Two identical loudspeakers that are 5.00 m apart and face toward each other are driven in phase by the same oscillator at a frequency of 875 Hz. The speed of sound in the room is 344 m/s. If you start out standing midway between the speakers, find the shortest distance you can walk toward either speaker in order to hear a minimum of sound.
0.0983m
To find the shortest distance you can walk toward either speaker in order to hear a minimum of sound, we need to calculate the phase difference between the sound waves emitted by the two speakers at your position.
The phase difference depends on the path length difference (ΔL) between the two speakers. We can calculate the path length difference using the formula:
ΔL = d * sin(θ)
where:
d = distance between the speakers (5.00 m)
θ = angle from the midpoint to your position (measured from the line connecting the speakers)
Since we want to find the minimum sound, we need the sound waves from the two speakers to interfere destructively, which requires a phase difference of π radians (180 degrees).
The formula for phase difference (Δφ) is given by:
Δφ = 2π * ΔL / λ
where:
λ = wavelength of the sound waves
The wavelength (λ) can be calculated using the formula:
λ = v / f
where:
v = speed of sound in the room (344 m/s)
f = frequency of the sound waves (875 Hz)
Let's calculate the necessary values step-by-step:
1. Calculate the wavelength (λ):
λ = v / f
= 344 / 875
≈ 0.3939 m
2. Calculate the maximum phase difference needed for destructive interference:
Δφ = π radians
= 180 degrees
3. Calculate the path length difference needed for the desired phase difference:
ΔL = (λ * Δφ) / (2π)
= (0.3939 * π) / (2π)
= 0.19695 m
4. Calculate the angle (θ) from the midpoint to your position using the path length difference:
θ = arcsin(ΔL / d)
= arcsin(0.19695 / 5.00)
≈ 0.4003 radians
The shortest distance you can walk toward either speaker in order to hear a minimum of sound is the distance that corresponds to half of the wavelength at the given frequency:
d_min = λ / 2
= 0.3939 / 2
≈ 0.1969 m
Therefore, you need to walk approximately 0.1969 meters toward either speaker to hear a minimum of sound.
To find the shortest distance you can walk to hear a minimum of sound, you need to consider the concept of destructive interference. Destructive interference occurs when two sound waves of the same frequency and opposite phase cancel each other out, resulting in minimum or no sound at a particular location.
In this case, you have two identical loudspeakers that are 5.00 m apart and facing toward each other. They are driven in phase by the same oscillator at a frequency of 875 Hz. The speed of sound in the room is 344 m/s.
To determine the minimum distance, you need to calculate the path difference between the sound waves from the two loudspeakers.
The path difference can be calculated using the formula:
Δx = d × sin(θ)
where:
Δx is the path difference,
d is the distance between the two speakers (5.00 m in this case), and
θ is the angle between the line connecting the midpoint between the speakers and your location and the line connecting the midpoint between the speakers and one of the speakers.
In order to experience destructive interference, the path difference needs to be an integer multiple of the wavelength. Since we want to find the minimum distance, we want the path difference to be equal to half a wavelength.
The wavelength (λ) of a sound wave can be calculated using the formula:
λ = v / f
where:
λ is the wavelength,
v is the speed of sound in the room (344 m/s in this case), and
f is the frequency of the sound wave (875 Hz in this case).
Let's calculate the wavelength first:
λ = 344 m/s / 875 Hz = 0.393 m
Now let's calculate the path difference for destructive interference:
Δx = (0.393 m / 2) × sin(θ)
Since we want the minimum distance, the angle θ should be the smallest angle that results in destructive interference. This occurs when the sine of the angle is equal to -1 (sin θ = -1).
Therefore:
-1 = sin(θ)
To find the smallest value of θ that satisfies this condition, we can take the inverse sine (or arcsine) of -1:
θ = arcsin(-1) = -π/2
However, since we are dealing with distances, we take the absolute value of the angle:
θ = | -π/2 | = π/2
Now let's calculate the path difference:
Δx = (0.393 m / 2) × sin(π/2) = 0.393 m / 2 = 0.1965 m
So, the shortest distance you can walk toward either speaker in order to hear a minimum of sound is approximately 0.1965 meters.