Two identical loudspeakers are 3.4 m apart. A person stands 5.8 m from one speaker and 3.6 m from the other. What is the lowest frequency at which destructive interference will occur at this point? The speed of sound in air is 343 m/s. The answer is in Hertz [Hz]
the difference in distance from the speakers is half the wavelength of the lowest frequency
1.8
To determine the lowest frequency at which destructive interference will occur at the given point, we need to find the frequency at which the path difference between the two speakers is equal to an odd multiple of half the wavelength.
The path difference between the two speakers can be calculated using the formula:
Δx = |x2 - x1|
Where:
Δx is the path difference
x2 is the distance from the second speaker to the point
x1 is the distance from the first speaker to the point
In this case, x2 = 3.6 m and x1 = 5.8 m. Thus, we have:
Δx = |3.6 - 5.8|
= |-2.2|
= 2.2 m
The wavelength can be calculated using the formula:
λ = v / f
Where:
λ is the wavelength
v is the speed of sound in air (343 m/s)
f is the frequency
We need to find the frequency when the path difference is equal to an odd multiple of half the wavelength. Hence:
Δx = (2n - 1) * (λ / 2)
Substituting the values, we have:
2.2 = (2n - 1) * (343 / 2f)
Rearranging the equation to solve for f, we get:
f = (2n - 1) * (343 / (2 * 2.2))
To find the lowest frequency, we need to substitute n = 1 in the equation:
f = (2 - 1) * (343 / (2 * 2.2))
= 343 / 4.4
≈ 78.03 Hz
Therefore, the lowest frequency at which destructive interference will occur at the given point is approximately 78.03 Hz.
To find the lowest frequency at which destructive interference will occur at the given point, we need to calculate the path difference between the two speakers.
The path difference is the difference in distance traveled by sound waves from each speaker to the given point. Destructive interference occurs when the path difference is equal to an integer multiple of the wavelength.
First, let's calculate the path difference. The person is 5.8 m away from one speaker and 3.6 m away from the other speaker. Therefore, the path difference is:
Path Difference = Distance to Second Speaker - Distance to First Speaker
= 3.6 m - 5.8 m
= -2.2 m
Since the path difference is negative, it means the second speaker is ahead of the first speaker.
Next, we need to calculate the wavelength corresponding to the path difference. The formula to calculate wavelength is:
Wavelength = Speed of Sound / Frequency
We can rearrange this formula to find the frequency:
Frequency = Speed of Sound / Wavelength
For destructive interference to occur, the path difference should be equal to an integer multiple of the wavelength, which can be expressed as:
Path Difference = N * Wavelength
where N is an integer.
Now, let's calculate the wavelength corresponding to the path difference:
Wavelength = Path Difference / N
= -2.2 m / N
To find the lowest frequency, we need to consider the largest possible wavelength. In this case, N will be equal to 1 to have the smallest possible wavelength. Now, substitute the values back into the frequency formula:
Frequency = Speed of Sound / Wavelength
= 343 m/s / (-2.2 m / 1)
≈ -155.9 Hz
Note that the negative sign indicates the phase difference between the two speakers, which leads to destructive interference at this point.
Therefore, the lowest frequency at which destructive interference will occur at this point is approximately 155.9 Hz.