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 firgure 6.34. The Path difference in the two waves at P is
`Delta = S_(1)P - S_(2) P = dcos theta`
= `3 lambda cos theta`
We know for constructive interference at P. The path difference must be an integral multiple ofwavelength `lambda`. Thus for a maxima at P, we have
`3 lambda cos theta = 0, 3lambda cos theta = lambda, 3lambda cos theta = 2lambda, 3 lambda cos theta = 3lambda` or `theta = (pi)/(2)` or `theta^(-1) (1)/(3)` or `theta = (cos^(-1)) (2)/(3)` or `theta = 0`
Therearefourpoints A, B, C and D oncircleat which `theta = 0` or `(pi)/(2)`
and there are two points in each quadrant at `theta = (cos^(-1)) (1)/(3)` and `theta `= `theta = (cos^(-1))(2)/(3)` at which constructive interference takes place. Thus there are total twelve points on circle at which maxima occurs