Figure shows two coherent sources `S_(1)` and `S_(2)` which emit sound of wavelength `lambda` in phase. The separation between the sources is `lambda.` 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

From previous question, we can say that for a point P on the circle shown in figure shown. The path difference in the two waves at P is
`triangle= S_1P - S_2 P = d cos theta = 3 lambda cos theta`
We know for constructive interference at P. The path difference must be an integral multiple of wavelength `lambda`. Thus for a maxima at P, we have
`3lambda cos theta = 0 , " " 3 lambda cos theta = lambda ,`
`3 lambda cos theta = 2 lambda , " " 3 lambda cos theta = 3 lambda ,`
or, `theta= (pi)/(2)" or " theta=cos^(-1)"" (1)/(3)`
or, `theta= cos^(-1)""(1)/(3) " or " theta = 0`
There are four points A,B,C and D on circle at which `theta=0 " or " (pi)/(2)` and there are two points in each quadrant at `theta = cos^(-1)""(1)/(3) " and "theta = cos^(-1)""(2)/(3)` at which constructive interference takes place. Thus there are total twelve points on circle at which maxima occurs.