To understand the wave equation in a more physical way,
we need to remember a little algebra. Recall how the polynomial
could be factored into (b-a)(b+a)? We can do a similar
thing with the wave equation by rewriting it as a big operator
acting on p(x,t) as
and then factoring as
We see immediately that there are two operators acting on on p(x,t):
, it turns out, represents
a forward going wave in the +x direction, and
, it turns out, represents
a backward going wave in the -x direction.