A source emitting sound of frequency `180 Hz` is placed in front of a wall at distance of `2 m` from it . A detector is also placed in front of the wall at the same distance from it. Find the minimum distance between the source and the detector for which the detector detects a maximum of sound. Speed of sound in air `= 360 m//s`.

Options

A. `3 m`
B. `2 m`
C. `3//2 m`
D. zero

Answer

A

Explanation

The situation is shown in Fig. S
Suppose the detector is placed at a distance of `x metre` from the source . The wave received from the source travels a distance `x metre`. The wave reaching the detector after reflection from the wall has travelled a distance of ` 2 [ sqrt((2)^(2) + x^(2)//4)] metre`. Thus the path difference between the two waves at detector is
`Delta = {2 sqrt((2)^(2) + (x^(2))/(4) - x)}` (i)
Constructive interference will take place when `Delta = lambda , 2 lambda,...`
Thus the minimum distance `x` for which a maximum occurs at detector , the path difference will be
`Delta = lambda` (ii)
The wavelength is
`lambda = (v)/(n) = (360)/(180) = 2 m`
`Delta = 2 sqrt(2^(2) + (x^(2))/(4)) - x = 2`
` sqrt( 4 + (x^(2))/(4)) = 1 + (x^(2))/(4) + x)`
` x = 3 m`
Thus , the detector should be placed at a distance of `3 cm` from the source to detect a maximum sound.