A soure of sound of frequency 165 hz is placed in front of a wall at a distance 2 m from it. A dtector is also placed in front of the wall at the same distance from it. Find the minimum distance between the source and detector for which maximum sound is recorded int he detector . the speed of sound is 330 m/s

Explanation

The situation is shown in figure. Suppose the detector is placed at a distance of x meter from the source. The direct wave received from the source travels a distance of x meter. The wave reaching the detector after reflection from the wall has travelled a distance of `2 [ (2)^(2) + x^(2)//4]^(1//2)` metre. The path difference between the two waves is
`Delta ={2 [ (2)^(2) + (x^(2))/(4) ]^((1)/(2))-x}`
metre.
Constructive interference will take place when `Delta = lambda , 2 lambda , . . . . . ` The minimum distance x for a maximum corresponds to
`Delta = lambda " ". . . (i)`
Thus wavelength is `lambda = (v)/(f) = (330 m//s)/(165 s^(-1)) = 2`
Thus, by (1) 2 `[ (2)^(2) + (x^(2))/(4)]^(1//2) - x = 2 `
or, `[ 4 + (x^(2))/(4) ]^(1//2) = 1 x + (x)/(2)`
or, `4 + (x^(2))/(4) = 1 + (x^(2))/(4) + x or, 3 = x `
Thus, the detector should be placed at a distance of 3m from the source. Note that there is no abrupt phase change.