Two speakers, Speaker A and Speaker B, are placed 3.0 meters apart and emit sound waves with a frequency of 500 Hz. The speed of sound in the medium is 340 m/s.
a) Calculate the wavelength of the sound wave emitted by each speaker.
b) Determine whether the sound waves from the two speakers will interfere constructively or destructively at point P located 2.5 meters away from Speaker A.
c) If the sound waves from the two speakers interfere constructively at point P, calculate the phase difference between the waves from Speaker A and Speaker B at that point.
a) The formula for the wavelength of a wave is given by:
wavelength = speed / frequency
For Speaker A:
wavelength_A = speed / frequency_A
Substituting the given values:
wavelength_A = 340 m/s / 500 Hz
Simplifying:
wavelength_A = 0.68 meters
For Speaker B, the wavelength would be the same since they emit sound waves of the same frequency.
wavelength_B = 0.68 meters
b) To determine whether the sound waves from the two speakers will interfere constructively or destructively at point P, we need to compare the path length difference to half a wavelength.
The path length difference between the waves from Speaker A and Speaker B can be calculated as follows:
path length difference = distance_BP - distance_AP
Given that point P is 2.5 meters away from Speaker A:
distance_AP = 2.5 meters
And the distance from point P to speaker B (distance_BP) can be calculated using the Pythagorean theorem:
distance_BP = sqrt(3.0^2 + 2.5^2)
distance_BP = 3.91 meters
Calculating the path length difference:
path length difference = 3.91 meters - 2.5 meters
path length difference = 1.41 meters
Since the path length difference is smaller than the wavelength (0.68 meters), the sound waves from the two speakers will interfere destructively at point P.
c) Since the sound waves interfere destructively, there will be a phase difference of half a wavelength between the waves from Speaker A and Speaker B at point P.
phase difference = 0.5 * wavelength
Substituting in the given wavelength:
phase difference = 0.5 * 0.68 meters
phase difference = 0.34 meters
Therefore, the phase difference between the waves from Speaker A and Speaker B at point P is 0.34 meters.
Note: The phase difference represents the displacement in the wave cycle at a given point.