Two loudspeakers, A and B, shown in figure, are driven by the same amplifier and
ID: 2138060 • Letter: T
Question
Two loudspeakers, A and B, shown in figure, are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00m to the right of Speaker A. The frequency of the sound waves produced by the loudspeakers is 206Hz. 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 to point P.
Two loudspeakers, A and B, shown in figure, are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00m to the right of Speaker A. The frequency of the sound waves produced by the loudspeakers is 206Hz. 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 to point P. For what values of x will destructive interference occur at point P? For what values of x will constructive interference occur at point P?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.
Assume speed of sound c = 343 m/s, then wavelength lambda = c/f = 1.665 m and lambda/4 = 0.41626 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 1.03 - 0.41626 and 1.03 + 0.41626 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 = 0.83252 m to either side. No more points can fit in that space. The locations are at 1.03 - 0.83252 and 1.03 + 0.83252 m.
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