Two loudspeakers are 2.42 m apart. A person stands 3.42 m from one speaker and 2.82 m from the other. What is the lowest frequency at which destructive interference (sound cancellation) will occur at this point?
To determine the lowest frequency at which destructive interference will occur, we can use the formula for the path difference between two sources. The path difference is the difference in the distances traveled by the waves from the two speakers.
In this case, the person is equidistant from both speakers, but at different distances. Let's assume the speed of sound as approximately 343 m/s (at room temperature).
The path difference (Δx) can be calculated as follows:
Δx = distance to the first speaker - distance to the second speaker
Δx = 3.42 m - 2.82 m
Δx = 0.6 m
Destructive interference occurs when the path difference (Δx) is equal to an odd multiple of half the wavelength (λ/2).
So, we can set up the following equation to find the lowest frequency:
Δx = (n + 1/2) * (λ/2)
Here, n is an odd integer (1, 3, 5,...), representing multiples of half the wavelength. Rearranging the equation, we have:
λ = 2 * Δx / (n + 1/2)
Let's substitute the given values into the equation to find the lowest frequency:
λ = 2 * 0.6 m / (1 + 1/2)
λ = 2 * 0.6 m / (3/2)
λ = 2 * 0.6 m * (2/3)
λ = 2.4 m
The lowest frequency occurs when the wavelength is at its maximum value, which is twice the distance between the speakers. Using the formula for the speed of sound, we can find this frequency (f):
343 m/s = λ * f
f = 343 m/s / 2.4 m
f ≈ 142.92 Hz
Therefore, the lowest frequency at which destructive interference will occur at the given point is approximately 142.92 Hz.