The general solution to the three dimensional wave equation is \begin{equation}u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r} \end{equation} where $F$ and $G$ are arbitrary functions. I want to understand the behavior of this function when $r$ is very small. The lecture notes here spell out sample behaviors in more detail than I have seen before. Specifically, I am looking at example 18.15, pages 1016-7, and figure 18.11.
This example considers a ball of radius 1 about the origin instantaneously illuminated with amplitude 1 for $r \leq 1$ as an initial disturbance $u(r,0$); and initial velocity $0$. Changing some notation ($r \to ||{\bf x}||$) the solution is given as
The behavior of this solution is sketched out in the figure:
For some values of $t$, this solution seems to allow arbitrarily negative amplitudes, and a divergence at the origin. I have two main questions about this:
In what sense are these large amplitudes 'real'? Clearly we do not have infinite displacement for any physical waves, but for physical systems, do we have some kind of very large displacement (greater than that of the initial conditions) when we are close to the origin?
What does this mean for the inward traveling spherical waves? Say such a wave has total energy $E$. Its amplitude grows stronger as its radius decreases, but what happens when it crosses the boundary $r=1$? It would seem its amplitude grows arbitrarily large. When $r$ is very small, the amplitude must be very large to keep $E$ constant. Is there anything wrong with this picture or is it sound?