Two speakers are 2m apart. An observer is 3m Infront of one speaker.
a) if both are driven by a single oscillator at 300Hz, what is the phase difference at the observer
b) what is the frequency closest to 300Hz for which the observer hear minimal sound?Take speed of sound in air as 343m/s
To find the answers to these questions, we need to understand the concepts of phase difference and interference in waves.
a) The phase difference between two points in a wave can be calculated using the formula:
Phase Difference = (Path Difference / Wavelength) * 2π
In this case, the two speakers are 2 meters apart, and the observer is 3 meters in front of one speaker. The path difference is the difference in the distances traveled by the waves from the speakers to the observer.
Let's calculate the path difference:
Path Difference = distance traveled from one speaker to the observer - distance traveled from the other speaker to the observer
= 3m - 0m (since the observer is in front of one speaker)
Path Difference = 3m
Now, we need to calculate the wavelength of the sound wave. The formula to calculate wavelength is:
Wavelength = Speed of Sound / Frequency
Given that the frequency is 300Hz and the speed of sound in air is 343m/s:
Wavelength = 343m/s / 300Hz
= 1.143 meters
Substituting the values into the phase difference formula:
Phase Difference = (Path Difference / Wavelength) * 2π
= (3m / 1.143m) * 2π
≈ 5.24 radians
Therefore, the phase difference at the observer is approximately 5.24 radians.
b) To find the frequency closest to 300Hz at which the observer hears minimal sound, we need to consider the concept of interference. When two sound waves with the same frequency interfere constructively, the sound is loud. When they interfere destructively, the sound is minimal.
For destructive interference, the path difference should be a multiple of half-wavelength. In this case, the path difference should be a multiple of half-wavelength to minimize the sound.
Let's calculate the path difference for destructive interference:
Path Difference = n * (λ/2)
Where n is an integer.
Substituting the values:
3m = n * (1.143m / 2)
Simplifying the equation:
6m = n * 1.143m
n = 6 / 1.143
n ≈ 5.24
To find the frequency for minimal sound, we need to calculate the corresponding wavelength using the speed of sound:
Wavelength = Speed of Sound / Frequency
Substituting the values:
1.143m = 343m/s / f
1.143f = 343m/s
f ≈ 300.4Hz
Therefore, the frequency closest to 300Hz for which the observer hears minimal sound is approximately 300.4Hz.