How Waves Overlap, and Why Common Sense Works! Principle of Superposition for Linear Equations

Why do wave amplitudes add when they overlap? More specifically, why does it make sense to add the displacement of each wave at every point in space and time in order to find the resultant wave? We know that waves of the same type will interfere with each other in this way.

In this video, we will look at the Principle of Superposition. It explains why when two waves overlap, we can simply add their displacements at each point to find the resultant wave. This is what we would find reasonable due to our common sense. But common sense isn't always correct - so why is it accurate here?

To understand this, we need to realize that the wave equation (the classical governing equation for describing all sorts of waves) is linear in wave displacement. In other words, the displacement of a wave (u) only appears as a single factor of u everywhere in the equation. No other powers of u, no functions of u such as ln(u). This linearity ensures that if we take any two known solutions of the wave equation, then adding them together produces yet another solution to the wave equation - in this case the resultant wave. However, if the wave equation was nonlinear (i.e. had powers of u other than 1), then this would not be the case. The sum of two existing solutions would NOT be a solution in itself.

Luckily, the universe seems to behave linearly very often, and any linear system can use the principle of superposition to find solutions that are formed by summing other existing solutions. As a result, wave interference becomes an easy-to-understand topic, but this has a much deeper reason than common sense!

The Principle of Superposition actually has a lot more mathematical detail and rigor behind it. Linear systems that follow the Principle of Superposition can be defined in terms of two properties: the linear functions must be additive and homogeneous. In other words, the function of two variables added together must be equal to the sum of the functions of the two individual variables (additivity), and a constant multiplied by the function of a variable must be equal to the function of the constant multiplied by the variable.

For more information on the Superposition Principle, as well as an idea of other systems where this applies, check out the following link: https://en.wikipedia.org/wiki/Superpo...

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