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, 1.2 m from the speaker.
What is the lowest frequency that will produce constructive interference at your location?
fail
To determine the lowest frequency that will produce constructive interference at your location, you need to consider the wavelength of sound and the phase difference between the two speakers.
First, let's calculate the phase difference between the two speakers. The phase difference depends on the angle between the line connecting the two speakers and the line connecting the speaker and your location.
In this case, since you are standing directly in front of one of the speakers, the angle between the line connecting the two speakers and the line connecting the speaker and your location is 0 degrees. This means there is no phase difference between the two speakers.
Next, let's calculate the wavelength of sound. The wavelength (λ) can be determined using the formula:
λ = v / f
where λ is the wavelength, v is the speed of sound in air, and f is the frequency.
The speed of sound in air is approximately 343 meters per second.
Now, we can find the lowest frequency that will produce constructive interference. Constructive interference occurs when the path difference between two waves is an integer multiple of the wavelength. In this case, the path difference is the distance between the speakers, which is 0.85 meters.
So, the path difference is:
Path difference = distance between speakers = 0.85 meters
For constructive interference, the path difference must be an integer multiple of the wavelength:
Path difference = n * λ
where n is an integer.
Since there is no phase difference between the speakers, the path difference can be used to find the wavelength:
0.85 meters = n * λ
Now, we can substitute the formula for the wavelength:
0.85 meters = n * (v / f)
Solving for f, the frequency:
f = v / (n * 0.85 meters)
To find the lowest frequency that will produce constructive interference at your location, we need to find the smallest integer value of n that satisfies the above equation. In this case, n = 1, as we are looking for the lowest frequency.
Now, substitute the values into the equation:
f = 343 meters per second / (1 * 0.85 meters)
f ≈ 403.53 Hz
Therefore, the lowest frequency that will produce constructive interference at your location is approximately 403.53 Hz.