Two loudspeakers emit sound waves along the x-axis. A listener in front of both speakers
hears a maximum sound intensity when speaker 2 is at the origin and speaker 1 is at x = 0.50
m. If speaker 1 is slowly moved forward, the sound intensity decreases and then increases,
reaching another maximum when speaker 1 is at x = 0.90 m.
(a) What is the frequency of the sound? Assume vsound = 340 m/s.
(b) What is the phase difference between the speakers?
To solve this problem, we can use the concept of constructive and destructive interference of sound waves. Constructive interference occurs when the waves from both speakers meet in phase, resulting in maximum intensity. Destructive interference occurs when the waves from both speakers meet out of phase, resulting in minimum intensity.
(a) To find the frequency of the sound, we need to use the formula for the phase difference between the speakers. The phase difference can be calculated using the equation:
Δφ = 2π * (Δx / λ)
Where:
Δφ = phase difference
Δx = difference in position between the speakers
λ = wavelength
We are given the positions of the speakers: speaker 2 is at x = 0 and speaker 1 is at x = 0.50 m and x = 0.90 m. The difference in position between the two speakers at the first maximum is Δx = 0.50 m - 0 m = 0.50 m. At the second maximum, the difference in position is Δx = 0.90 m - 0 m = 0.90 m.
Let's assume that the distance between two consecutive maximum points is equal to one wavelength. Therefore, λ = 0.90 m - 0.50 m = 0.40 m.
Now, we can calculate the phase difference at both maximum points using the formula:
Δφ = 2π * (Δx / λ)
At the first maximum (Δx = 0.50 m), Δφ = 2π * (0.50 m / 0.40 m) = 2π * 1.25 = 2.5π rad
At the second maximum (Δx = 0.90 m), Δφ = 2π * (0.90 m / 0.40 m) = 2π * 2.25 = 4.5π rad
Since the phase difference represents the number of cycles, we can see that the phase difference increases by 2π (1 cycle) between the first and second maximum points. Therefore, there is one full wavelength between the two maximum points.
Now, we can calculate the frequency of the sound using the formula:
v = fλ
Where:
v = speed of sound = 340 m/s
λ = wavelength
340 m/s = f * 0.40 m
To solve for f, we rearrange the equation: f = (340 m/s) / (0.40 m) = 850 Hz
Therefore, the frequency of the sound is 850 Hz.
(b) Since there is one full wavelength between the two maximum points, the phase difference between the speakers is 2π rad or 360°.
The phase difference between the speakers is 360°.