Question

Figure shows two coherent sources $S_1$ and $S_2$ which emit sound of wavelength $\lambda$ in phase. The separation between the source is $3\lambda$. A circular wire of large radius is placed in such a way that $S_1{S}_2$ lies in its plane and the midpoint of $S_1{S}_2$ is at the centre of the wire. Find the number of angular positions $\theta (\theta <{90}^{\circ })$ on the wire for which constructive interference takes place.

Answer 1

Clearly, the path difference $\Delta x=S_{1}P-S_{2}P$
From $\Delta S_1{S}_{2}R,$
$\cos \theta =\frac{\Delta x}{3\lambda }-----\left(1\right)$
For constructive interference, path difference, $\Delta x=n\lambda \Rightarrow \Delta x=0,\lambda ,2\lambda ,3\lambda ,4\lambda ....\left(2\right)$

From (1) and (2) and knowledge of trigonometry,
$\cos \theta \le 1---\left(2\right)$

Hence, only four values of $\Delta x=0,{\lambda }_{1}2\lambda ,3\lambda$ are possible.

But, for $\Delta x=3\lambda ,\cos \theta =1\Rightarrow \theta ={90}^{\circ }$ which is not possible (from the data given).

Therefore, we can take only three values of $\Delta x$ which will give three values of $\theta (<{90}^{\circ })$.