Answer:
The fundamental frequency at which the sound of speakers at the microphone produce constructive interference is 801.076458 Hz
Explanation:
For a given arrangement having constructive interference, we have;
R₁ - R₂ = 2·x = 0 + n·λ
The distance from one speaker to the microphone. R₁ = 4.50 m
The distance between the two speakers = 2.00 m
The angle formed between the direction from the microphone to the speaker closest and the directional path between the speakers = 90°
Therefore, by Pythagoras's theorem, the distance from the speaker furthest from the microphone, to the microphone, 'R₂' is given as follows;
R₂ = √(4.50² + 2.00²) = √(24.25) = 4.9244289009 ≈ 4.924
∴ R₂ ≈ 4.9244289 m
R₂ - R₁ = 4.9244289 m - 4.5 m = 0.4244289 m
For constructive interference, R₂ - R₁ =0.4244289 m = n·λ
For n = 1, we have;
R₂ - R₁ =0.4244289 m = n·λ = 1 × λ = λ
λ = 0.4244589 m
f = v/λ = 340 m/sec/(0.4244289 m) ≈ 801.076458 Hz
Therefore the lowest possible fundamental frequency at which the speakers produce constructive interference, f = 801.076458 Hz
The frequency at which the speakers produce a constructive interference is 801.076 Hz.
Given here,
[tex]R_1[/tex]- The distance from one speaker to the first microphone = 4.5 m
[tex]R_2[/tex] - The distance between microphone and second speaker = ?
The distance between the speakers = 2 m
From Pythagorean theorem,
[tex]R_2= \sqrt {4.50^2 + 2.00^2}\\\\R_2 = \sqrt{24.25} \\\\R_2= 4.924[/tex]
So,
[tex]R_2 - R_1 = 4.9244 \, {\rm m} - 4.5 \, {\rm m} \\\\R_2 - R_1 = 0.4244 \, \rm m[/tex]
For constructive interference,
[tex]R_2 - R_1 = m\lambda[/tex]
Since, m = 1
λ = 0.4244 m
Now, frequency can be calculated by the formula,
[tex]f = \dfrac v\lambda[/tex]
So,
[tex]f = \dfrac {340{\rm \ m/sec}}{(0.4244\; \rm m) }\\\\f = 801.076{\rm \ Hz}[/tex]
Therefore, the frequency at which the speakers produce a constructive interference is 801.076 Hz.
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