# 叠加原理

{{简述}说明线性系统的物理解是线性的基本物理原理}}

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 and only for waves with small amplitudes relative to their wavelengths.

[[文件：Anas platyrhynchos小鸭在倒影水.jpg|拇指|右|来自远源的几乎平面波s（对角线）和来自s的尾迹的波的叠加。线性仅在水中近似成立，并且仅适用于相对于其波长振幅较小的波。]]

Rolling motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: translation without rotation, and rotation without translation.

[[文件：Rolling animation.gif|右|拇指|滚动运动是两个运动的叠加。车轮的滚动运动可以描述为两个独立运动的组合：平移旋转和旋转无平移。]]

Rolling motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: translation without rotation, and rotation without translation.


The superposition principle,[1] 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).

“‘叠加原理’”，[2]也称为“‘叠加性质’”，指出，对于所有的线性系统，两个或多个刺激引起的净反应是每个刺激单独引起的反应的总和。因此，如果输入“A”产生响应“X”，输入“B”产生响应“Y”，则输入（“A”+“B”）产生响应（“X”+“Y”）。

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 $\displaystyle{ F(x) }$ that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties; additivity and homogeneity

A function $\displaystyle{ F(x) }$ that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties; additivity and homogeneity

$\displaystyle{ F(x_1+x_2)=F(x_1)+F(x_2) \, }$ Additivity

$\displaystyle{ F(x_1+x_2)=F(x_1)+F(x_2) \, }$Additivity

< math > f (x _ 1 + x _ 2) = f (x _ 1) + f (x _ 2) ，</math > 可加性

$\displaystyle{ F(a x)=a F(x) \, }$ Homogeneity

$\displaystyle{ F(a x)=a F(x) \, }$Homogeneity

$\displaystyle{ F(a x)=a F(x) \, }$Homogeneity

for scalar a.

for scalar .

This principle has many applications in physics and engineering 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, 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 behaviour.

This principle has many applications in physics and engineering 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, 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 behaviour.

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.

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.

## Relation to Fourier analysis and similar methods与傅里叶分析及类似方法的关系

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.

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 and phase.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.

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 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.

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 waves. For example, in electromagnetic theory, ordinary light is described 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 superposition of the behavior of these simpler plane waves.

Fourier analysis is particularly common for waves. For example, in electromagnetic theory, ordinary light is described 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 superposition of the behavior of these simpler plane waves.

## Wave superposition波叠加

Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a standing wave.

[[文件：驻波2.gif |拇指|右|在同一介质中以相反方向传播的两个波线性组合。在这个动画中，两个波的波长相同，振幅之和产生驻波.]] <！--见下文！-->

Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a standing wave.

two waves permeate without influencing each other

two waves permeate without influencing each other

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. 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.

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. 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.

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 top.)

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 top.)

### Wave diffraction vs. wave interference模板:Anchor模板:Anchor波衍射与波干涉模板:锚定模板:锚定

With regard to wave superposition, Richard Feynman wrote:[3]

With regard to wave superposition, Richard Feynman wrote:

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Other authors elaborate:[5]

Other authors elaborate:

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Yet another source concurs:[7]

Yet another source concurs:

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### Wave interference波的干涉

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-cancelling headphones, the summed variation has a smaller amplitude 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.

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-cancelling headphones, the summed variation has a smaller amplitude 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.

green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.

green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.

 combined waveform combined waveform 组合 < br > 波形 文件:Interference of two waves.svg File:Interference of two waves.svg 文件: 两个 waves.svg 的干涉 wave 1 wave 1 第一波 wave 2 wave 2 第二波 < br > Two waves in phase Two waves in phase 两个波同相位 Two waves 180° out of phase Two waves 180° out of phase 两波相位差为180 °

### Departures from linearity偏离线性

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.

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.

### Quantum superposition态叠加原理

In quantum mechanics, a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function, 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.[8]

In quantum mechanics, a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function, 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.

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

$\displaystyle{ |\psi_i\rangle = \sum_{j}{C_j}|\phi_j\rangle, }$

$\displaystyle{ |\psi_i\rangle = \sum_{j}{C_j}|\phi_j\rangle, }$

where the $\displaystyle{ C_j\in \textbf{C} }$ and the $\displaystyle{ |\phi_j\rangle }$ belongs to an orthonormal basis set. The equivalence class of $\displaystyle{ |\psi_i\rangle }$ allows a well-defined meaning to be given to the relative phases of the $\displaystyle{ C_j }$.[9]

where the $\displaystyle{ C_j\in \textbf{C} }$ and the $\displaystyle{ |\phi_j\rangle }$ belongs to an orthonormal basis set. The equivalence class of $\displaystyle{ |\psi_i\rangle }$ allows a well-defined meaning to be given to the relative phases of the $\displaystyle{ C_j }$.

There are some likenesses between the superposition presented in the main on this page, and quantum superposition. Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of [quantum] superposition ... 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 [italics in original]."[11]

There are some likenesses between the superposition presented in the main on this page, and quantum superposition. Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of [quantum] superposition ... 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 [italics in original]."

## Boundary value problems边值问题

A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation

A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation

$\displaystyle{ F(y)=0 }$

$\displaystyle{ F(y)=0 }$

with some boundary specification

with some boundary specification

$\displaystyle{ G(y)=z }$

$\displaystyle{ G(y)=z }$

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.

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:

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:

$\displaystyle{ F(y_1)=F(y_2)=\cdots=0\ \Rightarrow\ F(y_1+y_2+\cdots)=0 }$

$\displaystyle{ F(y_1)=F(y_2)=\cdots=0\ \Rightarrow\ F(y_1+y_2+\cdots)=0 }$

< math > f (y _ 1) = f (y _ 2) = cdots = 0 right tarrow f (y _ 1 + y _ 2 + cdots) = 0 </math >

while the boundary values superpose:

while the boundary values superpose:

$\displaystyle{ G(y_1)+G(y_2) = G(y_1+y_2) }$

$\displaystyle{ G(y_1)+G(y_2) = G(y_1+y_2) }$

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.

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 :

Consider a simple linear system :

$\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2}), x(0)=x_{0}. }$

$\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2}), x(0)=x_{0}. }$

By superposition principle, the system can be decomposed into

By superposition principle, the system can be decomposed into

$\displaystyle{ \dot{x}_{1}=Ax_{1}+Bu_{1}, x_{1}(0)=x_{0}. }$

$\displaystyle{ \dot{x}_{1}=Ax_{1}+Bu_{1}, x_{1}(0)=x_{0}. }$

$\displaystyle{ \dot{x}_{2}=Ax_{2}+Bu_{2}, x_{2}(0)=0. }$

$\displaystyle{ \dot{x}_{2}=Ax_{2}+Bu_{2}, x_{2}(0)=0. }$

with

with

$\displaystyle{ x=x_{1}+x_{2}. }$

$\displaystyle{ x=x_{1}+x_{2}. }$

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

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

$\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2})+ \phi (c^Tx), x(0)=x_{0}. }$

$\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2})+ \phi (c^Tx), x(0)=x_{0}. }$

$\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2})+ \phi (c^Tx), x(0)=x_{0}. }$

where $\displaystyle{ \phi }$ is a nonlinear function. By the additive state decomposition, the system can be ‘additively’ decomposed into

where $\displaystyle{ \phi }$ is a nonlinear function. By the additive state decomposition, the system can be ‘additively’ decomposed into

$\displaystyle{ \dot{x}_{1}=Ax_{1}+Bu_{1}+ \phi (y_{d}), x_{1}(0)=x_{0}. }$

$\displaystyle{ \dot{x}_{1}=Ax_{1}+Bu_{1}+ \phi (y_{d}), x_{1}(0)=x_{0}. }$

$\displaystyle{ \dot{x}_{2}=Ax_{2}+Bu_{2}+ \phi (c^Tx_{1}+c^Tx_{2})- \phi (y_{d}), x_{2}(0)=0. }$

$\displaystyle{ \dot{x}_{2}=Ax_{2}+Bu_{2}+ \phi (c^Tx_{1}+c^Tx_{2})- \phi (y_{d}), x_{2}(0)=0. }$

with

with

$\displaystyle{ x=x_{1}+x_{2}. }$

$\displaystyle{ x=x_{1}+x_{2}. }$

This decomposition can help to simplify controller design.

This decomposition can help to simplify controller design.

## Other example applications其他示例应用程序

• 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, related technique in circuit analysis, see Superposition theorem.
• 电气工程中，在线性电路中，输入（施加的时变电压信号）通过线性变换与输出（电路中任何位置的电流或电压）相关。因此，输入信号的叠加（即总和）将产生响应的叠加。在此基础上使用Fourier analysis尤其常见。另一方面，电路分析中的相关技术，参见叠加定理
• 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. Thus, the superposition principle can be used to simplify the computation of fields which 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.
• 物理学中，麦克斯韦方程暗示电荷电流的分布（可能是时变的）通过线性变换与电场磁场有关。因此，叠加原理可用于简化由给定电荷和电流分布产生的场的计算。这一原理也适用于物理学中出现的其他线性微分方程，如热方程
• In mechanical engineering, 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).[13] Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[14]
• 机械工程中，叠加用于解决组合荷载作用下梁和结构的挠度，当效应为线性时（即每个荷载不影响其他荷载的结果，并且每个荷载的效应不会显著改变结构系统的几何结构）。[15] Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[16]
• The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by linearization.
• 当用线性化分析非线性系统已知解的微小偏差时，可以应用叠加原理。

## History历史

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 and then by Joseph Lagrange. Later it became accepted, largely through the work of Joseph Fourier.[17]

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 and then by Joseph Lagrange. Later it became accepted, largely through the work of Joseph Fourier.

## References参考文献

1. The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London
2. The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London
3. Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [1]
4. Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [2]
5. N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [3]
6. N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [4]
7. Tim Freegarde, Introduction to the Physics of Waves, Cambridge University Press, Nov 8, 2012. [5]
8. Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. 62
9. Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623.
10. Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623.
11. Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK, p. 14.
12. Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK, p. 14.
13. Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192
14. Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785
15. Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192
16. Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785
17. Brillouin, L. (1946). Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York, p. 2.

• Haberman, Richard (2004). Applied Partial Differential Equations. Prentice Hall. ISBN 978-0-13-065243-0.