superposition

Superposition is the process of combining the impacts of two (or more) waves at the same time and place. This provides us the sum of the two waves’ effects. Although the theory of superposition works for non-material waves, “effect” denotes “displacement” for material waves (such as electromagnetic waves, the pressure interpretation of sound waves, and matter waves). The following is how we express superposition mathematically: 

∆ytotal(x,t)=∆y1(x,t)+∆y2(x,t), where y1and y2 are the only wave 1 and 2 displacements from equilibrium.

∆ ytotal describes the actual displacement of the medium, as seen in the diagram below.

Superposition Principle

The superposition principle states:

1.When two or more waves collide at a spot, the displacement at that point equals the sum of the individual waves’ displacements.

 2. Positive or negative wave displacements are possible. When the displacements are vectors, the sum is calculated by adding them together.

Superposition is a fundamental concept that can be used to describe a variety of phenomena such as interference, diffraction, and standing waves. It works with any sort of wave (sound waves, water surface waves, electromagnetic waves, etc.) but only under particular conditions, which we’ll go over later.

The waves disperse and are unaffected once they travel beyond the point of intersection.

Condition For Superposition 

The principle of superposition can be applied to any type of wave as long as the following conditions are met:

 1 The waves being superposed are of the same type (e.g., all are electromagnetic waves). 

2.The medium through which the waves are propagating behaves linearly, i.e., when part of the medium has twice the displacement, the restoring force is twice as strong. 

3.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 a suitable approximation for waves on water.

4.The superposition resembles another wave of the same frequency if the waves are also coherent, that is, if they all have the same frequency and a constant phase difference. 

Coherence 

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

Constructive and Destructive 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.

 1.There is constructive interference when the two waves are in phase: the resultant amplitude is big, equal to the sum of the two wave amplitudes.

2.There is destructive interference when two waves are in phase: the resulting amplitude is small, equal to the difference between the two wave amplitudes.

The two waves are neither in phase nor antiphase in general. The resultant amplitude is somewhere in between the two extremes

Limit of Superposition

As we know  combining the effects of two waves is the most obvious thing to do, it’s worth pausing to evaluate if that’s the only way we could have done it. In fact, we could have blended the waves in far more complex ways. We can’t use the principle of superposition taught here for very large water or sound waves. Shock waves, such as those produced by explosions or sonic booms, are instances of waves for which the superposition principle fails.

It’s important to remember that the principle of superposition is an experimentally proven fact, not one that can be obtained just from logical reasoning. We are fortunate that the principle of superposition is suitable for “small” waves.

Conclusion 

The response across every element in a linear bilateral network with more than one source is the sum of the responses obtained from each source evaluated separately, while all other sources are replaced by their internal resistances, according to the superposition theorem.