Speakers A and B are vibrating in phase. They are directly facing each other, are 7.4 m apart, and are each playing a 72.0 Hz tone. The speed of sound is 343 m/s. On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker A?
m (nearest)
m (middle)
m (farthest)
Calculate wavelength. When the difference in distances from each speaker are multiples of wavelength, construtcive interference occurs. For instance, if the wavelenght were 1 meter, then when distance A (speaker A to the point) = distance B -+ n Lambda
n= 0, 1, 2, ... constructive interference occurs. The easy solution is in the center, n=0. I suspect you will find a +- 1 solution.
To find the distances of the three points where constructive interference occurs, we first need to calculate the wavelength of the sound wave being produced by the speakers.
The formula for wavelength is given by:
λ = v / f
Where λ is the wavelength, v is the speed of sound, and f is the frequency.
In this case, the frequency f is 72.0 Hz and the speed of sound v is 343 m/s.
Substituting the values into the formula, we get:
λ = 343 m/s / 72.0 Hz
Calculating this, we find that the wavelength is approximately 4.76 m.
Now, let's consider the constructive interference equation:
distance A = distance B ± nλ
where n is an integer representing the number of wavelengths by which the path length from speaker A is longer (or shorter) than the path length from speaker B.
We need to find the three points where constructive interference occurs, so we'll start with n = 0 (the center point) and then consider n = ±1.
For n = 0, the distance A from speaker A to the point is equal to distance B. Since the speakers are 7.4 m apart, the distance A will be halfway between the speakers, which is 7.4 m / 2 = 3.7 m.
For n = ±1, we'll need to consider cases where distance A is greater or smaller than distance B by one wavelength.
When distance A is greater than distance B by one wavelength, we get:
distance A = distance B + λ
Since we already know distance A = 3.7 m, we can calculate distance B:
distance B = distance A - λ = 3.7 m - 4.76 m
Calculating this, we find that distance B is approximately -1.06 m. However, this negative value doesn't make sense physically, so we discard this solution.
When distance A is smaller than distance B by one wavelength, we get:
distance A = distance B - λ
Using the same reasoning as above, we calculate distance B:
distance B = distance A + λ = 3.7 m + 4.76 m
Calculating this, we find that distance B is approximately 8.46 m.
So, the three distances of the points where constructive interference occurs are:
Nearest point: 3.7 m
Middle point: 8.46 m
Farthest point: This would be 3.7 m beyond the farthest point from speaker B, which is at a distance of 7.4 m from speaker B. Therefore, the farthest point is 7.4 m + 3.7 m = 11.1 m from speaker A.