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:
$\mathrm{P}{S_1}^2=\mathrm{P}{Q}^2+Q{S}^2={\left(R\sin \theta \right)}^2+{\left(R\cos \theta -1.5\lambda \right)}^2$
$\mathrm{P}{S_1}^2=\mathrm{P}{Q}^2+\mathrm{Q}{S_1}^2={\left(R\sin \theta \right)}^2+{\left(R\cos \theta +1.5\lambda \right)}^2$
Path difference between the sound waves reaching point P is given by:
$$\begin{gathered} {\left(S_1\mathrm{P}\right)}^2-{\left(S_2\mathrm{P}\right)}^2=\left[{\left(R\sin \theta \right)}^2+{\left(R\cos \theta +1.5\lambda \right)}^2\right]-\left[{\left(R\sin \theta \right)}^2+{\left(R\cos \theta -1.5\lambda \right)}^2\right] \\ ={\left(1.5\lambda +R\cos \theta \right)}^2-{\left(R\cos \theta -1.5\lambda \right)}^2 \\ =6\lambda R\cos \theta \end{gathered}$$

$$\begin{gathered} \left(S_1\mathrm{P}-S_2\mathrm{P}\right)=\frac{6\mathrm{\lambda }R\cos \theta }{2R} \\ =3\mathrm{\lambda }\cos \theta \end{gathered}$$
Suppose, for constructive interference, this path difference be made equal to the integral multiple of $\lambda$.
Hence,
$$\begin{gathered} \left(S_1\mathrm{P}-S_2\mathrm{P}\right)=3\mathrm{\lambda }\cos \theta =n\mathrm{\lambda } \\ \Rightarrow \cos \theta =\frac{n}{3} \\ \Rightarrow \theta ={\cos }^{-1}\left(\frac{n}{3}\right) \end{gathered}$$
where, n = 0, 1, 2, ...

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