Two identical loudspeakers are some distance apart. A person stands 5.70 m from one speaker and 3.00 m from the other. What is the second lowest frequency at which constructive interference will occur at this point? The speed of sound in air is 343m/s.
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To find the second lowest frequency at which constructive interference will occur at the given point, we can use the formula for the path difference between two sources of sound in order to determine when the waves will be in phase.
The formula for the path difference can be expressed as:
Δx = |d2 - d1|
where:
Δx is the path difference,
d1 is the distance from the first speaker to the point of interest,
d2 is the distance from the second speaker to the point of interest.
In this case, we are given that the person stands 5.70 m from one speaker and 3.00 m from the other. Therefore:
d1 = 5.70 m
d2 = 3.00 m
Substituting these values into the equation, we have:
Δx = |3.00 m - 5.70 m|
Δx = |-2.70 m|
Δx = 2.70 m
Now, we need to determine the frequency at which the path difference is equal to an integer number of wavelengths. This is because at this condition, constructive interference will occur.
The formula for the wavelength of a sound wave is:
λ = v/f
where:
λ is the wavelength,
v is the speed of sound in air (343 m/s),
f is the frequency.
We can rearrange this formula to solve for the frequency:
f = v/λ
To find the second lowest frequency, we need to consider the path difference as the second harmonic, which corresponds to two wavelengths. Thus, we have:
Δx = 2λ
Substituting this into the formula, we get:
2.70 m = 2λ
Therefore, the wavelength is:
λ = 2.70 m / 2
λ = 1.35 m
Now, substituting the values of the speed of sound and the calculated wavelength into the frequency formula, we can solve for the second lowest frequency:
f = 343 m/s / 1.35 m
f ≈ 254.81 Hz
Therefore, the second lowest frequency at which constructive interference will occur at the given point is approximately 254.81 Hz.
Please note that this is an approximation and the actual answer may vary slightly due to rounding errors.