Interference — Question 2

Question 2

Geometrical spreading limits an acoustic cancellation

Two coherent point sources have equal emission strength, so each received pressure amplitude is $C/r$. Their distances to a receiver are $3.0$ and $4.0\,\mathrm m$. Their relative source phase can be adjusted freely at fixed frequency.

1. Find the largest and smallest resultant pressure amplitudes in terms of $C$.

2. Find the minimum-to-maximum mean-square pressure ratio. How must the farther source strength change for a perfect null?

Question 2 - Solution

Geometrical spreading limits an acoustic cancellation

(a) Method and calculation

The two received amplitudes are $P_1=C/3$ and $P_2=C/4$. Collinear phasors give the extrema:$$P_{\max}=\frac C3+\frac C4=\frac{7C}{12},\qquad P_{\min}=\left|\frac C3-\frac C4\right|=\frac C{12}.$$

(b) Interpretation and further calculation

Mean-square pressure of a sinusoid is amplitude squared divided by two, so$$\frac{\langle p^2\rangle_{\min}}{\langle p^2\rangle_{\max}}=\left(\frac17\right)^2=\frac1{49}.$$To equalize received amplitudes, $C_2/4=C_1/3$, hence $C_2=(4/3)C_1$. Opposite received phases can then cancel.

Physical check

Phase adjustment cannot cancel an amplitude mismatch. A pressure-strength ratio is not the same as a power ratio.