superposition principle

Information about superposition principle

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.

Vector fields

Enlarge picture
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.


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.

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

Linear Differential Equations

If and satisfy a linear homogeneous differential equation, then any linear combination of and will also satisfy that equation. Using linear operators, the proof of the principle of superposition is trivial.

See also

Notes

1. ^ H.A. Kramers, p. 62

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. 
The term superposition can have several meanings:

In physics and mathematics it may refer to the overlapping of waves, or to the overlapping of solutions to linear differential equations:
  • The combination of sound or light waves

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Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.
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A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case.
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In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalisations given at the end of the article.
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differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders.
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Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,[1] (later ABET [2]
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wave is a mode of energy transfer from one place to another, often with little or no permanent displacement of the particles of the medium (i.e. little or no associated mass transport); instead there are oscillations around almost fixed positions.
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Light is electromagnetic radiation of a wavelength that is visible to the eye (visible light). In a scientific context, the word "light" is sometimes used to refer to the entire electromagnetic spectrum.
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polarization (Brit., polarisation) is the property of electromagnetic waves, such as light, that describes the direction of the transverse electric field. More generally, the polarization of a transverse wave describes the direction of oscillation in the plane
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quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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Quantum superposition is the application of the superposition principle to quantum mechanics. The superposition principle is the addition of the amplitudes of waves from interference. In quantum mechanics it is the amplitudes of wavefunctions, or state vectors, that add.
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In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential
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vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space.

Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction
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In physics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. The vector directs from the reference point to the current position.
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wave is a mode of energy transfer from one place to another, often with little or no permanent displacement of the particles of the medium (i.e. little or no associated mass transport); instead there are oscillations around almost fixed positions.
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Interference is the addition (superposition) of two or more waves that results in a new wave pattern.

As most commonly used, the term interference usually refers to the interaction of waves which are correlated or coherent with each other, either because they
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amplitude is a nonnegative scalar measure of a wave's magnitude of oscillation, that is, the magnitude of the maximum disturbance in the medium during one wave cycle.

Sometimes this distance is called the peak amplitude
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The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the electric field E
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The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time.

General-audience description

Suppose one has a function u which describes the temperature at a given location (x,
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boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equations which also satisfies the boundary conditions.
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Electrical engineering (sometimes referred to as electrical and electronic engineering) is an engineering field that deals with the study and/or application of electricity, electronics and electromagnetism.
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A linear circuit is an electric circuit in which, for a sinusoidal input of frequency f, any output of the circuit is also sinusoidal with frequency f. Note that the output need not be in phase with the input.
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quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to
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Hydrogeology (hydro- meaning water, and -geology meaning the study of the Earth) is the part of hydrology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust, (commonly in aquifers).
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drawdown.
  • In subsurface hydrogeology, drawdown is the change in hydraulic head observed at a well in an aquifer, typically due to pumping a well as part of an aquifer test or well test.

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A water well is an artificial excavation or structure put down by any method such as digging, driving, boring, or drilling for the
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An aquifer is an underground layer of water-bearing permeable rock or unconsolidated materials (gravel, sand, silt, or clay) from which groundwater can be usefully extracted using a water well.
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