Figure shows two coherent sources `S_(1)` and `S_(2)` which emit sound of wavelength `lambda` in phase. The separation between the sources is `3lambda.` A circular wire of large radius is placed in such a way that `S_(1)S_(2)` lies in its plane and the middle point of `S_(1)S_(2)` is at the centre of the wire. Find the angular positions `theta,` on the wire for which constructive interference takes place.

Explanation

Path difference:
`Delta x = S_(1)P - S_(2)P = 3 lambda cos theta`
For maxima:
`Delta x = n lambda rArr 3 lambda cos theta = n lambda`
`cos theta = (n)/(3)`
`n = 0, theta = 90^(@)`
`n = 1, theta = cos^(-1)((1)/(3))`
`n = 2, theta = cos^(-1)((2)/(3))`
`n = 3, theta = 0^(@)`
In quarter circle, at four places intensity will be maximum. In entire circle, intensity will be maximum at `12` places.
Note:

If `R gt gt d,`
Path difference:
`Delta x = S_(1)P - S_(2)P = d cos theta`