### The Superposition Principle

In Physics, the principle of superposition says roughly that if two particular sorts of behavior are allowed in a physical system separately, then if you try to cause both simultaneously, the result will be the sum of the two individual behaviors (where sum is defined in a sensible way, mathematically) called the superposition of the two. This principle is not true in all (or perhaps even most) physical systems, but it is often at least approximately true^{1}. The "principle" as I've just stated it undoubtedly sounds vague. That is because it is a very general property that exhibits itself in many different sorts of systems, which are then said to "obey the superposition principle". We can make the definition more exact by relating it to the mathematical idea of linearity, which I discuss in the last section.

When it applies, the superposition principle is extremely useful, because it allows us to take a complex situation and think of it as the sum of many, simpler parts. By being able to break down a complex problem into many simple problems, we are able to understand a lot more than we otherwise would be able to. In fact, most of our understanding of the physical world concerns situations where the superposition principle is true, and we are only recently starting to understand systems without the superposition, in the fields of nonlinear dynamics and chaos.

Now let's talk about a specific example of superposition to make this idea more concrete.

### An Example of Superposition

The best example I can think of to illustrate the idea of superposition here is that of waves traveling on an idealized string. If this string obeys the superposition principle, and you can make two different wave packets (meaning two waves of certain shapes), A and B, on the string, then the principle tells you that when the wave packets overlap the result will just be as though one wave packet was stacked on top of the other one; the height of the result is the sum of the heights of the individual wave forms, in other words. At that point the two wave packets are said to be "superposed"^{2}. Furthermore, once they no longer overlap, each wave packet will continue on as if the other were never there. Here is a crude picture of what this would look like using a series of "freeze frames" at successive times:

A ______ B
| | ---> <--- /\
| | / \
____| |_________________________________/ \_____
_____/|
| | /\
| |/ \
_____________________| \_______________________
/|
/ |
___/ |
| \
| \
______________________| \________________________
/\
/ \
/ \
| |
| |
_______________________| |__________________________
|\
| \
| \____
/ |
/ |
_______________________/ |_______________________
|\_____
/\ | |
/ \| |
______________________/ |______________________
B ______ A
<--- /\ | | --->
/ \ | |
____/ \_____________________________________| |____

I hope that diagram is clear. Please /msg me if it isn't and try to be as specific as possible about what confuses you. The first and last diagrams are supposed to be long before and after the two wave packets cross.

### Applicability of Superposition

Perhaps one of the most important examples of where the principle of superposition is exactly true is the theory of electromagnetic fields in vacuum (the classical theory, anyway^{3}). If two different electric and magnetic fields can exist in a certain region, then the sum of those fields is also a possible behavior. Also, if a certain configuration of electric charges and electrical currents cause one field, and a second set of charges and currents cause a different field, then if both sets of charges and currents are present it will cause the sum of the two fields. It is this last property that makes the superposition principle particularly useful in studying electromagnetism. When you have a complicated source (some set of currents and charges), you can think of that source as the sum of many simpler sources. You can then figure out the effect of each of those sources separately. Afterward, you can add up the resulting fields to get the total field that the total source will cause. The simpler sources can be point charges, sheets of charge, or other simple charge distributions, and you can think of many similarly simple current sources. This technique can make the situation much easier to figure out. Another approach using the same idea is using a multipole expansion. Even if the source can't be described exactly by a few simple sources, this will often give a good approximate description.

In many other cases, the superposition principle holds for a good approximate description of the system in question. Most systems^{4} that have a stable equilibrium configuration will behave linearly for small deviations from that equilibrium[2], and disturbances from equilibrium should stay small (since it's a stable equilibrium), so you can describe them approximately with a theory that obeys the superposition principle as long as the motion does not start out too large. What is "too large" is defined by how far the system can go from equilibrium before nonlinear behavior becomes important, which depends on the details of the system in question. This sort of approximate validity is usually the case for electromagnetic fields inside of a medium, like glass or water. The superposition principle will be true for small field strengths, but for strong fields it will not hold true.

One example of a situation in which the superposition principle holds approximately is an electromagnet with an iron core. Putting electrical current into the electromagnet causes an electromagnetic field in the core. The core, which is often a ferromagnetic material, is there because the field caused in it by the current is stronger than the field that would be caused in empty space. If you add two small currents in the electromagnet, the resulting field will be approximately the sum of the fields that would have been caused by each current alone, obeying the superposition principle; however, if you add a large current the increase in magnetic field within the core will not be as large as you would expect from the superposition principle, because eventually the iron core begins reach "saturation". This is the point where nonlinearities in the behavior of the core have become important.^{5}

The most common context in which to discuss superposition is the study of waves. A linear wave theory is one that describes waves in a system in which the superposition principle holds exactly or at least approximately. Systems described by a linear wave theory include electromagnetic waves (a.k.a. light), gravitational waves, sound waves, and mechanical vibrations. Waves occur in a system that is close to stable equilibrium, so as I discussed before this often means that the superposition principle will hold true as long as the waves are small enough. When we have the superposition principle to rely on, even the motion of complicated wave packets can be understood, because we can understand them as nothing more than the sum of many simple, periodic waves. As long as we can understand how the simple waves behave, then we can add up the results to understand how the complex wave packet behaves. When two different waves come together, the result of superposing the two waves is usually called interference. In systems that only approximately obey the superposition principle, when the waves get too large and we no longer have simple superpositions, we enter the realm of waves in nonlinear media, which are much more difficult to understand and predict. Most of the time at a high school or undergraduate level you will only ever talk about waves in systems that obey the superposition principle, because those ones can be understood so much more simply and completely.

The superposition principle is useful to understand a huge variety of systems. The reason it's so useful is that it allows us to break down a complex problem into many simple problems and solve it piece by piece, since we can get the result just by adding up the results of the individual pieces. Practically all widely studied, well understood systems obey the superposition principle; the concept of superpositon is central to most of the study of Physics. Systems that don't are said to be "nonlinear" and are studied in nonlinear dynamics and chaos, but usually you can't say nearly as much about them as you can about linear systems with superposition. Superposition is the consequence of a mathematical property called linearity, which I'll finish up by describing.

### Mathematically Defining Superposition

I've said that systems that obey the superposition principle are called linear systems. One way this is often defined mathematically is to say that the quantities of the system you're interested in behave according to an equation of motion that is a linear differential equation:

a_{n}(t)*(d^{n}y/dt^{n}) + … + a_{2}(t)*(d^{2}y/dt^{2}) + a_{1}(t)*(dy/dt) + a_{0}(t)*y = 0

where d^{n}y/dt^{n} is the n^{th} derivative of y(t) written in Leibniz notation. The important feature of that equation is that if a function y_{B}(t) is a solution and a function y_{C}(t) is a solution then the function y_{D}(t) = y_{B}(t) + y_{C}(t) is also a solution. y_{D} is the superposition of y_{B} and y_{C}. While the above is an ODE, it could just as well have been a PDE. To put it more generally and formally, a linear system is one in which the solutions (the possible behaviors) form a vector space. This is true of the solutions of the differential equation above.

- How can something be approximately true? Isn't truth a binary thing? What I mean here is that you can use a model which obeys the superposition principle as a quantitatively good approximate description of the system.
- We say "superpose" to mean adding one to the other in this certain way, as opposed to if they were "superimposed", which would just be drawing one on top of the other.
- In Quantum Electrodynamics, higher order loop corrections add photon-photon interactions which make even the vacuum nonlinear[1]. So, for very strong fields, we would no longer have the superposition principle for EM fields. Still, it is a very good approximation of the true behavior in most situations. Also, we still would have the superposition principle for the Hilbert space of the quantum system.
- Generally here I'm speaking of a conservative system where the second derivatives of the potential energy at equilibrium are non-vanishing. A system where the quadratic part of the energy vanishes at equilibrium will be approximately quartic near equilibrium and, thus, have different behavior.
- In actuality, nonlinearities may be important well before saturation, unless the magnetic susceptibility curve is very straight. But in all cases, the core will reach saturation eventually and nonlinear effects are definitely important there.

Sources:

- Phys. Rev. D
**2**, 2341 (1970)
- Goldstein, Poole, and Safko, Classical Mechanics 3rd Ed. (Addison Wesley, San Fransisco, 2002)
- Howard Georgi, The Physics of Waves (Prentice Hall, Englewood Cliffs, New Jersey, 1993)
- Serway, Physics (Suanders College Publishing, Philadelphia, 1996)