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  (Figure 1) . Assume that the speed of sound is 344 m/s.

Part A

For what frequencies does their sound at the speakers produce constructive interference?

Express your answer in terms of n (an integer factor).

Part B

For what frequencies does their sound at the speakers produce destructive interference?

Express your answer in terms of n (a positive integer factor).

Explanation / Answer

here,

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.

From pythgoras theoram,
Distance to farther speaker, D = sqrt(2^2 + 4.5^2) = 4.924 m

so path difference , w = 4.924 - 4.50 = 0.424 m

so constructive interference are observed when this path difference is an integer number of wavelengths, because the waves will arrive at the mike in phase.

frequency , f = speed of sound/wavelength
so the lowest frequency that will produce an antinode at the mike is the one that makes frequency,
f1 = c/w
f1 = 334/0.424
f1 = 787.736 Hz

next interference will be w = 0.424/2 = 0.212 m

f2 = 334/0.212
f2 = 1602 Hz

and so on, with genereal equation, f = n*c/w --------------(2)

Part B:

For destructive interference the path difference must be
p = (n-0.5)w

because that will make the waves arrive at the mike 180° out of phase.
Therfore,
frequency, f = c*(n-0.5)/w ( Where, n is integer)

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