A. Force **F**_{1} applied to a certain object produces acceleration **a**_{1} and force **F**_{2} applied to the same object produces acceleration **a**_{2}. When force **F** = **F**_{1} + **F**_{2} is applied to the object, the resulting acceleration is **a** = **a**_{1} + **a**_{2} (these are vector sums).

B. Two objects have weights *W*_{1} and *W*_{2}. When they are placed on a scale together, the recorded weight is *W*_{1} + *W*_{2}. (This implies the additivity of mass—valid up to a point!)

C. At a certain location, the electric field arising from a collection of charges is the vector sum of the fields that would be produced at that location by the charges acting separately. The same principle holds for magnetic fields.

Example A may seem “obviously” true. Example B may seem so, too (since it is consistent with ordinary experience), but in fact, the masses of two objects are not strictly additive when the objects exert forces on one another—forces that can change the energy and therefore the mass of the pair of objects. Masses are additive in classical Newtonian theory, but not in relativity theory. The truth of Example C at first seems more doubtful because fields occupy space. Can they coexist in space without “bothering” one another? In classical electromagnetic theory (the theory of James Clerk Maxwell), the answer is yes. Fields of arbitrary strength *can* share space without mutual “disturbance.” In quantum theory, not quite. So both relativity theory and quantum theory slightly “besmirch” the superposition principle.

Consider some examples where the superposition principle is not true.

D. You pour a pint of water into a quart jar. The jar is half full. You separately pour a pint of alcohol into another quart jar. That jar, too, is half full. Now you pour a pint of water and a pint of alcohol together into a quart jar. The jar doesn’t quite fill up. One pint + one pint in this case is not equal to one quart.

E. You and a friend both earn $50,000 per year and pay the same tax. Another friend earns $100,000 per year. Her tax is more than twice yours.

F. Put a neutron on an imaginary super-sensitive scale and measure its mass to be *m*_{n}. On that same scale put a proton and measure its mass to be *m*_{p}. Now combine the neutron and proton to form a deuteron, and measure its mass. The deuteron’s mass *m*_{d} is not *m*_{p} + *m*_{n}. It is less.

So, even in physics, the superposition principle is not universally valid. Yet it does have widespread validity and is appealing for its utter simplicity: Effect (A + B) = Effect A + Effect B. It supports our view that nature usually chooses the simplest rules (or is kind enough to let us mere humans figure out the rules).

Since electric and magnetic fields can be superposed, it follows that electromagnetic waves can be superposed. Think of light from two stars intersecting somewhere in the depths of space. Each beam continues inexorably on its way, undeflected and unchanged by having interpenetrated the other. Of vital practical importance on Earth is the superposition of radio waves, without which radio communication would be impossible. Think of the electromagnetic disturbances in the space around you. If they were visible you would see a gaudy and ever changing splash of every conceivable color. If they were audible you would hear a cacophony of sound. Yet every component of this turmoil, with its own frequency, polarization, and direction of propagation, retains its unique identity, undisturbed by sharing the same space with thousands of other signals.

In the seventeenth century, long before the full import of superposition was appreciated for electromagnetic waves, Christian Huygens made use of the idea of superposition in describing the propagation of a single wave. He asserted that each point on an advancing wave front could be regarded as the source of a new wavelet, and that the subsequent position of the wave front was determined by the superposition of all of the tiny wavelets. This application of the idea of superposition, summing individual contributions from an advancing wave to get the total wave, is known now as Huygens’ principle. In a modernized form that takes proper account of the phase of the wavelets, it provides a very useful approach for understanding many wave phenomena, including interference and diffraction.

For water waves and other waves in material media, the principle of superposition is usually valid to good approximation, but it is never precisely valid. Looking at the surface of a pond on a perfectly still day, for instance, you can see that waves emanating from two dropped pebbles proceed on their way, crossing one another and hardly losing their individuality. The condition required for superposition of material waves is that the wave disturbance be relatively weak—gentle waves on water, for example, but not breaking waves at the beach; or mild sound waves, but not shock waves. If two shock waves cross one another, the resulting wave at their point of intersection is not the sum of the two separate waves, and they do not continue on their way uninfluenced by the encounter.

The development of the quantum theory of radiation in the 1930s revealed that not even for electromagnetic waves is the principle of superposition precisely valid. If two photons meet in space there is a chance—an exceedingly small chance—that they will interact and be deflected by the encounter. This violates the principle of superposition, for the resulting radiation flow is not the same as for the two photons passing separately by. This process, known as the scattering of light by light, has been measured in the laboratory, but it is too improbable to have any perceptible effect on radio communications or on any of the other practical examples of superposed electromagnetic waves or photons.^{1} It is interesting, however, to know that if photons are crowded together densely enough, they can begin to act like bits of matter.

1^{} The scattering of light by light was first studied theoretically by Gregory Breit and John Wheeler in 1934, and finally observed in the laboratory in 1997, after laser beams of sufficient intensity were available.