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.