nterference, Diffraction & the Principle of Superposition

Interference of waves in phase and out of phase.

Interference takes place when waves interact with each other, while diffraction takes place when a wave passes through an aperture. These interactions are governed by the principle of superposition. Interference, diffraction, and the principle of superposition are important concepts to understanding several applications of waves.
Interference & the Principle of Superposition
When two waves interact, the principle of superposition says that the resulting wave function is the sum of the two individual wave functions. This phenomena is generally described as interference.

Consider a case where water is dripping into a tub of water. If there's a single drop hitting the water, it will create a circular wave of ripples across the water. If, however, you were to begin dripping water at another point, it would also begin making similar waves. At the points where those waves overlap, the resulting wave would be the sum of the two earlier waves.

This holds only for situations where the wave function is linear, that is where it depends on x and t only to the first power. Some situations, such as nonlinear elastic behavior that doesn't obey Hooke's Law, would not fit this situation, because it has a nonlinear wave equation. But for almost all waves that are dealt with in physics, this situation holds true.

It might be obvious, but it's probably good to also be clear on this principle involves waves of similar type. Obviously, waves of water will not interfere with electromagnetic waves. Even among similar types of waves, the effect is generally confined to waves of virtually (or exactly) the same wavelength. Most experiments in involving interference assure that the waves are identical in these respects.
Constructive & Destructive Interference
The picture to the right shows two waves and, beneath them, how those two waves are combined to show interference.

When the crests overlap, the superposition wave reaches a maximum height. This height is the sum of their amplitudes (or twice their amplitude, in the case where the initial waves have equal amplitude). The same happens when the troughs overlap, creating a resultant trough that is the sum of the negative amplitudes. This sort of interference is called constructive interference, because it increases the overall amplitude. Another, non-animated, example can be seen by clicking on the picture and advancing to the second image.

Alternately, when the crest of a wave overlaps with the trough of another wave, the waves cancel each other out to some degree. If the waves are symmetrical (i.e. the same wave function, but shifted by a phase or half-wavelength), they will cancel each other completely. This sort of interference is called destructive interference, and can be viewed in the graphic to the right or by clicking on that image and advancing to another representation.

In the earlier case of ripples in a tub of water, you would therefore see some points where the interference waves are larger than each of the individual waves, and some points where the waves cancel each other out.
Diffraction
A special case of interference is known as diffraction and takes place when a wave strikes the barrier of an aperture or edge. At the edge of the obstacle, a wave is cut off, and it creates interference effects with the remaining portion of the wave fronts. Since nearly all optical phenomena involve light passing through an aperture of some kind - be it an eye, a sensor, a telescope, or whatever - diffraction is taking place in almost all of them, although in most cases the effect is negligible. Diffraction typically creates a "fuzzy" edge, although in some cases (such as Young's double-slit experiment, described below) diffraction can cause phenomena of interest in their own right.
Consequences & Applications
Interference is an intriguing concept and has some consequences that are worth note, specifically in the area of light where such interference is relatively easy to observe.

In Thomas Young's double-slit experiment, for example, the interference patterns resulting from diffraction of the light "wave" make it so that you can shine a uniform light and break it into a series of light and dark bands just by sending it through two slits, which is certainly not what one would expect. Even more surprising is that performing this experiment with particles, such as electrons, results in similar wave-like properties. Any sort of wave exhibits this behavior, with the proper set-up.

Perhaps the most fascinating application of interference is to create holograms. This is done by reflecting a coherent light source, such as a laser, off of an object onto a special film. The interference patterns created by the reflected light are what result in the holographic image, which can be viewed when it is again placed in the right sort of lighting.