Principle of Superposition of Waves

Introduction

Imagine you and your friend are holding a piece of string from both ends. Once the string is stretched tight, both of you start moving it up and down with your hand from your respective ends. We know that plucking of the string would lead to oscillations/harmonic waves. But, since this activity is being carried out from both ends, these waves would pass through each other at some point, i.e., superposition of waves, leading to interference. The concept at play here is the Principle of Superposition of Waves.

Conditions for Superposition of Waves

We can apply the principle of superposition of waves to any set of waves as long as we ensure that the following conditions are met:

• Waves of the same type are superimposed.
• The medium through which the waves travel acts linearly, which means that when a portion of the medium has twice the displacement, it has twice the restoring force. When the amplitudes are tiny, this is frequently the case. For example, little ripples on a pond whose amplitude is far less than their wavelength are suitable for water waves.
• The superposition resembles another wave of the same frequency if the waves are coherent- if they all have the same frequency and a constant phase difference.

Coherence of Waves

Superposition occurs at every point when waves from two sources spread out and cross across a region of space, resulting in interference. If the sources are coherent, the superposition of waves from two sources results in a visible fixed (stationary) interference pattern. This indicates that the sources’ waves have the same frequency and the constant phase difference between them.

This can only be accomplished in the case of electromagnetic waves by dividing waves from a single coherent source, such as a laser (as in Young’s double-slit experiment) or diffracting light from a single bulb to illuminate two slits. This is why we rarely notice light interference in our daily lives.

In a practical setting, coherent sound waves can be generated by connecting two or more speakers to the same signal generator.

Mathematical Derivation 

The Principle of Superposition of Waves says that the resultant displacement of a group of waves in a medium at a given point equals the vector sum of the individual displacements produced by each wave at that point.

To derive the superposition of waves equation, we will consider two waves traveling in opposite directions with a constant phase difference between them. At any given time, we can see waveforms in the string. The algebraic total of the displacements owing to each wave is the net displacement of every element of the string at any given time.

The two traveling waves would be denoted as y1(x,t) and y2(x,t). When these two waves meet and overlap, the resultant displacement is denoted as y(x,t).

So mathematically speaking, 

 y(x,t )= y1(x,t) + y2(x,t)

Now, as per the principle of superposition, the two overlapping waves can be added algebraically to obtain the resultant wave. To move further, let us take the wave functions of the traveling waves as –

y1= f1(x-vt)

y2= f2(x-vt)

.

.

.

.

yn= fn(x-vt)

So, the function of the resultant wave can be given as 

y= f1(x-vt)+f2(x-vt)+f3(x-vt)+………fn(x-vt)

Now, consider a wave traveling along a stretched string with the equation y1(x, t) = A*sin (kx – ωt), and another wave, y2(x, t) = A*sin (kx – ωt +φ), which is moved from the first by a phase φ.

Based on the above equations, we can infer that these waves have the same angular frequency and wavenumber. This means that even their wavelength and amplitude A would be the same.

To find out the displacement, we would apply the principle of superposition to these waves and obtain the following equation

y(x,t) = A*sin (kx – ωt) + A*sin (kx – ωt +φ)

By applying some basic fundamentals of trigonometric equations, we can simplify the above-mentioned equation to the following

y(x,t) = 2*A*cos(φ/2)*sin(kx – ωt +φ/2)

The resulting wave is a sinusoidal wave traveling in the positive X direction, with a phase angle half that of the component waves and an amplitude equal to [2*cosφ/2] times the original waves’ amplitudes.

Types Of Interference

The amplitude of the resultant displacement (i.e., the peak value of displacement as it oscillates through time) is determined by the phase difference between the two waves when two coherent waves are superposed.

Constructive Interference 

Constructive interference occurs when two waves superimpose in the same phase. In mathematical terms, it is observed when the phase difference between two waves is an even multiple of 180 degrees or 𝝅. And when the amplitude of the consequent wave is equal to the total of the amplitudes of the separate waves, which in turn results in the maximum intensity of light. A perfect real-life example of this type of interference is when an observer is placed equidistant from two speakers positioned next to each other. This way, the path difference is zero, and the net sound heard by the user would be of the highest intensity. 

Destructive Interference

Destructive interference occurs when two waves are superimposing each other in the opposite phases, i.e., in antiphase. So, mathematically speaking when the phase difference between two waves is an odd multiple of 180 degrees or 𝝅. When two waves in antiphase collide with each other, the crest of one wave superimposes the trough of another wave. The difference in the individual amplitudes of the waves is equal to the resulting amplitude of the wave that suffers destructive interference.

Mathematical Interpretation of Constructive and Destructive Interference

We know that the intensity of any wave is directly proportional to the square of the amplitude of the resultant wave.

ɪ∝ A2

ɪ= kA2

So, we can say that 

ɪ1=kA12

ɪ2=kA22

Both the wave having phase difference φ

ɪ = ɪ1+ɪ2+2*√ɪ1*√ɪ2  *cosφ

ɪ = kA12 + kA22 + 2*√kA1*√kA2*cosφ

Now, for the case of Constructive Interference.,

                                             ɪ=ɪmax    when cosφ=1

                                                    Where φ=2n𝝅 and n=0,1,2,3…

                                                So, ɪmax = ɪ1+ɪ2+2*√ɪ1*√ɪ2  *cosφ

                                                          Put  φ=2n𝝅 or cosφ=1

                                              This can be further simplified as 

                                                  ɪmax=(√ ɪ1+√ ɪ2)2=k(A1+A2)2

Now, for the case of Destructive Interference

                                                         ɪ=ɪmin     when cosφ=-1

                                                  Where φ=(2n-1)𝝅 and n=0,1,2,3…

                                                  So, ɪmax = ɪ1+ɪ2+2*√ɪ1*√ɪ2  *cosφ

                                                              Put  φ=φ=(2n-1)𝝅 or cosφ=-1

                                                  This can be further simplified as 

                                                            ɪmin=(√ ɪ1-√ ɪ2)2=k(A1-A2)2

Conclusion

In this Article, we understood the fundamentals of the principle of superposition of waves and how it plays an essential role in our day-to-day activities. Then, we went through the superposition of waves equation to understand the concept from a mathematical point of view. Once the equation was derived, we took two possible scenarios and understood the reasoning behind them