Two loudspeakers are placed above and below each other and driven by the same source at a frequency of 3.70 102 Hz.An observer is in front of the speakers at point O, at the same distance from each speaker. What minimum vertical distance upward should the top speaker be moved to create destructive interference at point O? (Let h = 3.02 m.)
the drawing is necessary
To create destructive interference, the path difference between the two speakers should be half a wavelength.
The wavelength (λ) of the sound wave can be found using the formula:
λ = v / f
where v is the speed of sound and f is the frequency of the source. The speed of sound in air is approximately 343 m/s.
λ = 343 m/s / (3.70 × 10^2 Hz)
λ ≈ 0.929 m
The path difference between the two speakers should be half a wavelength, so the top speaker should be moved upward by half of the wavelength.
Path difference = λ / 2
Path difference ≈ 0.929 m / 2 ≈ 0.4645 m
Therefore, the minimum vertical distance upward the top speaker should be moved is approximately 0.4645 meters.
To determine the minimum vertical distance upward that the top speaker needs to be moved to create destructive interference at point O, we need to consider the concept of path difference.
Destructive interference occurs when the waves from the two speakers arrive at point O completely out of phase, meaning the crests of one wave coincide with the troughs of the other wave. This results in the cancellation of the waves at that point.
The path difference between the two waves is equal to an integer multiple of the wavelength. In this case, the wavelength can be calculated using the formula:
wavelength = speed of sound / frequency.
The speed of sound can be assumed to be approximately 343 m/s.
Let's calculate the wavelength:
wavelength = 343 m/s / (3.70 x 102 Hz) = 0.928 m
Now, if we want destructive interference to occur, the path difference between the waves should be equal to an odd multiple of half-wavelengths. So, we can set up the following equation:
path difference = (2n + 1) * (wavelength / 2)
where n is an integer.
In this case, the path difference should be equal to the distance between the two speakers, which is the given value of h = 3.02 m.
(2n + 1) * (wavelength / 2) = 3.02 m
Simplifying the equation:
(2n + 1) * 0.928 m = 3.02 m
Solving for n:
2n + 1 = 3.02 m / 0.928 m
2n + 1 = 3.26
2n = 3.26 - 1
2n = 2.26
n = 2.26 / 2
n = 1.13
As n should be an integer, the value of n becomes 1.
Therefore, the minimum vertical distance upward that the top speaker should be moved to create destructive interference at point O is one half-wavelength.
Substituting the wavelength value, we find:
minimum vertical distance = 0.5 * wavelength
minimum vertical distance = 0.5 * 0.928 m
minimum vertical distance = 0.464 m
So, the top speaker should be moved upward by at least 0.464 meters.