How does the solution
behave? Well, we know that subtracting a number from the argument of a function shifts that function to the right. For example, suppose that is a Gaussian function of . Replacing with will shift the function to the right by a distance :
So when we’re considering , if we were to plot this function vs. at and also at , the version will be the same as the version except shifted to the right by a distance . If is a Gaussian, then, the plot of at the two times would look like:
Since the function has moved in the positive direction by a distance in time , it must be moving at a speed of . In fact, we call the quantity the phase velocity. The solutions to the wave equation are waves, and is the speed at which the wave is moving.
I mentioned briefly that there is another possible solution to the scalar wave equation, namely:
Hopefully you can see that this will be a wave moving to the left (negative direction) with speed . 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 .