Two point sources of sound are placed at a distance d and a detector moves on a straight line parallel to the line joining the sources as shown in figure. At a distance D away from sources. Initially Detector is situated on the line so that it is equidistant from both the sources. Find the displacement of detector when it detects `n^(th)` maximum sound and also find its displacement when it detects `n^(th)` minimum sound.

Explanation

The situation is shown in figure

Let us consider the situation when detector move by a distance x as shown. Let at this position the path diference between the waves from `S_1" and "S_2` to detector by `triangle` then we have
`triangle= S_2D-S_1D cong S_2Q [" Where "S_1Q " is perpendicular on line "S_2D]`
Here if `theta` is small angle as `D gt gt d`, we have
`S_2D=d sin theta cong d tan theta = d(x)/(D)`
Thus at the position of detector, path deference is
`triangle= (dx)/(D)" ""............."(1)`
The expression for path difference in equation (1) is an important formula for such problems. Students are advised to keep this formula in mind for future use. When detector was at point O, path difference was zero and it detects a maxima, now if detector detects `n^(th)` maximum then its path difference at a distance x from O can be given as
`triangle n lambda " or " (dx)/(D)= n lambda " or "x= (n lambda D)/(d)`
Similarly if detector detects `n^(th)` minima then the path difference between two waves at detector can be given as
`triangle =(2n+1 (lambda)/(2))" or "`
`(dx)/(D)= (2n+1) (lambda)/(2)" or " x= ((2n+1)lambda D)/(2d)`.