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In physics and systems theory , the superpositionprinciple [1], also known as superposition property ,

states that, for all linear systems, the net response at agiven place and time caused by two or more stimuli isthe sum of the responses which would have been

caused by each stimulus individually. So that if input  Aproduces response  X and input B produces response Y 

then input ( A + B) produces response ( X + Y ).

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y The superposition principle holds because, by

definition, a linear system must be additive.

Superposition may sometimes imply linearity,

depending on whether homogeneity is included or

implied in the definition of superposition.

y In the field of electrical engineering, where the  x and  y

signals are allowed to be complex-valued (as is

common in signal processing), a linear system must

satisfy the superposition property, which requiresthe system to be additive and homogeneous.[2][3] An

additive system satisfies F ( x1 +  x2) =  F ( x1) +  F ( x2). A

homogeneous system satisfies F (a x) = aF ( x), where a

is a scalar. Often the additivity and homogeneity

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Superposition of 

almost plane

waves (diagonallines) from a

distant source

and waves from

the wake of theducks. Linearity

holds only

approximately

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y This principle has many applications in physics and

engineering because many physical systems can bemodeled as linear systems. For example, a beam can

be modeled as a linear system where the input

stimulus is the load on the beam and the output

response is the deflection of the beam. Becausephysical systems are generally only approximately

linear, the superposition principle is only an

approximation of the true physical behavior; it

provides insight for typical operational regions forthese systems.

y The superposition principle applies to an y linear

system, including algebraic equations, linear

differential equations, and systems of equations of 

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 similar methods

y By writing a very general stimulus (in a linear system)

as the superposition of stimuli of a specific, simple

form, often the response becomes easier to compute,

y

For example, in Fourier analysis, the stimulus iswritten as the superposition of infinitely many

sinusoids. Due to the superposition principle, each of 

these sinusoids can be analyzed separately, and its

individual response can be computed. (The responseis itself a sinusoid, with the same frequency as the

stimulus, but generally a different amplitude and

phase.) According to the superposition principle, the

response to the original stimulus is the sum (or

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yFourier analysis is particularly common

for waves. For example, in

electromagnetic theory, ordinary light isdescribed as a superposition of plane

waves (waves of fixed frequency,

polarization, and direction). As long as

the superposition principle holds (which

is often but not always; see nonlinear

optics), the behavior of any light wave

can be understood as a su er osition of 

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Application to waves

y W aves are usually described by variations in some parameter throughspace and timefor example, height in a water wave, pressure in asound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave, and the wave itself 

is a function specifying the amplitude at each point.y In any system with waves, the waveform at a given time is a function of 

the sources (i.e., external forces, if any, that create or affect the wave)and initial conditions of the system. In many cases (for example, in theclassic wave equation), the equation describing the wave is linear. W hen this is true, the superposition principle can be applied. Thatmeans that the net amplitude caused by two or more waves traversingthe same space, is the sum of the amplitudes which would have beenproduced by the individual waves separately. For example, two wavestraveling towards each other will pass right through each other withoutany distortion on the other side.

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Wave interference

Combined 

wavef orm

wave 1

wave 2

Two waves in phase

Two waves 180° out

of phase

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yThe phenomenon of interference between

waves is based on this idea. When two or

more waves traverse the same space, the

net amplitude at each point is the sum of 

the amplitudes of the individual waves. In

some cases, such as in noise-cancelling

headphones, the summed variation has a

smaller amplitude than the componentvariations; this is called dest ruct ive 

inter  fer ence. In other cases, such as in

Line Array, the summed variation will

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Departures from linearity

yIn most realistic physical situations,the equation governing the wave is

only approximately linear. In these

situations, the superposition principle

only approximately holds. As a rule,

the accuracy of the approximationtends to improve as the amplitude of 

the wave gets smaller. For examples

of phenomena that arise when the

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Quantum superposition

y In quantum mechanics, a principal task is to

compute how a certain type of wave propagates

and behaves. The wave is called a

wavefunction, and the equation governing thebehavior of the wave is called Schrödinger's

wave equation. A primary approach to

computing the behavior of a wavefunction is to

write that wavefunction as a superposition

(called "quantum superposition") of (possibly

infinitely many) other wavefunctions of a

certain typestationary states whose behavior 

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Boundary value problems

y A common type of boundary value problem is (to put it

abstractly) finding a function  y that satisfies some equation

y F ( y) = 0 with some boundary specification

y G( y) =  z For example, in Laplace's equation with Dirichletboundary conditions, F would be the Laplacian operator in a

region R, G would be an operator that restricts  y to the

boundary of  R, and  z would be the function that  y is required to

equal on the boundary of  R.

y In the case that  F and G are both linear operators, then the

superposition principle says that a superposition of solutions to

the first equation is another solution to the first equation:

y while the boundary values superpose:

y G( y1) + G( y2) = G( y1 +  y2) Using these facts, if a list can be