A pair of in-phase stereo speakers is placed side by side, 0.85 m apart. You stand directly in front of one of the speakers, 1.1 m from the speaker. What is the lowest frequency that will produce constructive interference at your location?
Well, let's calculate the lowest frequency for constructive interference, but brace yourself, because this might get a little phasy.
To achieve constructive interference between the two speakers, the path length difference between the two speakers should be an integer multiple of the wavelength. In this case, you're standing in front of one speaker, and to create constructive interference, the difference in the path length between the two speakers should be equal to the wavelength.
So, we can use the equation: Δl = (m * λ), where Δl is the path length difference, m is an integer, and λ is the wavelength.
In this case, the path length difference is given by: Δl = 0.85 m.
Since you're standing directly in front of one speaker, the path length difference can be calculated as: Δl = 2d = 2 * 1.1 m = 2.2 m.
Now, we can equate these two expressions to find the wavelength: 0.85 m = m * λ.
To get the lowest frequency, we need to find the maximum wavelength, which will happen when m = 1. So, we have 0.85 m = 1 * λ.
Now, solving for λ, we find λ = 0.85 m.
To find the lowest frequency, we can use the equation: v = f * λ, where v is the velocity of sound in air. Assuming the velocity of sound in air is 343 m/s, we can now plug in the values:
343 m/s = f * 0.85 m.
Solving for f, we get f = 343 m/s / 0.85 m ≈ 403 Hz.
So, the lowest frequency that will produce constructive interference at your location is approximately 403 Hz. Let the phasing begin!
To determine the lowest frequency that will produce constructive interference at your location, we need to calculate the path difference between the two speakers at that frequency.
The path difference is the difference in distance that the sound waves from each speaker travels to reach your location. Constructive interference occurs when the path difference is an integer multiple of the wavelength of the sound.
In this case, the path difference is the difference in distance traveled by the sound waves from each speaker:
Path Difference = Distance from Speaker 2 - Distance from Speaker 1
Since you stand directly in front of Speaker 1, the distance from Speaker 1 is 0.
Therefore, the path difference simplifies to:
Path Difference = Distance from Speaker 2 = 0.85 m
Now, let's find the wavelength corresponding to the path difference.
The general formula relating wavelength (λ), frequency (f), and the speed of sound (v) in air is:
v = f * λ
The speed of sound in air is approximately 343 m/s.
We can rearrange the formula to solve for the wavelength:
λ = v / f
Now, we can substitute the given values and solve for the wavelength:
λ = 343 m/s / f
To determine the lowest frequency that will produce constructive interference, we want to find the value of f that makes the path difference equal to an integer multiple of the wavelength:
Integer * λ = Path Difference
Substituting the values, we have:
Integer * (343 m/s / f) = 0.85 m
Simplifying further, we get:
Integer * 343 m/s = 0.85 m * f
Dividing both sides by 0.85 m to isolate f:
Integer * (343 m/s / 0.85 m) = f
Now, we can substitute the speed of sound and solve for the lowest frequency (f):
f = Integer * (343 m/s / 0.85 m)
Given that the path difference is 0.85 m, the lowest frequency that will produce constructive interference at your location is given by:
f = Integer * (343 m/s / 0.85 m)
To determine the lowest frequency that will produce constructive interference at your location, we need to consider the wavelength and path difference between the speakers.
Constructive interference occurs when the path difference between two wave sources is equal to a whole number of wavelengths. In the case of the stereo speakers, when the path difference between the two speakers is a multiple of the wavelength, constructive interference occurs.
Given:
Distance between the speakers (d) = 0.85 m
Your distance from one of the speakers (D) = 1.1 m
The path difference between the two speakers can be calculated using the formula:
Path difference (δ) = D * sin(θ)
Where θ is the angle between the line connecting the two speakers and the line connecting one of the speakers to your location. In this case, θ can be approximated as θ = tan^(-1)(d/D).
Let's calculate θ first:
θ = tan^(-1)(d/D) = tan^(-1)(0.85/1.1) ≈ 38.02 degrees
Now we can calculate the path difference:
δ = D * sin(θ) = 1.1 * sin(38.02 degrees)
Using this path difference, we can find the wavelength (λ) that corresponds to constructive interference by relating it to the path difference:
Path difference (δ) = m * λ
Where m is an integer indicating the number of wavelengths. Since we are looking for the lowest frequency (longest wavelength) that produces constructive interference, we can assume m = 1.
Therefore:
λ = δ / m = δ = 1.1 * sin(38.02 degrees)
Finally, we can calculate the lowest frequency (f) using the formula:
f = c / λ
Where c is the speed of sound. In this case, we will assume the speed of sound in air at room temperature is approximately 343 m/s.
f = 343 / λ
Now substitute the value of λ from the previous calculation to get the answer.
For lowest-order constructive interference, the two distances from speaker to listener must differ by one wavelength.
Wavelength = sqrt[0.85^2 + 1.1^2] = 1.39 m
Frequency = (sound speed)/(wavelength) 340m/s/1.39m = 245 Hz