Принцип суперпозиции в линейных системах

Fundamental physics principle stating that physical solutions of linear systems are linear
the superposition principle in linear systems

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. Superposition can be defined by two simpler properties: additivity

and homogeneity

for scalar a. 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 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).

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. 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.

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 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 the top.)

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 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, 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 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, 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, 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 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 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. 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.

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. 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.