Superposition of Waves: Waves are an integral part of our lives. From sound to light, waves are a medium to transfer matter and energy from one place to another. While waves travel through space and other media, they often pass through each other and interact with each other. Imagine you are sitting in a car, and the person in the car nearby sounds the horn.

You will hear the horn’s noise coming directly from the horn, and the horn’s sound reflected from the surroundings. Let us understand how two or more waves behave when they meet or pass through each other. To proceed, we first need to understand the principle of superposition of waves.

Principle of Superposition

For simplicity’s sake, we will use transverse waves to examine the superposition principle in this article. The principle of superposition states that when two or more waves move in the same region of space, the resultant disturbance is equal to the algebraic sum of the individual disturbances due to each wave. Let us understand this with the help of two transverse wave pulses travelling on a string.

When two wave pulses travelling in opposite directions cross each other, they maintain their original identities after they have crossed. Each wave pulse behaves as if the other wave pulse isn’t there. However, when they overlap, the wave pattern formed is significantly different from either of the two travelling waves. This happens as a result of the principle of superposition.

In the above diagram, the displacement of the string will be due to the displacements produced due to each pulse, i.e. the net displacement of the string will be the algebraic sum of the displacements due to each pulse. Since displacement is a vector, it can be positive and negative. Consider the principle of superposition for two wave pulses of the same size but inverted with respect to each other and moving in opposite directions:

We can see that the displacements of the waves will cancel each other at the point where they meet, and after that, the waves continue onwards as if they never crossed paths.

Derivation for the Expression of Superposition of Waves

Let us consider \(y_{1}(x, t)\) and \(y_{2}(x, t)\) be the displacement of the two waves in a given medium. When the waves arrive in a region simultaneously, they will overlap. The net displacement of the disturbance generated in the medium can be calculated using the principle of superposition. Thus, 

\(y(x, t)=y_{1}(x, t)+y_{2}(x, t)\)

When we have two or more waves moving in a given medium, then the equation of the resultant waveform will be the sum of the wavefunctions of the individual waves.

Let the wave functions of the waves moving through the medium be:

\(y_{1}=f_{1}(x-v t)\)

\(y_{2}=f_{2}(x-v t)\)

\(y_{3}=f_{3}(x-v t)\)

…………..

……………

\(y_{n}=f_{n}(x-v t)\)

The wave function that describes the disturbance of the medium can be given as:

\(y=f_{1}(x-v t)+f_{2}(x-v t)+\ldots f_{n}(x-v t)\)

\(y=\sum f_{i}(x-v t)\)

The following phenomena can be explained using the superposition principle:

1. When two waves with the same frequency move with the same speed and opposite directions in a specific medium, they superpose to yield stationary waves.
2. When two waves with the same value of frequency travel through a specific medium with the same speed in the same direction, they superimpose to yield an interference pattern of waves.
3. When two waves with different frequencies travel in a specific medium with the same speed in the same direction, they superimpose to yield beats.

To see the superposition principle in action, we will use it to demonstrate the phenomenon of interference. 

What is Interference?

When two waves with the same frequency travel in the same direction simultaneously, then, due to the principle of superposition of waves, the resultant disturbance at any point in the medium is equal to the sum of disturbances of the two waves. At a given position, the resultant displacement of the two waves may be greater than their individual displacements or lesser.

Consider two transverse waves travelling on a stretched string. Let the angular frequency of both the waves be \(\omega\), their wave number be \(k\), and their wavelength is \(\lambda .\) Thus, the wave speeds of the two waves are equal. Let the amplitudes \(A\) of the two waves be the same, and if the waves are going along the positive direction of the \(x\)-axis, then the two waves only differ in their initial phase by a phase difference \(\phi\). Thus, the equations of the two waves can be written as:

\(y_{1}(x, t)=A \sin (k x-\omega t)\)

\(y_{2}(x, t)=A \sin (k x-\omega t+\phi)\)

Applying the principle of superposition, we get:

\(y(x, t)=y_{1}(x, t)+y_{2}(x, t)=A \sin (k x-\omega t)+A \sin (k x-\omega t+\phi)\)

(using the trigonometric identity: \(\left.\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right)\)

\(y(x, t)=A\left[2 \sin \left[\frac{(k x-\omega t)+(k x-\omega t+\phi)}{2}\right] \cos \left(\frac{\phi}{2}\right)\right]\)

\(y(x, t)=2 A \cos \left(\frac{\phi}{2}\right) \sin \left(k x-\omega t+\frac{\phi}{2}\right)\) —–(1)

This is the equation of a travelling wave moving along the positive \(x-\)axis with the same frequency and wavelength as the individual waves generated on the string. The initial phase of the wave is \(\frac{\phi}{2}\). The amplitude of the given wave, from equation \((1)\)

\(a=2 A \cos \left(\frac{\phi}{2}\right)\)

Thus, the amplitude is the function of the phase difference.

If we substitute \(\phi=0\) into the equation \((1)\), the waves are said to be in phase, and the equation becomes:

\(y(x, t)=2 A \sin (k x-\omega t)\) ——(2)

The amplitude of the resultant wave is maximum, i.e. \(a=2A\)

Constructive interference of waves

Equation \((2)\) represents the constructive interference of the two waves. The amplitude of the waves will add up in the resultant wave. The intensity of the resulting wave is maximum.

If we substitute \(\phi=\pi\) into the equation \((1)\), the waves are said to be out of phase, and the equation becomes:

\(y(x, t)=0\) ——-(3)

The amplitude of the resultant wave is minimum, i.e. \(a=0\)

Destructive interference of waves

Equation \((3)\) represents the destructive interference of the two waves. The amplitude of the waves will get subtracted in the resultant wave. The intensity of the resulting wave is minimum, in this case, zero! This is kind of bizarre, isn’t it? Two waves travelling on a string that produces an undisturbed string! This phenomenon is used in Active Noise Cancellation (ANC) in headphones. Headphones with ANC have an additional mini-speaker that generates waves that destructively interfere with the ambient noise in order to remove that noise.

Summary

The principle of superposition states that when two waves move in the same region of space, the resultant disturbance is equal to the algebraic sum of the individual disturbances due to each wave. Thus, when two wave pulses travelling in opposite directions cross each other, they maintain their original identities after they have crossed. The wave function that describes the disturbance of the medium due to the superposition of several waves can be given as \(y=\sum f_{i}(x-v t)\)

When two waves with the same value of frequency travel in the same direction simultaneously, then due to the principle of superposition of waves, the resultant displacement at any point in the medium is equal to the sum of individual displacements of the two waves. At a given position, the resultant displacement of the two waves may be greater than their individual displacements, or they may be lesser. In the case of constructive interference, the amplitude and hence intensity of the resulting wave will be maximum. In contrast, in the case of destructive interference, the amplitude and intensity of the resulting wave will be minimum.

Frequently Asked Questions

Q.1. Two waves with similar frequencies moving with the same speed and opposite directions in a specific medium are superimposed. What kind of waves are obtained?
Ans: Stationary waves

Q.2. Define the principle of superposition of waves.
Ans: The principle of superposition of waves states that when two waves move in the same region of space, the resultant disturbance of the medium is equal to the algebraic sum of the disturbances due to each wave.

Q.3. Give the condition for constructive interference of the two waves.
Ans: For constructive interference, the phase difference between the two waves should be zero.

Q.4. What is the amplitude of the resulting wave in case of destructive interference?
Ans: The amplitude of the resulting wave is minimum if the individual amplitudes are unequal, zero if they are equal.

Q.5. When two waves with different frequencies travel in a specific medium with the same speed in the same direction. What kind of waves are obtained.
Ans: Beats are formed when two waves with different frequencies travel at the same speed and in the same direction in a specific medium.

We hope this detailed article on the superposition of waves helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!