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 answer these questions, we need to understand a few key concepts related to sound waves: the relationship between frequency, wavelength, and speed of sound.
a) Wavelength (λ) is the distance between two consecutive points in a sound wave that are in phase. In this case, the two speakers A and B are emitting identical, synchronized sound waves at 240 kHz.
To find the wavelength, we can use the formula:
λ = v / f
Where:
λ is the wavelength,
v is the speed of sound, and
f is the frequency of the sound wave.
Since we do not know the speed of sound at this point, we cannot calculate the wavelength directly. We will come back to this question after finding the speed of sound.
b) The speed of sound (v) is the rate at which sound propagates through a medium. It can be calculated using the formula:
v = λf
Where:
v is the speed of sound,
λ is the wavelength, and
f is the frequency of the sound wave.
To find the speed of sound, we need to determine the frequency and wavelength. We already know the frequency from the information provided (240 kHz).
To find the wavelength, we can use the information about the distances from the two speakers to point D. The distance from speaker A to D is 3.10 m, and the distance from speaker B to D is 2.40 m.
Since the speakers are emitting synchronized sound waves, the difference in distances from the speakers to point D corresponds to a difference in the phase of the sound waves. This difference causes interference patterns and changes in sound intensity.
The difference in distance is equal to the number of wavelengths between the two speakers (A and B) at point D. In other words, the path difference between speaker A and speaker B is equal to a whole number of wavelengths.
3.10 m - 2.40 m = 0.70 m
Since the difference of 0.70 m corresponds to a whole number of wavelengths, we can use this information to find the wavelength.
Divide the path difference by the number of wavelengths to find the wavelength:
0.70 m / 1 wavelength = 0.70 m
Since the path difference is equal to one wavelength, the wavelength is 0.70 m.
Now that we know the frequency (240 kHz) and the wavelength (0.70 m), we can calculate the speed of sound (v):
v = λf
v = (0.70 m)(240,000 Hz)
v = 168,000 m/s
Therefore, the speed of sound is 168,000 m/s.
Now we can revisit the first question and calculate the wavelength using the speed of sound obtained:
λ = v / f
λ = 168,000 m/s / 240,000 Hz
λ = 0.70 m
Therefore, the wavelength of the sound is 0.70 m.