Respuesta :
Answer:
4 m, 1.71 m and 6.29 m
Explanation:
Let L = 8 m be the distance between the two speakers. Let x be the distance from speaker A of constructive interference. The distance to speaker B from the point of constructive interference is thus x₁ = L - x.
There is constructive interference when the distance x₁ - x = nλ where n = is an integer and λ = wavelength L - x
x₁ - x = nλ
L - x - x = nλ
L - 2x = nλ
x = (L - nλ)/2 = (L - nv/f)/2. where v = speed of wave = 343 m/s and f = frequency = 75 Hz
The distance from A where constructive interference would occur starts from when
n = 0
x₂ = (L - nv/f)/2 = (8 - 0 × 343/75)/2 = (8 - 0)/2 = 8/2 = 4 m
n = 1
x₃ = (L - nv/f)/2 = (8 - 1 × 343/75)/2 = (8 - 4.57)/2 = 3.43/2 = 1.71 m
when n = 2
x₄ = (L - nv/f)/2 = (8 - 2 × 343/75)/2 = (8 - 9.14)/2 = -1.15/2 = -0.57 m
So the value at n = 2 is not included.
The third point occurs at x₅ = L - x₃ where x₃ = 1.71 m is the distance away from point B where constructive interference also occurs. (since it is symmetrical about the point x₂ = 4 m
x₅ = L - x₃ = 8 - 1.71 = 6.29 m
Answer:
The distances of the three points from speaker A is 1). 1.713 m, 2). 4 m,
3). 6.287 m.
Explanation:
Here we have
Speed of sound, v = fλ
Where:
f = Frequency of sound and
λ = Wavelength of the sound
Therefore, λ = v/f = [tex]\frac{343 \hspace{0.09cm}m/s}{75.0 \hspace{0.09cm}Hz}[/tex] = 4.573 m
The two speakers are 8.0 m apart
Let X be a point from speaker A on the line where we have constructive interference. Therefore,
(L - X) - X = n·λ
Which gives [tex]X= \frac{L - n\cdot \lambda}{2}[/tex]
Therefore, we have, when n = 0,
[tex]X= \frac{8 - 0\cdot 4.573}{2} = 4 m[/tex]
When n = 1 we have
[tex]X= \frac{8 - 1\cdot 4.573}{2} = 1.71 \hspace {0.09cm} m[/tex], which is the distance from speaker A, since from the nature of the calculation, if we selected X to be from speaker B, then there will be a point of constructive interference at 1.71 m from speaker B
In other words since there is a point of constructive interference at the mid point, we will have constructive interference at λ/2 on either side of the mid point
Therefore, the three points are;
4 - (4.573 m)/2, 4, 4+(4.573 m)
The distances of the three points from speaker A is
1). 1.713 m,
2). 4 m,
3). 6.287 m.
