What Is the Principle of Superposition?

Meg Higa

In physics and engineering, the principle of superposition is the additive property of any linear function or system. Given the calculated or measured outcome of an input variable, if one or more separate, additional variables are simultaneously applied, the resulting net outcome will equal the addition of each variable’s respective individual outcomes. Simply stated, its basic concept can be expressed as follows: if input A results in output X, and input B’s output is Y, then the superposition of both input A+B will result in the corresponding output X+Y. One of the reasons for the term “superposition” is that the principle applies to a specific place and time. Given the changing state of active systems, superimposed input and output are positional events and measurements.

The principle of superposition can be applied to linear mathematical functions, such as quadratic equations.
The principle of superposition can be applied to linear mathematical functions, such as quadratic equations.

The principle of superposition can be applied to linear mathematical functions, such as algebraic equations. When any of the input variables are affected by scalars, such as with the constant coefficients of math’s quadratic equations, the function is said to be both linear and homogeneous. For the example above, if known scalars 1 and 2 are applied to the input variables 1A+2B, superposition carries over to the output 1X+2Y. The combined output is often called the sum.

Many mechanical and electrical products, systems and processes are designed to be linear. If a knob is turned clockwise, volume correspondingly increases. In all but the simplest of devices however, most systems are complex and affected by many variables. They are rarely, absolutely linear. While the principle of superposition is a convenient and helpful tool for modeling and analyzing systems, it is regarded as only an optimal approximation of real life operational conditions.

Among the linear systems to have most benefited from application of the superposition principle are those which employ wave energy. Sound, light and other electromagnetic radiation waves also have strongly additive properties. The form of a wave itself can be described as a linear equation. According to the principle, two or more waves of a particular height or amplitude occupying the same space and time will transform into a single wave whose amplitude is the sum of its original constituent waves’ amplitudes. Similarly, light of the wavelength for the color red when superimposed with that of green will be additively transformed to a wavelength corresponding to the color yellow.

This principle of superposition is the underlying technology of noise-cancellation headphones. A microphone analyzes the waveform of ambient sound, such as the low rumble of an airplane engine. A speaker recreates the same waveform, and prior to adding this sound to the system, it is shifted in temporal phase. When the amplitude of the engine’s sound wave crests at a representative value of 1, it coincides with the added sound’s trough, the equivalent value of -1. Their sum effect is zero.

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