What is the simplest way to see that this abstract plane wave is "traveling" at the speed of light?

Question

In EMFT notation,

$$ {\bf E} = E_0 \sin(\omega (t - \sqrt{\epsilon_0 \mu_0} z)) \hat{{\bf a}}_x \\ {\bf H} = H_0 \sin(\omega (t - \sqrt{\epsilon_0 \mu_0} z)) \hat{{\bf a}}_y $$

I am having trouble picturing this plane wave and in particular confirming that it is travelling along $z$-axis at the speed of light. I've accepted $u = 1/\sqrt{\epsilon_0 \mu_0} = $ speed of light. Knowing that, what is the trick to show that this wave is travelling at that speed? What does that look like? Links to a simple visualization are okay.

Answer 1

You can consider the movement of a point of constant phase. For instance, the minimum of the magnitude of the $E$ field. This will happen when the phase is a multiple of $2 \pi$. For instance, take the point where the phase is zero:

$\implies t-z/u = 0 \quad \implies z(t) = ut,$

Now, by using the definition of speed as the derivative with respect to time, we have that the wave is travelling at the speed $u$, the speed of light. Since this is true for all different phases of the wave, we can see that the equation describes a motion through time of the whole sine wave as a translation at speed $u$ along the $z$-axis.

The same goes for the $H$ field.

Answer 2

The periodicity of the equation makes its real behaviour hard to see. For more clarity, realize that a real plane wave will be made up of many Such modes${}^{1}$:

$$E(z,t) = \sum_{\omega} A_{\omega}\sin\left(\omega(t - \sqrt{\epsilon_{0}\mu_{0}}z)\right)$$

It turns out that if the E-field starts out with an initial value${}^{2}$ $E(z)$, you can always solve for a set of $A_{\omega}$ that satisfy this initial condition. Now, we want to study the speed of this wave. Well, from the nature of the above equation, it should be clear that if $E(z,t)$ takes some value, then the field will also take that same value at some later value $E(z + C, t + C\sqrt{\epsilon_{0}\mu_{0}})$. The speed at which this unchanging value moves is:

$$\begin{align} v &= \frac{\delta z}{\delta t}\\ &= \frac{z + C - z}{t + C\sqrt{\epsilon_{0}\mu_{0}} - t}\\ &= \frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}\\ &= c \end{align}$$

${}^{1}$ The sum will be replaced with an integral in the continuum case, but I want to make this as simple as possible

${}^{2}$ And the field satisfies appropriate boundary conditions