What it is

In mathematics

It is an important algebraic property of systems of linear equations: if there are several solutions, every linear combination of these solutions is still a solution of the system. That is: the solutions of a linear system can be multiplied by arbitrary constants and added algebraically to each other, and what is obtained is still a solution of the same system.

In physics

The same principle applies to all physical systems that can be described by (systems of) linear equations. Generally the "equations" in question are the dynamical equations of the system (e.g. the equations of motion), and the "solutions" are the possible time evolutions of the system (for example the possible motions).

More formally, we can say that in all situations in which the physical quantities of a system are described by functions of time and position f(r,t), solutions of linear differential equations, the superposition principle holds. These equations have the formidable property that if two functions f₁(r,t) and f₂(r,t) are solutions of the differential equation, a linear combination of them, that is, a function of the type

${\displaystyle \lambda f_1(\mathbf{𝐫},t)+\mu f_2(\mathbf{𝐫},t)}$ with ${\displaystyle \lambda ,\mu \in \mathbb{ℝ}}$

is still a solution.

Examples and applications

• The superposition principle tells us, for example, that if two simple motions of a linear system are known, they can be combined to obtain more complex motions that are equally allowed. By "motion" we mean here:
  • an actual movement, such as an oscillation, or a translation for a mechanical system;
  • in an electrical system it could be the value of a current flowing through a given element of the circuit;
  • in an acoustic system it could be the local value of the pressure with respect to the reference value of atmospheric pressure; etc.
• Conversely, given a complex motion, one can try to express it as a combination of elementary motions. This topic is discussed further in the sections normal modes, Fourier's theorem and wave synthesis.
  • For example, these animations illustrate how a simple linear combination of two simple oscillations in a string generates a more complex oscillation (see also the section on the normal modes of a string).

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• Applied to waves, the superposition principle expresses the fact that, at every point in space where two waves arrive simultaneously, the overall oscillation is given by the algebraic sum of the oscillations of the two incident waves taken separately.
  • This property, common to many types of waves, at least in the regime in which they are not too intense, accounts, for example, for the experience that several sounds can be heard simultaneously: the overall sound is the result of the linear superposition of the individual sounds.
  • Waves generate complex patterns in the regions of space where they meet, but they apparently "ignore" each other, emerging from these zones exactly as they entered them (see the animation below).

Why principle and not law?

"Superposition" is the physical property of waves that corresponds mathematically to the algebraic sum of wave functions. For example, given two sea waves ψ₁ and ψ₂, their algebraic sum is a third wave ψ₃, whose height is given, point by point, by the sum of the heights of the two waves above the water surface.

${\displaystyle {\psi }_3={\psi }_1+{\psi }_2}$.

So far we have used no physical law and no physical principle. We have only built a wave mathematically starting from two other waves.

Now let us turn to physics, which tells us that the motion of a wave obeys a certain physical law: the wave equation. This means that both the wave ψ₁ and the wave ψ₂ satisfy the wave equation. For simplicity, let us write this as:

${\displaystyle F({\psi }_1)=0}$

${\displaystyle F({\psi }_2)=0}$

What can we conclude about the wave ψ₃: will it also satisfy the wave equation, or does it remain a mathematical object with no connection to physics?

${\displaystyle F({\psi }_3)=0???}$

In reality the answer is: it depends. It depends on the type of waves, on the regime in which they superpose, on the possible interactions between the waves, and so on. Which means we might be forced to use different wave equations in different circumstances.

Reality is complicated, but the human brain loves simple and powerful models, so physicists immediately identified the simplest case, which is also the most common one, in which the wave produced by superposing two waves is still a wave.

${\displaystyle F({\psi }_3)=0!!!}$

In short, if this assumption holds, given ψ₁ and ψ₂, ψ₃ will also behave as a wave and, conversely, given ψ₃, we can always break it down into a combination of ψ₁ and ψ₂.

Powerful, isn't it?

However, since this is not a law of nature but an assumption, which must be verified or falsified case by case, we speak of a physical principle and not of a physical law.

For a less elementary treatment, see the wave equation.

Why it is so important

The superposition principle holds strictly only for systems governed by linear physical laws. Of course, nonlinear systems also exist in nature (indeed, they are the majority, and show a great variety of interesting behaviours). So why does the superposition principle remain so useful?

1. Linear algebra is a very well known field of mathematics, and there is a vast set of theorems and techniques that can also be used in the study of physical systems, both classical and quantum.
   • E.g.: given an electric circuit, the superposition principle allows us to break it down into simpler circuits and, once these have been solved, to recombine the solutions to obtain that of the original circuit.
   • E.g.: the state of an atom or a molecule, however complex its time evolution, can always be described in quantum mechanics as a superposition of (infinitely many) stationary states, that is, states invariant in time.
2. It is therefore clear that, even in the presence of nonlinear systems, one of the first approaches often consists in "linearizing" the equations, that is, in looking for a linear approximation to the equations of motion that makes it possible to apply the many techniques of linear algebra.
   • From a physical point of view, the linearized equations provide an approximate "first-order" description for all those systems (and they are the vast majority) for which "small causes correspond to small effects", that is, "a small external excitation corresponds to a small response of the system". For example, linearizing an oscillation produces a particularly simple equation, whose solution is known as harmonic motion.
   • Whenever the study of a system is replaced by that of its linear model, it is very important to define clearly within which parameters the model describes the system well. Referring to the previous point, we can certainly say that very often the system departs from the model as soon as the excitations are no longer "small", but at other times the nonlinearities behave more subtly.

For waves, the superposition principle stops holding when the waves are very intense: for example very loud sounds near an explosion, which become shock waves, or sea waves breaking on the shore. In these cases the waves really do interact with each other (see non-linear effects).

Check your understanding

Further reading

• Conditions of applicability of the superposition principle: can waves always be superposed? See wave equation
• The superposition principle at work in interference, in the operation of antennas, and in the formation of complex waves
• The superposition principle in the analysis of complex signals: normal modes
• The superposition principle in the synthesis of complex signals: Fourier's theorem
• Related questions and answers:
  • Is it possible to deflect a sound with another sound?