Two identical loudspeakers are 2 m apart. A person stands 4.6 m from one speaker and 3.4 m from the other. What is the lowest frequency at which destructive interference will occur at this point?
Destructibe interference will occur if the difference between the distances to the two speakers is n + 1/2, where n is an integer zero or higher. When n = 0, you get the longest wavelength and lowest frequency with destructive interference.
Therefore 4.6 - 3.4 = 1.2 = (wavelength)/2
wavelngth = 2.4 m
freq
Destructive interference will occur if the difference between the distances to the two speakers is n + 1/2, where n is an integer equal to zero, or higher. When n = 0, you get the longest wavelength and lowest frequency with destructive interference.
Therefore 4.6 - 3.4 = 1.2 = (wavelength)/2
wavelength = 2.4 m
frequency = (speed of sound)/2.4 m
= 140 Hz (approx.)
To determine the lowest frequency at which destructive interference will occur at the given point, we can use the equation for the path length difference between the two speakers:
ΔL = |r2 - r1|
Where ΔL is the path length difference, r1 is the distance from the first speaker, and r2 is the distance from the second speaker.
Given:
Distance between speakers = 2 m
Distance from the first speaker (r1) = 4.6 m
Distance from the second speaker (r2) = 3.4 m
Substitute these values into the equation:
ΔL = |3.4 - 4.6|
ΔL = |-1.2|
ΔL = 1.2 m
Since destructive interference occurs when the path length difference is equal to a whole number of wavelengths, we need to divide the path length difference by the wavelength to find the lowest frequency.
The wavelength (λ) is given by the formula:
λ = 2d
Where d is the distance between the speakers.
Substitute the value for d into the equation:
λ = 2 * 2
λ = 4 m
Now, divide the path length difference by the wavelength:
Frequency (f) = ΔL / λ
f = 1.2 / 4
f = 0.3 Hz
Therefore, the lowest frequency at which destructive interference will occur at the given point is 0.3 Hz.
To determine the lowest frequency at which destructive interference will occur at the given point, we need to calculate the frequency that will result in a phase difference of 180 degrees between the waves from the two loudspeakers.
To achieve destructive interference, the path length difference between the two speakers must be equal to an odd multiple of half the wavelength.
Let's break down the solution using the following steps:
Step 1: Find the path length difference.
The path length difference (ΔL) between the two speakers can be calculated as follows:
ΔL = |Distance from Speaker 1 - Distance from Speaker 2|
Given that the person stands 4.6 m from one speaker and 3.4 m from the other speaker, we have:
ΔL = |4.6 m - 3.4 m| = 1.2 m
Step 2: Calculate the wavelength.
The wavelength (λ) can be calculated using the formula:
λ = 2d
Given that the speakers are 2 m apart, we have:
λ = 2 × 2 m = 4 m
Step 3: Determine the frequency.
Using the formula for the speed of sound, we have:
v = f × λ,
where v is the speed of sound (which is approximately 343 m/s).
Rearranging the formula, we get:
f = v / λ
Plugging in the values:
f = 343 m/s / 4 m = 85.75 Hz
Therefore, the lowest frequency at which destructive interference will occur at the given point is approximately 85.75 Hz.