Superposition principle

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This article is about the superposition principle in linear systems. For other uses, see Superposition.
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Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. Linearity holds only approximately in water.
Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. Linearity holds only approximately in water.

In linear algebra, the superposition principle refers to two related ideas in linear systems.

The first version[1] of the superposition principle states that, for a linear system of the form F(x)=0 (sometimes called homogeneous), any sum (or more generally, linear combination) of solutions to the system is also a solution to the same system:

IF   \; 0=F(x_1)=F(x_2)=\cdots ,   THEN   \; F(x_1+x_2+\cdots)=0.

An important application of this is to waves, which are often described by wave equations of this form. For example, in electromagnetic theory, ordinary light is described as a superposition of waves of different frequency and polarization, moving in different directions.

The second version of the superposition principle states that, for a linear system of the form F(x)=r, where x is some sort of stimulus and r is some sort of response (or vice-versa), the superposition (i.e., sum) of stimuli yields a superposition of responses:

F(x_1+x_2+\cdots)=F(x_1)+F(x_2)+\cdots.

In other words, in a linear system,

The net response at a given place and time caused by two or more independent stimuli is the sum of the responses which would have been caused by each stimulus individually.

This has many applications in physics and engineering, because many physical systems can be modeled as linear systems.

The superposition principle (in either version) applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object which satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum.

Contents

[edit] Applications

[edit] First version of the superposition principle

  • The phenomenon of interference between waves is based on the superposition principle. Waves are usually described by variations in some parameter through space and time (for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave). The superposition principle says that the net variation in that parameter caused by two or more waves traversing the same space, is the sum of the variations of that parameter which would have been produced by the individual waves separately.
  • In other cases, such as in Line Array, the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference.
  • In the theory of waves, it is often advantageous to write a wave as a superposition of sinusoidal waves, with a constant frequency and direction (and, where applicable, polarization). The behavior of each of the component waves is often simpler to analyze, thus facilitating the computation of how the wave propagates and behaves through time. This is equivalent to Fourier analysis.

[edit] Second version of the superposition principle

  • By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific, more easily analyzable form, often the response becomes easier to compute, using the superposition principle. In this sense, the superposition principle is the underlying basis for Fourier analysis (in which the stimulus is written as the superposition of infinitely many sinusoids), and Green's function analysis (in which the stimulus is written as the superposition of infinitely many impulse functions, and the response is then a superposition of impulse responses).
  • In electrical engineering, in a linear circuit, the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses. The use of Fourier analysis on this basis is particularly common. For another, related technique in circuit analysis, see Superposition theorem.
  • In physics, Maxwell's equations imply that the (possibly time-varying) distributions of charges and currents are related to the electric and magnetic fields by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields which arise from given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the heat equation.
  • In mechanical engineering, superposition is used to solve for beam and structure deflections of combined loads when the effects are linear (i.e., each load does not effect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).[3]

[edit] Other uses

  • The superposition principle can be applied to some linear boundary value problems with linear non-zero boundary conditions.
  • The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by linearization.
  • In music, theorist Joseph Schillinger used a form of the superposition principle as one basis of his Theory of Rhythm in his Schillinger System of Musical Composition.

[edit] See also

[edit] References

  1. ^ The terms first version and second version are non-standard, and in fact unique to this article.
  2. ^ Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. 62 ISBN 978-0486667720
  3. ^ Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 ISBN 0072520361