A pair of speakers emits coherent sound waves in phase at a frequency of 440 Hz. A microphone is placed such that it is 1.0 m from one speaker on a line that is perpendicular to the line between the speakers. Which of the following distances reflects a possible distance between the speakers if there is complete destructive interference at the location of the microphone? The speed of sound is 340 m/s.
A) 1.0 m
B) 0.77 m
C) 2.76 m
D) 1.3 m
E) 1.77 m

Question

Quizplus · Accepted Answer

For complete destructive interference, the difference in the distances traveled by the sound waves from the two speakers to the microphone must be an odd multiple of half the wavelength. Let d be the distance between the speakers. Then, the difference in the distances to the microphone from the two speakers is given by √(d² + 1.0²) - √(d² + (1.0 - x)²), where x is the distance along the line between the speakers to the point of interference. At the point of complete destructive interference, this difference must be equal to (2n+1)λ/2, where n is any integer. So, we have:

√(d² + 1.0²) - √(d² + (1.0 - x)²) = (2n+1)λ/2

Substituting λ = v/f = 340/440 = 0.773 m and x = 0, we get:

√(d² + 1.0²) - d = (2n+1)λ/2

Squaring both sides and simplifying, we get:

d² - 2dn + (nλ)² - 1.0² = 0

This is a quadratic equation in d, with solutions:

d = nλ ± √(n²λ² + 1.0²)

For complete destructive interference, we need the smaller solution, which is:

d = nλ - √(n²λ² + 1.0²)

Putting n = 1 and λ = 0.773 m, we get d ≈ 2.76 m, which is option (C).

A Pair of Speakers Emits Coherent Sound Waves in Phase