A non-mathematical introduction to the wave equation

Waves pop up everywhere in physics. They're most obvious at the beach, but waves are also used to describe light, pendulums, and all sorts of other things. Because waves can describe so much in physics, it's important to know what it actually means when you talk about waves.

The medium is not the wave

When physicists talk about waves, they mean something very specific. THey mean that some material is moving in a specific way. Waves at the beach are the best way to visualize this for me. Water waves have a very obvious medium: water. The waves are not the water, they're the way the water moves up and down over time. And the water doesn't just move up and down randomly; a peak on the water seems to travel towards the beach.
This is a very key point. Waves are just the way that some type of medium is moving. Light waves, for example, are just motion in the electromagnetic field.
So what causes the wave to move like it does? The answer to that relies on two separate ideas: energy input and strain in the medium.

Energy input

For most mediums (like water), if you leave it alone the waves will all die out and it'll be still. There needs to be some transfer of energy into the medium in orer for a wave to start. At the beach, the energy to start a wave often comes from wind. For light, the energy to start a light wave (photon) usually comes from electrons bouncing around.
Not all mediums are like this. In outer space, the electromagnetic field will keep a light wave going forever. That only works once the light wave gets started, which still takes energy input.

Strain

Strain in a medium is the tendency for it to return to it's original position. In water, strain is provided by gravity. Because gravity pulls on all the water in the ocean, the water tries to keep an even level. If wind pushes some water up higher, gravity will try to pull it down. The energy in the peak will get transferred to nearby water molecules, and the wave will move.

The wave equation

The way that strain in a medium causes a wave to travel is described by the wave equation
\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x}
This equation shows how the time rate of change of the wave (the left hand side) is related to the strain in the medium (the right hand side). This equation is what's called a second-order differential equation, which just means we're taking derivatives twice (the \partialstuff).
To understand this equation, it helps to take a look at each side separately. To do this, we're going to look at snapshots of the wave in a couple of different ways.

Localizing space

Let's think about the left side of this equation first. The entire left side basically means "the way the medium changes with time". To get a feel for this, I like to imagine I'm treading water at the beach. Maybe I've swum out a few leagues and I'm just bobbing up and down with the water. You could plot my height above sea level over time, and that would give you one view of the ocean waves. We've made a little movie of the wave for a single point in space, and ignored all the other points.
The left hand side of the wave equation represents how fast my height at that point is changing (which is equivalent to saying that it's the second derivative of my height with respect to time).

Freezing time

The right hand side of this equation is another second derivative. This time it's a derivative with respect to space, instead of time (x instead of t). To understand this, think of freezing time instead of localizing space. If we take a snapshot of an ocean wave at a given time, it would be like a bunch of troughs and valleys in the water that don't change.
The second derivative on the right hand side of this equation represents the curvature of the water at any given point. That curvature is a pretty good measure of the strain the water is under. The spikier the water, the more the curvature, the more the strain.

Time and Space together

The wave equation is an assertion that curvature of a wave in space (the strain) is related to the way that the wave travels as time moves forward. The relation is given by the factor c^2 in the wave equation. The travel of some part of the wave through time is equal to the curvature of the wave at that point multiplied by speed of the wave in that medium. If we're talking about light waves, then c is the trusy old 3*10^8 m/s. If we're talking about sound waves or water waves, then c is going to be different. The exact value of c depends on what kind of medium we're talking about.

Conclusions

So that's the main thing that a wave is. A wave is just a way that strains in a medium move around, and you can describe that motion using a specific equation. The wave equation says that the curvature in the medium at a point is related to the rate at which the strain of the medium changes at that point.
If you know the properties of a medium and what the strain in the medium looks like now, you can calculate the curvature of the medium. That let's you figure out what the medium looks like later. You may also need some extra information, like the velocity of the wave now or the energy being put into the wave.