
The superposition principle, also known as superposition property, states that, for all
linear 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 input ''A'' produces response ''X'' and input ''B'' produces response ''Y'' then input (''A'' + ''B'') produces response (''X'' + ''Y'').
A
function that satisfies the superposition principle is called a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
. Superposition can be defined by two simpler properties:
additivity
and
homogeneity
for
scalar .
This principle has many applications in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
because many physical systems can be modeled 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. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques,
frequency domain linear transform methods such as
Fourier and
Laplace transforms, and
linear 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 to ''any'' linear system, including
algebraic equations,
linear differential equations, and
systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors,
vector fields, time-varying signals, or any other object that satisfies
certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a
vector 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).
Etymology
The word superposition is derived from the
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...
word "super", which means above, and the word "position", which means place.
Relation to Fourier analysis and similar methods
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, in
Fourier analysis, the stimulus is written 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 response is itself a sinusoid, with the same frequency as the stimulus, but generally a different
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...
and
phase.) 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, in
Green's function analysis, the stimulus is written as the superposition of infinitely many
impulse functions, and the response is then a superposition of
impulse responses.
Fourier analysis is particularly common for
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...
s. For example, in electromagnetic theory, ordinary
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
is described as a superposition of
plane waves (waves of fixed
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
,
polarization
Polarization or polarisation may refer to:
Mathematics
*Polarization of an Abelian variety, in the mathematics of complex manifolds
*Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
, 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 superposition of the behavior of these simpler
plane waves.
Wave superposition

Waves are usually described by variations in some parameters through space and time—for example, height in a water wave,
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
in a sound wave, or the
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classica ...
in a light wave. The value of this parameter is called the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...
of the wave and the wave itself is a
function specifying the amplitude at each point.
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 the classic
wave 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.)
Wave diffraction vs. wave interference
With regard to wave superposition,
Richard Feynman wrote:
Other authors elaborate:
Yet another source concurs:
Wave interference
The 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-canceling headphones, the summed variation has a smaller
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...
than the component variations; this is called ''destructive interference''. In other cases, such as in a
line array, the summed variation will have a bigger amplitude than any of the components individually; this is called ''constructive interference''.
Departures from linearity
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 articles
nonlinear optics and
nonlinear acoustics
Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dynamics (for sound waves in liquids and gases ...
.
Quantum superposition
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
, a principal task is to compute how a certain type of wave
propagates and behaves. The wave is described by a
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
, and the equation governing its behavior is called the
Schrödinger 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.
[Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. 62 ]
The projective nature of quantum-mechanical-state space makes an important difference: it does not permit superposition of the kind that is the topic of the present article. A quantum mechanical state is a ''ray'' in
projective Hilbert space, not a ''vector''. The sum of two rays is undefined. To obtain the relative phase, we must decompose or split the ray into components
where the
and the
belongs to an orthonormal basis set. The equivalence class of
allows a well-defined meaning to be given to the relative phases of the
.
There are some likenesses between the superposition presented in the main on this page and the quantum superposition. Nevertheless, on the topic of quantum superposition,
Kramers writes: "The principle of
uantumsuperposition ... has no analogy in classical physics." According to
Dirac: "''the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory''
talics in original"
Boundary value problems
A common type of boundary value problem is (to put it abstractly) finding a function ''y'' that satisfies some equation
with some boundary specification
For example, in
Laplace's equation with
Dirichlet boundary 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''.
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:
while the boundary values superpose:
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.
Additive state decomposition
Consider a simple linear system:
By superposition principle, the system can be decomposed into
with
Superposition principle is only available for linear systems. However, the
Additive state decomposition can be applied not only to linear systems but also nonlinear systems. Next, consider a nonlinear system
where
is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into
with
This decomposition can help to simplify controller design.
Other example applications
* 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. 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. For another, a related technique in circuit analysis, see
Superposition theorem.
* In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
,
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
Th ...
imply that the (possibly time-varying) distributions of
charges
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* ''Charge!!'', an album by The Aqu ...
and
currents are related to the
electric and
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
s 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 the
heat equation.
* In
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, superposition is used to solve for beam and structure
deflections 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). Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.
* In
hydrogeology
Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rock (geology), rocks of the Earth's crust (ge ...
, the superposition principle is applied to the
drawdown of two or more
water wells pumping in an ideal
aquifer. This principle is used in the
analytic element method The analytic element method (AEM) is a numerical method used for the solution of partial differential equations. It was initially developed by O.D.L. Strack at the University of Minnesota. It is similar in nature to the boundary element method (B ...
to develop analytical elements capable of being combined in a single model.
* In
process control, the superposition principle is used in
model predictive control.
* The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by
linearization
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linea ...
.
* In
music
Music is generally defined as the The arts, art of arranging sound to create some combination of Musical form, form, harmony, melody, rhythm or otherwise Musical expression, expressive content. Exact definition of music, definitions of mu ...
, theorist
Joseph Schillinger used a form of the superposition principle as one basis of his ''Theory of
Rhythm
Rhythm (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed ...
'' in his ''
Schillinger System of Musical Composition
The Schillinger System of Musical Composition, named after Joseph Schillinger (1895–1943) is a method of musical composition based on mathematical processes. It comprises theories of rhythm, harmony, melody, counterpoint, form and semantics, pur ...
''.
* In computing, superposition of multiple code paths, code and data, or multiple data structures is sometimes seen in
shared memory,
fat binaries, as well as
overlapping instructions in highly optimized
self-modifying code and
executable text.
History
According to
Léon Brillouin, the principle of superposition was first stated by
Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and then by
Joseph 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.
Later it became accepted, largely through the work of
Joseph Fourier.
[ Brillouin, L. (1946). ''Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices'', McGraw–Hill, New York, p. 2.]
See also
*
Additive state decomposition
*
Beat (acoustics)
*
Coherence (physics)
*
Convolution
*
Green's function
*
Impulse response
*
Interference
*
Quantum superposition
References
Further reading
*
Superposition of sound waves
External links
*
*
{{authority control
Mathematical physics
Waves
Systems theory