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Question:
If you are 3.00 m from speaker A directly to your right and 3.50 m from speaker B directly to your left. What is the shortest distance d you need to walk forward to be at a point where you cannot hear the speakers?
Answer:
The shortest distance d you need to walk forward to be at a point where you cannot hear the speakers is 5.625 m
Explanation:
Shortest distance is given by
For constructive interference of wave, we have
= 2nλ/2
For destructive interference of wave, we have
= (2n+1)λ/2
That is the difference in the number of wavelength between constructive and destructive interference is λ/2
We note that our positioning is 3.0 m from the first speaker and 3.5 m from the second, Which is of the form 2n and 2n + 1
Therefore when we walk forward away from the two speakers, we cannot hear the speakers at
Δr = λ/2 = r - r
Which is the distance between the points where we have constructive and destructive interference or where there is destructive interference between the waves
Where:
r = Distance from speaker A and
r = Distance from speaker B
λ/2 = (344 m/s /688 Hz)/2
= 1/4
For
λ/2 = r - r
we have
√(3.5² +d²) - √(3²+d²) = 1/4
∴ d = 5.625 m
