1,2$, the frequencies are $85$, $255$ and $425\,\mathrm{Hz}$.

(b) Interpretation and further calculation

At $170\,\mathrm{Hz}$, $\lambda=c/f=2.0\,\mathrm m$ and $\delta=2\pi$. Therefore$$P_{\mathrm{res}}=2P|\cos(\delta/2)|=2P.$$The phase comparison subtracts the two traveled distances; neither $3.0$ nor $5.0\,\mathrm m$ alone determines interference.

Physical check

Exact cancellation requires both the odd-half-cycle phase difference and equal received amplitudes, as explicitly assumed.