Question

Figure shown two coherent sources S₁ and S₂ which emit sound of wavelength λ in phase. The separation between the sources is 3λ. A circular wire of large radius is placed in such way that S₁,S₂ is at the centre of the wire. Find the angular positions θ on the wire for which constructive interference takes place.

Answer 1

Let the sound waves from the two coherent sources S₁ and S₂ reach the point P.

rework
OQ = R cosθ
OP = R cosθ
OS₂ = OS₁ = 1.5$$\lambda$$

From the figure, we find that:

$$P{S_1}^2=P{Q}^2+Q{S}^2={(R\sin \theta )}^2+{(R\cos \theta -1.5\lambda )}^2$$

$$P{S_1}^2=P{Q}^2+Q{S_1}^2={(R\sin \theta )}^2+{(R\cos \theta +1.5\lambda )}^2$$

Path difference between the sound waves reaching point P is given by:

$${\left(S_{1}P\right)}^2-{\left(S_{2}P\right)}^2=[{(R\sin \theta )}^2+{(R\cos \theta +1.5\lambda )}^2]-[{(R\sin \theta )}^2+{(R\cos \theta -1.5\lambda )}^2]$$ 

$$={(1.5\lambda +R\cos \theta )}^2-{(R\cos \theta -1.5\lambda )}^2$$ 

$$=6\lambda R\cos \theta$$

$$(S_{1}P-S_{2}P)=\frac{6\lambda R\cos \theta }{2R}$$ 

$$=3\lambda \cos \theta$$

Suppose, for constructive interference, this path difference be made equal to the integral multiple of $$\lambda$$ .

Hence ,

$$(S_{1}P-S_{2}P)=3\lambda \cos \theta =n\lambda$$ 

$$\Rightarrow \cos \theta =\frac{n}{3}$$ 

$$\Rightarrow \theta ={\cos }^{-1}\left(\frac{n}{3}\right)$$

where, n = 0, 1, 2, ...

⇒ θ = 0°, 48.2°, 70.5°and 90° are similar points in other quadrants.