The figure below shows two speakers, A and B, which 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 answer these questions, we need to use the concept of interference of sound waves. Let's break it down step by step:
a) To find the wavelength of the sound, we can use the interference pattern created by the two speakers. The distance from speaker A to D is 3.10m, and the distance from speaker B to D is 2.40m. We can assume that point D is where the person hears the minimum sound, which means point D is at a node (a point of destructive interference) between the two speakers.
The condition for destructive interference is that the path difference between the two speakers must be an odd multiple of half the wavelength. Therefore, we have:
2.40m - 3.10m = (2n + 1) * (λ/2), where n is an integer.
Simplifying this equation, we get:
-0.70m = (2n + 1) * (λ/2)
Since λ is the wavelength, we can now solve for it:
λ = (-0.70m) * (2 / (2n + 1))
λ = -1.40m / (2n + 1)
Here, n can be any integer, so we have a range of possible wavelengths. The exact value of λ will depend on the particular value of n.
b) To find the speed of sound, we can use the formula:
Speed = Wavelength * Frequency
We are given that the frequency is 240 kHz, which means 240,000 cycles per second. Now, we need to convert the frequency to Hz:
240 kHz = 240,000 Hz
Since we do not have the exact value of the wavelength from part a, we can take the average wavelength when n = 0. Substituting these values into the formula, we get:
Speed = (average wavelength) * (frequency)
Speed = (-1.40m / (2 * 0 + 1)) * (240,000 Hz)
Using this calculation, we can determine the speed of the sound.
Please note that the negative sign in the equation arises from the fact that we are calculating the distance between two nodes (destructive interference points) which are out of phase. If you need a positive distance, simply ignore the negative sign during the calculation.