We’ve derived a “wave equation” from Maxwell’s equations, but it’s not clear what it means, or even why this is called a wave equation. Let’s consider the abstracted form, which both electric and magnetic fields satisfy:
where is the “Laplacian” operator, defined on scalar functions by taking the gradient followed by the divergence, and extended linearly to vector fields. If we have a Cartesian coordinate system — and remember we’re working in good, old so it’s possible to pick just such coordinates, albeit not canonically — we can write
where is the -component of , and a similar equation holds for the and components as well. We can also write out the Laplacian in terms of coordinate derivatives:
Let’s simplify further to just consider functions that depend on and , and which are constant in the and directions:
We can take this big operator and “factor” it:
Any function which either “factor” sends to zero will be a solution of the whole equation. We find solutions like
where and are pretty much any function that’s at least mildly well-behaved.
We call solutions of the first form “right-moving”, for if we view as time and watch as it increases, the “shape” of stays the same; it just moves in the increasing direction. That is, at time we see the same thing at that we saw at — units to the left — at time . Similarly, we call solutions of the second form “left-moving”. In each family, solutions propagate at a rate of , which was the constant from our original equation. Any solution of this simplified, one-dimensional wave equation will be the sum of a right-moving and a left-moving term.
More generally, for the three-dimensional version we have “plane-wave” solutions propagating in any given direction we want. We could do a big, messy calculation, but note that if is any unit vector, we can pick a Cartesian coordinate system where is the unit vector in the direction, in which case we’re back to the right-moving solutions from above. And of course there’s no reason we can’t let be a vector-valued function. Such a solution looks like
The bigger is, the further in the direction the position vector must extend to compensate; the shape stays the same, but moves in the direction of with a velocity of .
It will be helpful to work out some of the basic derivatives of such solutions. Time is easy:
Spatial derivatives are a little trickier. We pick a Cartesian coordinate system to write:
We don’t really want to depend on coordinates, so luckily it’s easy enough to figure out:
which will make our lives much easier to have worked out in advance.