Interference and principle of superposition

Introduction

Interference is a phenomenon where two or more waves superimpose to form a resultant wave of greater, lower, or the same amplitude. This typically occurs when two waves from coherent sources (having the same frequency and constant phase difference) overlap. Constructive interference occurs when the waves are in phase, leading to an increase in amplitude. Conversely, destructive interference occurs when the waves are out of phase, reducing the amplitude. Interference is an essential concept in wave optics, explaining phenomena such as the colourful patterns seen in soap bubbles and thin films.

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This Story also Contains

1. Introduction
2. Some Solved Examples
3. Summary

When two waves of the same frequency, the same wavelength, and the same velocity (nearly equal amplitude) move in the same direction, Their superimposition results in interference. Due to interference, the resultant intensity of sound at that point is different from the sum of intensities due to each wave separately. This modification of intensity due to the superposition of two or more waves is called interference.

The displacement at any time due to any number of waves meeting simultaneously at a point in a medium is the vector sum of the individual displacements due to each one of the waves at that point at the same time.

if $y_1, y_2, y_3 \ldots \ldots$ are the displacements at a particular time at a particular position, due to individual waves, then the resultant displacement would be :

$y=y_1+y_2+y_3 \ldots \ldots$

Let at a given point two waves arrive with the phase difference $\phi$ and the equation of these waves are given by $y_1=a_1 \sin (\omega t), y_2=a_2 \sin (\omega t+\phi)$ then by the principle of superposition

$
y=y_1+y_2 \Longrightarrow A \sin (\omega t+\theta)
$

Where $A=\sqrt{a_1^2+a_2^2+2 a_1 a_2 \cos (\phi)}$ and $\tan (\theta)=\frac{a_2 \sin \phi}{a_1+a_2 \cos (\phi)}$

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Some Solved Examples

Example 1. The intensity of sound increases at night due to

1) Increase in density of air

2)Decrease in density of air

3)Low temperature

4)None of these

Solution:

The intensity of the wave, $I=\frac{1}{2} \rho \omega^2 A^2 v$
$\rho_{=\text {density }}$
$\mathrm{w}=$ angular frequency
A=Amplitude
$v=$ Wave speed
$I \alpha \rho$,

We know that at night amount of carbon dioxide in the atmosphere increases which raises the density of the atmosphere. Since intensity is directly proportional to density, the intensity of sound is higher at night.

Hence, the answer is the option (1).

Example 2: Two waves having equations $x_1=a \sin \left(\omega t+\phi_1\right) x_2=a \sin \left(\omega t+\phi_2\right)$,. If in the resultant wave, the frequency and amplitude remain equal to those superimposing waves. The phase difference between them is

1)$\frac{\pi}{6}$
2) $\frac{2 \pi}{3}$
3) $\frac{\pi}{4}$
4) $\frac{\pi}{3}$

Solution
Resultant Intensity - $\square$

$
I=I_1+I_2+2 \sqrt{I_1 I_2} \cdot \cos \phi
$

- wherein

$
\phi=\text { phase difference }
$

The superposition of waves does not alter the frequency of the resultant wave and the resultant amplitude

$
\begin{aligned}
& a^2=a^2+a^2+2 a^2 \cos \Phi=2 a^2(1+\cos \Phi) \\
& \cos \Phi=-1 / 2=\cos 2 \pi / 3 \\
& \therefore 2 \pi / 3
\end{aligned}
$

Hence, the answer is the option 2.

Example 3: Equations of motion in the same direction are given by and $y_2=2 a \sin (\omega t-k x-\theta)$. The amplitude of the medium particle will be

1) $2 a \cos \theta$
2) $\sqrt{2} a \cos \theta$
3) $4 a \cos \frac{\theta}{2}$
4) $\sqrt{2} a \cos \frac{\theta}{2}$

Solution:
Resultant Intensity -

$
I=I_1+I_2+2 \sqrt{I_1 I_2} \cdot \cos \phi
$

- wherein

$
\phi=\text { phase difference }
$

Resultant amplitude

$
A_R=2 a \cos \left(\frac{\Theta}{2}\right)=2 \times 2 a \cos \left(\frac{\Theta}{2}\right)=4 a \cos \left(\frac{\Theta}{2}\right)
$

Hence, the answer is the option 3.

Summary

Interference is a wave phenomenon resulting from the superposition of two or more coherent waves, leading to constructive or destructive patterns based on their phase relationship. Constructive interference increases amplitude, while destructive interference reduces it. Interference explains many natural patterns, such as those in thin films and diffraction gratings. It provides insight into wave behaviour and is used in applications like noise-cancelling headphones and holography. The conditions for interference depend on factors like wavelength, frequency, and phase difference.