Principle of superposition of waves – Restoring force

Waves encompass us, and their essence attempts to channel various peculiarities. Envision you in a boat and hear the alarm of a boat. In this situation, you’ll have the option to get sound waves straightforwardly from the boat alarm, just as the sound wave that gets reflected by the seawater. To comprehend this idea, let us focus on the central idea of the Superposition of Waves, along with the top to bottom information identified with the superposition theorem.

Superposition of waves

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)

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 travelling waves as –

y1= f1(x-vt)

y2= f2(x-v)

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 travelling along a stretched string with the equation 

y1(x, t) = Asin (kx – ωt), 

and another wave, y2(x, t) = Asin (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. 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) = Asin (kx – ωt) + Asin (kx – ωt +φ)

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

y(x,t) = 2Acos(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 [2cos(φ/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.

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.

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.