LOUDSPEAKER INTERFERENCE
Two small loudspeakers, A and B (Fig. 16.23), are driven by the same amplifier and emit pure sinusoidal waves in phase. (a) For what frequencies does constructive interference occur at point P? (b) For what frequencies does destructive interference occur? The speed of sound is 350 m/s.
IDENTIFY and SET UP:
The nature of the interference at P depends on the difference d in path lengths from point A to P and from point B to P. We calculate the path lengths using the Pythagorean theorem. Constructive interference occurs when d equals a whole number of wavelengths, while destructive interference occurs when d is a half-integer number of wavelengths. To find the corresponding frequencies, we use v = fλ.
EXECUTE:
The A-to-P distance is \left[(2.00 \mathrm{~m})^{2}+(4.00 \mathrm{~m})^{2}\right]^{1 / 2}= 4.47 m, and the B-to-P distance is \left[(1.00 \mathrm{~m})^{2}+(4.00 \mathrm{~m})^{2}\right]^{1 / 2}= 4.12 m. The path difference is d = 4.47 m – 4.12 m = 0.35 m.
(a) Constructive interference occurs when d = 0, λ, 2λ, ….or d = 0, v/f, 2v/f, …. = nv/f. So the possible frequencies are
(b) Destructive interference occurs when d = λ/2, 3λ/2, 5λ/2, …. or d = v/2f, 3v/2f, 5v2f,… . The possible frequencies are
\begin{aligned}f_{n} &=\frac{n v}{2 d}=n \frac{350 \mathrm{~m} / \mathrm{s}}{2(0.35\mathrm{~m})} \quad(n=1,3,5, \ldots) \\&=500 \mathrm{~Hz}, 1500 \mathrm{~Hz}, 2500\mathrm{~Hz}, \ldots\end{aligned}
EVALUATE: As we increase the frequency, the sound at point P alternates between large and small (near zero) amplitudes, with maxima and minima at the frequencies given above. This effect may not be strong in an ordinary room because of reflections from the walls, floor, and ceiling.