Name: Open and Closed-End Air Column
Uploaded: 2019-07-19
Duration: 4 min 32 s
Description: Many musical instruments consist of an air column enclosed inside of a hollow metal tube. We will learn about the open- and closed-end air column in this video.

Question

Give the analytical treatment of interference and establish the conditions for constructive and destructive interferences.

Accepted Answer

Consider ‘ $S$ ’ be the source of monochromatic light wave $S_{1}$ and $S_{2}$ are the two coherent sources emitted from single point. Initially, the wave emitted from $S_{1}$ and $S_{2}$ are in phase but when they meet at point where they are not in phase. Let be the phase difference of wave emitting from both sources and meeting at a point and $y_{1}$ and $y_{2}$ be the displacement of wave emitting from $S_{1}$ and $S_{2}$ then

Let there be a wave from a source which is given by

$y_{1} = a_{1} \sin (ω t - k x)$

Let another wave is reaching the same point which is given by

$y_{2} = a_{2} \sin (ω t - k x + ϕ)$

According to the superposition principle,

Resultant is given by $y = y_{1} + y_{2}$

$\Rightarrow y = a_{1} \sin (ω t - k x) + a_{2} \sin (ω t - k x + ϕ)$

Using the identity $\sin (A + B) = \sin A \cos B + \cos A \sin B$

$\Rightarrow y = a_{1} \sin (ω t - k x) + a_{2} \sin (ω t - k x) \cos ϕ + a_{2} \cos (ω t - k x) \sin ϕ$

$\Rightarrow y = (a_{1} + a_{2} \cos ϕ) \sin (ω t - k x) + a_{2} \cos (ω t - k x) \sin ϕ$

Let $R$ be a quantity(amplitude) such that

$R \cos θ = (a_{1} + a_{2} \cos ϕ) . . . . . . (i)$ and

$R \sin θ = a_{2} \sin ϕ . . . . . . (i i)$

$\Rightarrow y = R \cos θ . \sin (ω t - k x) + R \sin θ . \cos (ω t - k x)$

$\Rightarrow y = R \sin (ω t - k x + θ)$

This is the resultant wave equation.

Squaring and adding equation (i) and (ii) we get,

$R^{2} \cos^{2} θ + R^{2} \sin^{2} θ = {(a_{1} + a_{2} \cos ϕ)}^{2} + (a_{2} \sin ϕ)^{2} \Rightarrow R^{2} (\cos^{2} θ + \sin^{2} θ) = {a_{1}}^{2} + {a_{2}}^{2} \cos^{2} ϕ + 2 a_{1} a_{2} \cos ϕ + {a_{2}}^{2} s i n^{2} ϕ \Rightarrow R^{2} = {a_{1}}^{2} + {a_{2}}^{2} (\cos^{2} ϕ + s i n^{2} ϕ) + 2 a_{1} a_{2} \cos ϕ \Rightarrow R^{2} = {a_{1}}^{2} + {a_{2}}^{2} + 2 a_{1} a_{2} \cos ϕ$

The intensity is maximum when $\cos ϕ = 1$

or $ϕ = 0, 2 π, 4 π, 6 π . . . . . . . . . . 2 n π (w h e r e n = 0, 1, 2)$

$\Rightarrow I_{m a x} = {(a_{1} + a_{2})}^{2}$

For $a_{1} = a_{2} = a$

$I_{m a x} = 4 a^{2}$

Intensity is minimum when $\cos ϕ = - 1$

or $ϕ = π, 3 π, 5 π, 7 π . . . . . . . . . . (2 n + 1) π (w h e r e n = 0, 1, 2)$

$\Rightarrow I_{m i n} = {(a_{1} - a_{2})}^{2}$

$a_{1} = a_{2} = a$

$\Rightarrow I_{m i n} = 0$

The condition for constructive interference is that the phase difference between the two waves should be an even integral multiple of $π$ or $180^{\circ}$ . For destructive interference, the phase difference between the two waves is an odd integral multiple of $π$ or $180^{\circ}$ .

Give the analytical treatment of interference and establish the conditions for constructive and destructive interferences.

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