In the figure, two speakers separated by distance d1 = 1.70 m are in phase. Assu

Question

In the figure, two speakers separated by distance d1 = 1.70 m are in phase. Assume the amplitudes of the sound waves from the speakers are approximately the same at the listener's ear at distance d2 = 4.00 m directly in front of one speaker. Consider the full audible range for normal hearing, 20 Hz to 20 kHz and use 343 m/s for the speed of sound in air.

(a) What is the lowest frequency that gives minimum signal (destructive interference) at the listener's ear?


(b) What is the second lowest frequency that gives minimum signal?

(c) What is the third lowest frequency that gives minimum signal?
Hz

(d) What is the lowest frequency that gives maximum signal (constructive interference) at the listener's ear?
Hz

(e) What is the second lowest frequency that gives maximum signal?
Hz

(f) What is the third lowest frequency that gives maximum signal?
Hz

Explanation / Answer

(a) Path difference of the interfering waves, x = (d12 + d22)1/2 - d2 = (1.72 + 42)1/2 - 4 = 0.346 m

Condition for destructive interference: x = (n + 1/2) * (vs/f)

where vs is the speed of sound in air and n is a whole number.

=> frequency, f = (n + 1/2)vs / x = (n + 1/2) * 343/0.346 = 991.3(n + 0.5)

For lowest frequency of destructive interference, n = 0

So, lowest frequency, fo = 991.3 * 0.5 = 495.7 Hz

(b) Second lowest frequency, f1 = 991.3 * (1 + 0.5) = 1487 Hz

(c) Third lowest frequency, f2 = 991.3 * (2 + 0.5) = 2478.3 Hz

(d) Condition for constructive interference: x = n * (vs/f)

where vs is the speed of sound in air and n is a natural number.

=> frequency, f = nvs / x = n * 343/0.346 = 991.3n

So, lowest frequency for maximum signal, fo = 991.3 * 1 = 991.3 Hz

(b) Second lowest frequency, f1 = 991.3 * 2 = 1982.6 Hz

(c) Third lowest frequency, f2 = 991.3 * 3 = 2973.9 Hz

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