Two speakers, A and B, are emitting identical, synchronized (in-phase) sound waves at 240kHz. A person standing a point C hears a relatively loud sound. She then starts walking towards point D, and notices the sound gets quieter, reaching a minimum when she arrives at D. The distance from speaker A to D is 3.10m. The distance from speaker B to D is 2.40m.
a) From the information given above,what is the wavelength of the sound?
b) From the information given above, what is the speed of the sound?
To find the wavelength of the sound, we can use the formula:
wavelength = distance / number of cycles
Since both speakers are emitting identical, synchronized sound waves, the number of cycles from both speakers must be the same.
The number of cycles is equal to the distance traveled divided by the wavelength. We can set up an equation using this information:
Number of cycles = distance / wavelength
We know that the person hears a relatively loud sound at point C, which means the person is at a point where constructive interference occurs. Constructive interference happens when the path length difference between the two speakers is an integer multiple of the wavelength.
At point D, however, the person hears a minimum sound level, which means destructive interference occurs. Destructive interference happens when the path length difference between the two speakers is an odd multiple of half the wavelength.
The distance from speaker A to D is 3.10m, and the distance from speaker B to D is 2.40m.
Using this information, we can set up two equations:
3.1 = n * wavelength (1)
2.4 = (n + 0.5) * (wavelength / 2) (2)
where n represents the number of wavelengths that fit into the given distance.
The solution for n in equation (1) can be substituted into equation (2). Solving these equations will give us the wavelength.
Now, let's solve the equations:
From equation (1):
wavelength = 3.1 / n
Substituting this into equation (2):
2.4 = (n + 0.5) * (3.1 / (2 * n))
Simplifying:
2.4 = (n + 0.5) * 3.1 / (2n)
4.8n = 3.1n + 0.775
1.7n = 0.775
n ≈ 0.456
Substituting this value of n back into equation (1):
wavelength = 3.1 / 0.456
wavelength ≈ 6.80m
Therefore, the wavelength of the sound is approximately 6.80m.
To find the speed of sound, we can use the equation:
speed = frequency * wavelength
The frequency is given as 240kHz, which can be converted into 240,000 Hz.
Using the wavelength we found earlier:
speed = 240,000 * 6.80
speed ≈ 1,632,000 m/s
Therefore, the speed of sound is approximately 1,632,000 m/s.
To find the wavelength of the sound in part a), we can use the formula:
wavelength = speed of sound / frequency
Since speakers A and B are emitting identical sound waves, and the sound waves are synchronized (in-phase), we can consider them as a single source.
Let's denote the speed of sound as v, and the frequency of the sound as f. Therefore, the distance from speaker A to the point D is D_AD = 3.10 m, and the distance from speaker B to the point D is D_BD = 2.40 m.
When the person is at point C, she hears a relatively loud sound, which means the sound waves from both speakers are in-phase. This implies that the path difference from the two speakers to point C must be an integer multiple of the wavelength.
When the person moves towards point D, she notices the sound getting quieter until reaching a minimum. At this point, the sound waves from the two speakers are completely out of phase, meaning the path difference from the two speakers to point D must be an odd multiple of half the wavelength.
To calculate the wavelength, we can subtract the two path differences, D_AD and D_BD, at point C and point D. The path difference at point C is:
path difference at point C = D_AD - D_BD
Similarly, the path difference at point D is:
path difference at point D = D_AD - D_BD
Since the path difference at point C is an integer multiple of the wavelength, and the path difference at point D is an odd multiple of half the wavelength, we can set up the following equation:
path difference at point C - path difference at point D = (n * wavelength) - (m * (wavelength / 2)) = wavelength * (2n - m / 2) = wavelength * a
Where n and m are integers, and a = (2n - m / 2).
Now, we can determine the values for n and m that satisfy the condition. We know that a is an integer because the path difference must be an integral multiple of the wavelength. The smallest value for a while considering positive integers for n and m would be 1.
Therefore, we have:
wavelength = (path difference at point C - path difference at point D) / a
Calculating the values:
path difference at point C = D_AD - D_BD = 3.10 m - 2.40 m = 0.70 m
path difference at point D = 0 (since the person is at point D)
wavelength = (0.70 m - 0) / 1 = 0.70 m
So, the wavelength of the sound is 0.70 m.
For part b), we can calculate the speed of sound using the formula:
speed of sound = frequency * wavelength
We already know the frequency from the question, which is 240 kHz. However, we need to convert it to Hz since the SI unit is used for the speed of sound.
240 kHz = 240,000 Hz
Now, we can calculate the speed of sound:
speed of sound = frequency * wavelength = 240,000 Hz * 0.70 m = 168,000 m/s
Therefore, the speed of sound is 168,000 m/s.