Two small stereo speakers are driven in step by the same variable-frequency osci

Question

Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone arranged as shown in the figure . Assume that the speed of sound is 344 m/s.

For what frequencies does their sound at the speakers produce constructive interference? Express your answer in terms of n (an integer factor).

For what frequencies does their sound at the speakers produce destructive interference? Express your answer in terms of n (an integer factor).

Explanation / Answer

Figure is not available , so I assume distances as 2 and 4.50 m You need to find the path difference. That is, how much further must sound waves from the more distant speaker travel than the close speaker, to reach the mike. Use Pythagoras to find the distance of the further speaker: it is v(2.00²+4.50²)=4.924m so the path difference is 4.924-4.50=0.424m. You will get constructive interference when this path difference is an integer number of wavelengths, because the waves will arrive at the mike in phase. The speed of sound is 344m/s so the lowest frequency that will produce an antinode at the mike is the one that makes 0.424=? v=f? so f=v/? f=344/0.424=811.32Hz. The next one will be when 0.424m = 2? => ?=0.212m f=344/0.212=1622.64Hz and so-on according to f=344n/0.424 where n is an integer. So For Constructive it is f=344n/0.424 where n is an integer. For destructive interference the path difference must be (n-½)? because that will make the waves arrive at the mike 180° out of phase. f=340(n-½)/0.424

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