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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... A linear system is a model of a system based on some kind of linear operator. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

The principle of superposition is widely used in physics and engineering because many physical systems may be modelled as linear systems. The superposition principle is also applied when small deviations from a known solution to a nonlinear system are analyzed by linearization. For linear physical quantities, this implies 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. Consequently, 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. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the Science of Nature. ... Engineering is the application of scientific and technical knowledge to solve human problems. ... Linearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. ... Addition is one of the basic operations of arithmetic. ...

Two important classes of quantities that occur in linear systems are Vector Fields and Time-Varying Signals.

Vector fields

Superposition of almost plane waves from a distant source and waves from the duck and the ducklings. Linearity holds only approximately.

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 occurred._ Image File history File linksMetadata Download high resolution version (2992x2231, 1242 KB) en: Anas platyrhynchos, or Mallard or Wild Duck. ... Image File history File linksMetadata Download high resolution version (2992x2231, 1242 KB) en: Anas platyrhynchos, or Mallard or Wild Duck. ... In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ... Subfamilies Dendrocygninae Oxyurinae Anatinae Merginae For other uses, see Duck (disambiguation). ... The word linear comes from the Latin word linearis, which means created by lines. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... Look up displacement in Wiktionary, the free dictionary. ... A wave is a disturbance that propagates through space, often transferring energy. ... A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ...

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.

Applications

The superposition principle is often applied to physical systems described by differential equations such as the electromagnetic wave equation, the heat equation. Lasers used for visual effects during musical performance. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...

In electrical engineering, the superposition principle is used to solve problems in linear circuit analysis. This article treats electronics engineering as a subfield of electrical engineering, though this is not typical use in some areas. ...

In quantum mechanics, the superposition principle is applied to solutions of Schrödinger's wave equation. For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the space- and time-dependence of quantum mechanical systems. ...

See also

• Impulse response
• Green's function