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.

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Mathematical derivation of superposition of waves

To derive the superposition of waves equation, we will consider two waves travelling in opposite directions with a constant phase difference. 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 travelling 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)

As per the principle of superposition, now the algebraic addition of the two overlapping waves is done to obtain the resultant wave. To move further, let us take the wave functions of the travelling waves as –

y₁= f₁(x-vt)

y₂= f₂(x-vt)

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y_n= f_n(x-vt)

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

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

Now, consider a wave travelling 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.

We can infer that these waves have the same angular frequency and wavenumber based on the above equations. It 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 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 travelling in the positive X direction, with a phase angle half that of the component waves and an amplitude equal to [2*cos(φ/2)] times the amplitudes of the original waves.

Types Of Interference

Destructive Interference

Destructive Interference occurs when waves meet up to offset one another.

There are many intriguing wave peculiarities regarding nature that a singular wave can’t characterise. Moreover, to understand the damaging impedance peculiarity, we should examine the dependence on the mix of waves. To inspect these, we apply the rule of superposition which says:

“In the event that at least two waves are going in a medium, the subsequent wave work is the logarithmic all out of the singular wave’s work.”

Constructive Interference

When two waves travel a similar way and are in sync with one another, their sufficiency gets added, and the resultant wave is acquired. Here, the waves are said to have gone through a valuable impedance.

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:

1. Waves of the same type are superimposed.
2. 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.
3. 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.

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 their reasoning.