Two loudspeakers, A and B (see figure below), are driven by the same amplifier a

Question

Two loudspeakers, A and B (see figure below), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is L = 1.60 m to the right of speaker A. The frequency of the sound waves produced by the loudspeakers is 192 Hz. Consider point P between the speakers and along the line connecting them, a distance x to the right of speaker A. Both speakers emit sound waves that travel directly from the speaker to point P. (The speed of sound in air is 344 m/s.)

(a) For what values of x will destructive interference occur at point P? (Enter your answers from smallest to largest starting with the first answer blank. Enter NONE in any remaining answer blanks.)

(b) For what values of x will constructive interference occur at point P? (Enter your answers from smallest to largest starting with the first answer blank. Enter NONE in any remaining answer blanks.)

(c) Interference effects like those in parts (a) and (b) are almost never a factor in listening to home stereo equipment. Why not?

Explanation / Answer

A. Destructive interference will occur at all points where the path length difference is an integer number of wavelengths (including 0) plus 1/2 wavelength.
wavelength lambda = c/f = 344/192 = 1.7916 m and lambda/4 = 0.4479 m.

There are two locations between the speakers where there is interference. At the center the path lengths are equal. If you move lambda/4 from the center you increase one path by lambda/4 and decrease the other by the same amount, for a 1/2 lambda difference. If you then try to add 1 lambda to the difference to get to the next pair of points you have to move the points another lambda/2 which is beyond the speakers. So the locations are at 0.8 - 0.4479 and 0.8 + 0.4479 m.

B. For constructive interference you want a pathlength difference equal to lambda multiplied by an integer>=0. Then you have one constructive point at the center and, using similar logic as in A, two more at lambda/2 = 0.8958 m to either side. No more points can fit in that space. The locations are at 0.8 - 0.8958 and 0.8 + 0.8958 m.

C. Normal stereo listening does not produce obvious interference because the sounds are seldom single frequencies, are usually changing in frequency, and are reflected from floor, walls and ceiling introducing many additional near-random path length differences.

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