Two speakers are driven by a common oscillator at 870 Hz and face each other at a distance of 1.20 m. Locate the points along a line joining the two speakers where relative minima of pressure amplitude would be expected. (Use v = 343 m/s. Choose one speaker as the origin and give your answers in order of increasing distance from this speaker.)
im not sure how to go about this problem.
What happens when two speakers of a stereo are brought face to face? Why
To find the points along the line joining the two speakers where relative minima of pressure amplitude would be expected, we need to consider the phenomenon of interference.
Interference occurs when waves from different sources combine to form a resultant wave. In this case, the two speakers are driven by a common oscillator, which means they are producing waves with the same frequency and same phase.
The distance between the speakers is 1.20 m, which will affect the interference pattern. To determine the points where relative minima occur, we need to find the locations where the waves from the two speakers are completely out of phase (180 degrees), leading to destructive interference.
The formula for the distance between two consecutive relative minima in an interference pattern due to two sources is given by:
d = λ/2,
where d is the distance between the consecutive relative minima, and λ is the wavelength of the sound wave.
To find the wavelength, we can use the formula:
v = f × λ,
where v is the speed of sound, f is the frequency, and λ is the wavelength.
Given that the frequency is 870 Hz and the speed of sound is 343 m/s, we can calculate the wavelength:
λ = v/f = 343 m/s / 870 Hz = 0.394 m.
Now, we can calculate the distance between consecutive relative minima:
d = λ/2 = 0.394 m / 2 = 0.197 m.
Starting from one of the speakers, we can move in increments of 0.197 m to find the locations of the relative minima. The points along the line joining the two speakers with increasing distance from the origin speaker where relative minima of pressure amplitude would be expected are:
1. Origin Speaker (0.00 m)
2. 0.197 m
3. 0.394 m
4. 0.591 m
5. 0.788 m
6. 0.985 m
7. 1.182 m (Second Speaker; distance between the speakers)
Note: The distances mentioned above are measured from the origin speaker.
To find the points along the line where relative minima of pressure amplitude would be expected, we can use the concept of interference in waves. In this case, we have two speakers driven by a common oscillator and facing each other.
Step 1: Determine the wavelength of the sound wave.
The wavelength (λ) can be found using the formula:
λ = v/f
where v is the speed of sound and f is the frequency of the sound wave.
Given:
v = 343 m/s (speed of sound)
f = 870 Hz (frequency of the sound wave)
Plugging in the values, we get:
λ = 343 m/s / 870 Hz = 0.394 m
Step 2: Determine the distance between the two speakers.
Given that the speakers face each other at a distance of 1.20 m, let's call this distance "d".
Step 3: Determine the positions of the relative minima along the line.
The relative minima occur when the waves from the two speakers interfere destructively. Destructive interference occurs when the path difference between the waves from the two speakers is an odd multiple of half the wavelength (λ/2).
Let's assume the distance from one speaker to a point of relative minimum as x (measured from the origin speaker).
For destructive interference:
Path difference = (x - d) - x = -d
Path difference = -(odd multiple of λ/2)
Therefore, the conditions for destructive interference are:
d = (2n + 1)(λ/2)
Solving for the distance:
1.20 m = (2n + 1)(0.394 m/2)
2.40 m = (2n + 1)(0.394 m)
n = (2.40 m / 0.394 m) - 1
n ≈ 5.08 - 1 ≈ 4.08
So, we find that n is approximately 4.08, which means n can be either 4 or 5 (as we cannot have a fraction of an interferent minima).
Using the given definition of the 'origin speaker', we can now calculate the positions of the relative minima along the line. Plugging in the values of n, we get the following approximate positions:
When n = 4:
x = 0 + (4 * 0.394 m) = 1.576 m
When n = 5:
x = 0 + (5 * 0.394 m) = 1.970 m
Therefore, the points along the line where relative minima of pressure amplitude would be expected from the origin speaker are approximately 1.576 m and 1.970 m.