Superposition principle
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- This article is about the superposition principle in linear systems. For other uses, see Superposition
In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. The superposition principle applies to linear systems of algebraic equations, linear differential equations, or systems of linear differential equations. Two important classes of quantities that occur in linear systems are Vector Fields and Time-Varying Signals.
The principle of superposition is widely used in physics and engineering because many physical systems may be modeled as linear systems. For linear physical quantities, this means that
- The net result at a given place and time caused by two or more independent phenomena is the sum of the results which would have been caused by each phenomenon individually,
as commonly happens for waves. Thus, in electromagnetic theory, ordinary light is described as a superposition of waves of different length and polarization, moving in different directions; in quantum mechanics, the state of a system is modeled by a wave and can be expressed as a quantum superposition of various eigenstates.
Sometimes, it is possible to analyze the behavior of linear physical systems by considering the behavior of each component of the system separately, and then summing the separate results to find the total result. The superposition principle is also applied when small deviations from a known solution to a nonlinear system are analyzed by linearization.
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[edit] Vector fields
For vector fields, the principle of superposition states that the net displacement at a given place and time caused by two or more waves traversing the same space is the vector sum of the displacements which would have been produced by the individual waves separately. If the resultant sum is greater than either (displacement of an) individual wave, the event occurring when the waves meet is called constructive interference, and amplitude at that point is increased. When the resultant sum is less than either displacement, then destructive interference occurs, and overall amplitude decreases. If the superposition of waves brings the amplitude to zero, complete destructive interference has no answer.
[edit] Time-varying signals
For time-varying signals, the principle of superposition states that the total response at a given place and time caused by two or more signals propagating in the same space is the sum of the separate responses which would have been produced by the individual signals.
[edit] Applications
- The superposition principle is often applied to physical systems described by differential equations such as the electromagnetic wave equation, the heat equation.
- The superposition principle can be applied to some linear boundary value problems with linear non-zero boundary conditions.
- In electrical engineering, the superposition principle is used to solve problems in linear circuit analysis.
- In quantum mechanics, the superposition principle is applied to attain a general solution of Schrödinger's wave equation, a linear homogeneous differential equation[1]. The solution is a linear combination of particular solutions - possibly infinitely many.
- In hydrogeology, the superposition principle is applied to the drawdown of two or more water wells pumping in an ideal aquifer.
- In Music, theorist Joseph Schillinger used a form of the superposition principle as one of the basis’ of his "Theory of Rhythm" in his Schillinger System of Musical Composition.
[edit] Linear Differential Equations
If u and v satisfy a linear homogeneous differential equation, then any linear combination of u and v will also satisfy that equation. Using linear operators, the proof of the principle of superposition is trivial.
[edit] See also
[edit] Notes
- ^ H.A. Kramers, p. 62
[edit] References
- Haberman, Richard (2004). Applied Partial Differential Equations. Prentice Hall. ISBN 0-13-065243-1.
- Kramers, H.A. (1957). Quantum Mechanics. Dover. ISBN 978-0486667720.
