Question

32. Two loudspeakers, A and B , are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speak- ers emit sound waves that travel directly from the speaker to point Q. (see figure) (a) What is the lowest frequency for which constructive interference occurs at point Q? (b) What is the lowest frequency for which destructive interference occurs at point Q? 2.00 m

Answer 1

(a)

Constructive interference takes place when the path difference is n $\lambda$ where n=1,2,3 ...

Path difference at the point Q from the speakers A and B = 3 m - 1 m = 2 m

Therefore, n $\lambda$ = 2 m (or) $\lambda$ = (2 / n) m

We know, Frequency (f) = Velocity (v) / Wavelength ( $\lambda$ ) ------------------- (1)

Velocity of sound wave in air (v) = 344 m/s

Substituting the value of $\lambda$ and v in equation (1),

f = (344 m/s x n) / 2 m = 172 Hz x n where n=1,2,3 .....

Therefore, the lowest frequency at which constructive interference takes place at at the point Q is 172 Hz

(b)

Destructive interference takes place when the path difference is n/2 x $\lambda$ where n=1,3,5 ...

Path difference at the point Q from the speakers A and B = 3 m - 1 m = 2 m

Therefore, n/2 x $\lambda$ = 2 m (or) $\lambda$ = (4 / n) m

Substituting the value of $\lambda$ and v in equation (1),

f = (344 m/s x n) / 4 m = 86 Hz x n where n=1,3,5 .....

Therefore, the lowest frequency at which destructive interference takes place at the point Q is 86 Hz