# Properties of the scalar wave equation solutions

How does the solution

\$latex 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 \$latex g(x)\$ is a Gaussian function of \$latex x\$.  Replacing \$latex x\$ with \$latex (x – d)\$ will shift the function to the right by a distance \$latex d\$: So when we’re considering \$latex f(z – v_p t)\$, if we were to plot this function vs. \$latex z\$ at \$latex t = 0\$ and also at \$latex t = t_1\$,  the \$latex t = t_1\$ version will be the same as the \$latex t = 0\$ version except shifted to the right by a distance \$latex v_p t_1\$.  If \$latex f(x, t = 0)\$ is a Gaussian, then, the plot of \$latex f(z, t)\$ at the two times would look like: Since the function has moved in the positive \$latex z\$ direction by a distance \$latex v_p t_1\$ in time \$latex t_1\$, it must be moving at a speed of \$latex v_p\$.  In fact, we call the quantity \$latex v_p\$ the phase velocity.  The solutions to the wave equation are waves, and \$latex 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:

\$latex E_x = f(z + v_p t)\$

Hopefully you can see that this will be a wave moving to the left (negative \$latex z\$ direction) with speed \$latex 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 \$latex v_p\$.