Theory

Wave motion occurs when you have some displacement in a medium where the restoring force is stronger than the dissipative force. The restoring force brings the displaced part of the medium back to equilibrium. However because the dissipation is small there is energy left over, the restoration overshoots the equilibrium, and the process repeats from the other direction, oscillating. Usually the restoring force is due to the elasticity of the medium; if the medium must be continuous the disturbance will propagate through it.

Example: a raindrop falls on the surface of a still pond. The momentum of the droplet pushes the surface of the water downward. This ``hole'' in the water is filled by water rushing in from all the other directions, filling in the ``hole'' and lowering the surface around it. It so happens that water is not very viscous, flows quickly, and so this is an overcorrection: the ``hole'' in the water turns into a ``pile'' of water with a lower surface around it, and the process repeats again. After three or four oscillations the place where the droplet fell is smooth again, but the disturbance has spread out in a few concentric rings. It's important that the water flows quickly: you don't get this type of wave in, say, honey.

(Note that I have used imprecise and perhaps misleading language by discussing waves in a ``medium.'' This type of motion is described by simple mathematics which arise in many situations, not only in material objects; all of these are typically called waves. Most famously, light is an oscillation in the electric field, which can occur in the absence of matter. In this experiment the ``medium'' is a table of numbers, represented by colors on a computer screen.)

Two waves that pass each other, if they aren't too big, pass right through each other. (I'm in a room with two lamps; the light from one lamp doesn't knock the light from the other out of the way.) In our ``medium,'' the displacements simply add. Where both waves are making the water surface high, the net displacement is more than it would be due to one wave alone; where one wave raises the surface and another wave lowers it, the competition leaves the surface closer to equilibrium. This is called ``constructive'' and ``destructive'' interference. In regions with more than one wave, this interference produces complicated patterns.

A long time ago, Young realized that two coherent sources of spherical waves would produce a very simple interference pattern. In the plane midway between the two spheres, the interference is always constructive. As you move away from this plane, the interference becomes destructive, then constructive again, etc., as the difference between the path lengths changes by half a wavelength, a whole wavelength, and so on. It takes some geometry to figure out that if the wavelength is $\lambda$, the distance between the point sources is $d$, the distance from the plane connecting them is $L$, and the distance from the optical axis is $x$, the constructive interference occurs when

$$n\lambda = \pm xd/L$$

for $n=0,1,2,\ldots$ an integer. Destructive interference occurs when $n$ is a half-integer ( $\frac{1}{2},\frac{3}{2},\ldots$). (This is an approximation good when $x\ll L$ and $d\ll L$, that is, when $L$ is large and we are far away from the sources.)