We saw that the solutions to the scalar wave equation were of the form

where was any twice-differentiable function. I’m going to start using sinusoidal or harmonic functions for – functions like cosine, sine, and complex exponentials. Let’s start with cosine. We should certainly be allowed to replace with a cosine, as it is twice differentiable. If we do so, we have

I decided to go with the minus sign for the forward-propagating wave (wave traveling in the direction), and I slipped in a few new constants: out front, which is just the amplitude of the cosine wave; and a inside the cosine, which I’ll discuss below. Neither of these constants changes the fact that this harmonic wave is a solution to the wave equation, because it’s still twice-differentiable, and it still has the necessary grouping inside. I’m going to rewrite the harmonic wave a bit without really changing it. First I’ll bring the inside the parentheses:

The quantity in front of in a harmonic function is just the angular frequency , so we can rewrite this as

where we must have

You probably have some intuitive feel for the angular frequency , where is the ordinary frequency, often given in Hertz (). For example, as gets bigger, the harmonic function is oscillating faster in time. The period of oscillation – the time it takes for the oscillation to undergo one complete cycle – is just

Slightly newer is , which is called the wavenumber. Looking at our harmonic wave, notice that sits in front of , just as sits in front of . In fact, the wavenumber plays the same role as does, except that it applies to the wave’s oscillation in space, rather than time. So, as gets bigger, the harmonic wave is oscillating faster in space. We can think of the wavenumber as being an angular spatial frequency.

The wavelength is the spatial distance between peaks of the wave; i.e. it’s the distance over which the oscillation undergoes one complete cycle. It’s analogous to the period, but applied to space, rather than time. Similar to how the period , the wavelength

.

We can do a little manipulation here to come up with a very famous equation. We already know that

and we know , and . In the next section we will see that the phase velocity . Putting all of this together and rearranging things,

In words, wavelength times frequency equals the speed of light divided by the refractive index. This holds for all electromagnetic waves, including light.

Below are some animations to illustrate the connections between , , and .

Animation showing a wave with amplitude A = 1, , and :

Animation showing a wave with amplitude A = 2, , and :

Animation showing a wave with amplitude A = 1, , and :

(Notice how the frequency has increased, because the faster spatial oscillations translate into faster time oscillations when the wave passes at a fixed speed.)

Animation showing a wave with amplitude A = 1, , and :

(Notice how the frequency has increased, even though the spatial frequency have not, because the spatial oscillations are passing by faster.)