Thesuperposition principle,^[1] also known assuperposition property, states that, for alllinear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if inputA produces responseX, and inputB produces responseY, then input (A +B) produces response (X +Y).

Afunction${\displaystyle F(x)}$ that satisfies the superposition principle is called alinear function. Superposition can be defined by two simpler properties:additivity$${\displaystyle F(x_1+x_2)=F(x_1)+F(x_2)}$$andhomogeneity$${\displaystyle F(ax)=aF(x)}$$forscalara.

This principle has many applications inphysics andengineering because many physical systems can be modeled as linear systems. For example, abeam can be modeled as a linear system where the input stimulus is theload on the beam and the output response is thedeflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques,frequency-domainlinear transform methods such asFourier andLaplace transforms, andlinear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.

The superposition principle applies toany linear system, includingalgebraic equations,linear differential equations, andsystems of equations of those forms. The stimuli and responses could be numbers, functions, vectors,vector fields, time-varying signals, or any other object that satisfiescertain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as avector sum. If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist).

By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute.

For example, inFourier analysis, the stimulus is written as the superposition of infinitely manysinusoids. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a differentamplitude andphase.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.

As another common example, inGreen's function analysis, the stimulus is written as the superposition of infinitely manyimpulse functions, and the response is then a superposition ofimpulse responses.

Fourier analysis is particularly common forwaves. For example, in electromagnetic theory, ordinarylight is described as a superposition ofplane waves (waves of fixedfrequency,polarization, and direction). As long as the superposition principle holds (which is often but not always; seenonlinear optics), the behavior of any light wave can be understood as a superposition of the behavior of these simplerplane waves.

Waves are usually described by variations in some parameters through space and time—for example, height in a water wave,pressure in a sound wave, or theelectromagnetic field in a light wave. The value of this parameter is called theamplitude of the wave and the wave itself is afunction specifying the amplitude at each point.

In any system with waves, the waveform at a given time is a function of thesources (i.e., external forces, if any, that create or affect the wave) andinitial conditions of the system. In many cases (for example, in the classicwave equation), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.)

With regard to wave superposition,Richard Feynman wrote:^[2]

No-one has ever been able to define the difference betweeninterference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used.

Other authors elaborate:^[3]

The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is [a matter] of degree only, and basically, they are two limiting cases of superposition effects.

Yet another source concurs:^[4]

In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter [Fraunhofer diffraction] is, therefore, a continuation of Chapter 8 [Interference]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront.

The phenomenon ofinterference 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 innoise-canceling headphones, the summed variation has a smalleramplitude than the component variations; this is calleddestructive interference. In other cases, such as in aline array, the summed variation will have a bigger amplitude than any of the components individually; this is calledconstructive interference.

In 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 approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articlesnonlinear optics andnonlinear acoustics.

Inquantum mechanics, a principal task is to compute how a certain type of wavepropagates and behaves. The wave is described by awave function, and the equation governing its behavior is called theSchrödinger equation. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "quantum superposition") of (possibly infinitely many) other wave functions of a certain type—stationary states whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.^[5]

The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is aray inprojective Hilbert space, not avector.According toDirac: "if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state [italics in original]."^[6]However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himselfuses ket vector representations of states to decompose or split,for example, a ket vector${\displaystyle |{\psi }_i⟩}$ into superposition of component ket vectors${\displaystyle |{\varphi }_j⟩}$  as:$${\displaystyle |{\psi }_i⟩=\sum _j{C}_j|{\varphi }_j⟩,}$$ where the${\displaystyle C_j\in \mathbf{\text{C}}}$ .The equivalence class of the${\displaystyle |{\psi }_i⟩}$  allows a well-defined meaning to be given to the relative phases of the${\displaystyle C_j}$ .,^[7] but an absolute (same amountfor all the${\displaystyle C_j}$ ) phase change on the${\displaystyle C_j}$ does not affect the equivalence class of the${\displaystyle |{\psi }_i⟩}$ .

There are exact correspondences between the superposition presented in the main on this page and the quantum superposition.For example, theBloch sphere to representpure state of atwo-level quantum mechanical system(qubit) is also known as thePoincaré sphere representing different types of classicalpurepolarization states.

Nevertheless, on the topic of quantum superposition,Kramers writes: "The principle of [quantum] superposition ... has no analogy in classical physics"^{[citation needed]}.According toDirac: "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory [italics in original]."^[8]Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase,they actually mean phase translation symmetry derived fromtime translation symmetry, which is alsoapplicable to classical states, as shown above with classical polarization states.

A common type of boundary value problem is (to put it abstractly) finding a functiony that satisfies some equation$${\displaystyle F(y)=0}$$ with some boundary specification$${\displaystyle G(y)=z.}$$ For example, inLaplace's equation withDirichlet boundary conditions,F would be theLaplacian operator in a regionR,G would be an operator that restrictsy to the boundary ofR, andz would be the function thaty is required to equal on the boundary ofR.

In the case thatF andG 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:$${\displaystyle F(y_1)=F(y_2)=\cdots =0\Rightarrow F(y_1+y_2+\cdots )=0,}$$ while the boundary values superpose:$${\displaystyle G(y_1)+G(y_2)=G(y_1+y_2).}$$ Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems.

Consider a simple linear system:$${\displaystyle \dot{x}=Ax+B(u_1+u_2),x(0)=x_0.}$$ 

By superposition principle, the system can be decomposed into with$${\displaystyle x=x_1+x_2.}$$ 

Superposition principle is only available for linear systems. However, theadditive state decomposition can be applied to both linear and nonlinear systems. Next, consider a nonlinear system$${\displaystyle \dot{x}=Ax+B(u_1+u_2)+\varphi \left(c^{\mathsf{T}}x\right),x(0)=x_0,}$$ where${\displaystyle \varphi }$  is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into with$${\displaystyle x=x_1+x_2.}$$ 

This decomposition can help to simplify controller design.

• Inelectrical engineering, in alinear 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. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses.
• Inphysics,Maxwell's equations imply that the (possibly time-varying) distributions ofcharges andcurrents are related to theelectric andmagnetic fields by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as theheat equation.
• Inengineering, superposition is used to solve for beam and structuredeflections of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).^[9] Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.^[10]
• Inhydrogeology, the superposition principle is applied to thedrawdown of two or morewater wells pumping in an idealaquifer. This principle is used in theanalytic element method to develop analytical elements capable of being combined in a single model.
• Inprocess control, the superposition principle is used inmodel predictive control.
• The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed bylinearization.

According toLéon Brillouin, the principle of superposition was first stated byDaniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected byLeonhard Euler and then byJoseph Lagrange. Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution.^[11]

Later it became accepted, largely through the work ofJoseph Fourier.^[12]

• Additive state decomposition
• Beat (acoustics)
• Coherence (physics)
• Convolution
• Green's function
• Impulse response
• Interference
• Quantum superposition

•   Media related toSuperposition principle at Wikimedia Commons
•   The dictionary definition ofinterference at Wiktionary