Properties of the scalar wave equation solutions

How does the solution

$E_x = f(z - v_p t)$

behave? Well, we know that subtracting a number from the argument of a function shifts that function to the right. For example, suppose that $g(x)$ is a Gaussian function of $x$ . Replacing $x$ with $(x - d)$ will shift the function to the right by a distance $d$ :

So when we’re considering $f(z - v_p t)$ , if we were to plot this function vs. $z$ at $t = 0$ and also at $t = t_1$ , the $t = t_1$ version will be the same as the $t = 0$ version except shifted to the right by a distance $v_p t_1$ . If $f(x, t = 0)$ is a Gaussian, then, the plot of $f(z, t)$ at the two times would look like:

Since the function has moved in the positive $z$ direction by a distance $v_p t_1$ in time $t_1$ , it must be moving at a speed of $v_p$ . In fact, we call the quantity $v_p$ the phase velocity. The solutions to the wave equation are waves, and $v_p$ is the speed at which the wave is moving.

I mentioned briefly that there is another possible solution to the scalar wave equation, namely:

$E_x = f(z + v_p t)$

Hopefully you can see that this will be a wave moving to the left (negative $z$ direction) with speed $v_p$ . In fact, if we went back to the more general form of the Helmholtz equation, we would find solutions moving in all possible directions in 3d space at speed $v_p$ .