Superposition principle

The superposition principle applies to any linear system: the total effect of multiple inputs equals the sum of the effects each input would produce on its own. For a function \(F(x)\), this means that if input \(A\) gives response \(X\) and input \(B\) gives response \(Y\), then input \(A+B\) gives response \(X+Y\). Such a function is called linear, and superposition itself breaks down into two simpler rules: additivity, where \(F(x_1 + x_2) = F(x_1) + F(x_2)\), and homogeneity, where \(F(ax) = aF(x)\) for any scalar \(a\).

Because many physical systems are only approximately linear, the principle is itself an approximation of real behavior. Still, it is widely used in physics and engineering—for example, a beam’s deflection under a load can be treated as a linear system. Linear systems are mathematically tractable: they allow techniques like Fourier and Laplace transforms, linear operator theory, and straightforward analysis of algebraic equations, differential equations, and systems of those forms. The stimuli and responses involved can be numbers, functions, vectors, vector fields, or time-varying signals; when vectors or vector fields appear, superposition means vector addition. If superposition holds, it also holds for any linear operation applied to those functions, such as gradients, differentials, or integrals.

The principle makes it easier to compute responses by breaking a general stimulus into simpler parts. In Fourier analysis, a stimulus is written as a superposition of infinitely many sinusoids. Each sinusoid can be analyzed separately (its response is another sinusoid of the same frequency but different amplitude and phase), and the total response is the sum or integral of these individual responses. Similarly, in Green’s function analysis, the stimulus is treated as a superposition of impulse functions, and the response is a superposition of impulse responses. Fourier analysis is especially common for waves: ordinary light, for instance, is described as a superposition of plane waves of fixed frequency, polarization, and direction. As long as superposition holds (which is usually true, except in nonlinear optics), any light wave’s behavior can be understood from these simpler components.

Waves are variations in some parameter—like water height, sound pressure, or the electromagnetic field—across space and time; the value at each point is the wave’s amplitude. In many systems, the wave equation is linear, so the net amplitude from multiple waves is the sum of their individual amplitudes. For example, two waves traveling toward each other pass through without distortion. Regarding the distinction between wave interference and diffraction, Richard Feynman noted that no satisfactory definition exists—it is largely a matter of usage, with interference usually referring to a few sources and diffraction to many. Other authors add that the difference is one of convenience: if the superposed waves come from a few coherent sources, the effect is called interference; if they come from subdividing a wavefront into infinitesimal coherent wavelets, it is called diffraction. The difference is one of degree, not of fundamental physics.