Superposition principle

This article is about the superposition principle in linear systems. For other uses, see Superposition.

Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. In the Physics of Wave propagation (especially Electromagnetic waves, a plane wave (also spelled planewave) is a constant-frequency wave whose A wake is the region of Turbulence immediately to the rear of a solid body caused by the flow of Air or Water around the body For duck as a food see Duck (food; for other meanings see Duck (disambiguation. Linearity holds only approximately in water. The word linear comes from the Latin word linearis, which means created by lines.

In linear algebra, the superposition principle refers to two related ideas in linear systems. Linear algebra is the branch of Mathematics concerned with A linear system is a mathematical model of a System based on the use of a Linear operator.

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. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves For example, in electromagnetic theory, ordinary light is described as a superposition of waves of different frequency and polarization, moving in different directions. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 Frequency is a measure of the number of occurrences of a repeating event per unit Time. Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations

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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and

The superposition principle (in either version) applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. In Mathematics, an algebraic equation over a given field is an Equation of the form P = Q where P and Q In Mathematics, a linear differential equation is a Differential equation of the form Ly = f \ where the Differential In Mathematics simultaneous equations are a set of Equations containing multiple variables The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object which satisfies certain axioms. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum.

1 Applications
2 See also
3 References

Applications

First version of the superposition principle

The phenomenon of interference between waves is based on the superposition principle. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern 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). Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The electromagnetic field is a physical field produced by electrically charged objects. 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 some cases, such as in noise-cancelling headphones, the summed variation has a smaller amplitude than the component variations; this is called destructive interference. Noise-cancelling headphones reduce unwanted ambient Sounds (i Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system

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. The term line array means a speaker system that is made up of a varying number of vertically arranged units which give the effect of a single sound source with the same dimensions

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). A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. Frequency is a measure of the number of occurrences of a repeating event per unit Time. Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations 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. In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series

In quantum mechanics, the superposition principle is applied to attain a general solution of Schrödinger's wave equation, a linear homogeneous differential equation. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system ^[2] The solution is a linear combination of particular solutions—possibly infinitely many. See Quantum superposition. Quantum superposition is the fundamental law of Quantum mechanics.

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 mathematics Fourier analysis is a subject area which grew out of the study of Fourier series In Mathematics, Green's function is a type of function used to solve inhomogeneous Differential equations subject to boundary conditions The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. The impulse response of a system is its output when presented with a very brief input signal an impulse

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. Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of A linear circuit is an Electric circuit in which for a sinusoidal input voltage of Frequency f, any output of the circuit (current through any 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. In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series For another, related technique in circuit analysis, see Superposition theorem. The superposition theorem for electrical circuits states that the total current in any branch of a bilateral linear circuit equals the algebraic sum of the currents

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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges 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. The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time

In mechanical engineering, superposition is used to solve for beam and structure deflections of combined loads when the effects are linear (i. Mechanical Engineering is an Engineering discipline that involves the application of principles of physics for analysis Design, Manufacturing 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]

In hydrogeology, the superposition principle is applied to the drawdown of two or more water wells pumping in an ideal aquifer. Hydrogeology ( hydro- meaning water and -geology meaning the study of the Earth) is the area of Geology that deals with the distribution and In Water -related science and Engineering there are two similar but distinct definitions in use for drawdown. An aquifer is an underground layer of Water -bearing Permeable rock or unconsolidated materials ( Gravel, Sand, Silt, or Clay

In process control, the superposition principle is used in model predictive control. Process control is a Statistics and Engineering discipline that deals with Architectures mechanisms and Algorithms for controlling Model Predictive Control, or MPC is an advanced method of Process control that has been in use in the process industries such as Chemical plants and

Other uses

The superposition principle can be applied to some linear boundary value problems with linear non-zero boundary conditions. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints

The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by linearization. In Mathematics and its applications linearization refers to finding the Linear approximation to a function at a given point

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. Music is an Art form in which the medium is Sound organized in Time. Joseph Schillinger ( August 31, 1895 &ndash March 23, 1943) was a composer music theorist and composition teacher Rhythm (from Greek ῥυθμός - rhythmos, "any measured flow or movement symmetry" is the variation of the length and accentuation of

References

^ The terms first version and second version are non-standard, and in fact unique to this article. The impulse response of a system is its output when presented with a very brief input signal an impulse In Mathematics, Green's function is a type of function used to solve inhomogeneous Differential equations subject to boundary conditions Quantum superposition is the fundamental law of Quantum mechanics. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern
^ Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. Hendrik Anthony Kramers ( Rotterdam, February 2, 1894 &ndash Oegstgeest, April 24, 1952) was a Dutch Physicist 62 ISBN 978-0486667720
^ Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 ISBN 0072520361

Haberman, Richard (2004). Applied Partial Differential Equations. Prentice Hall. ISBN 0-13-065243-1.
Superposition of sound waves

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