characteristic coordinate method to solve wave equation

Question

In solving wave equation $u_{tt}=c^2u_{xx}$ by characteristic coordinate it is chosen that $\epsilon =x+ct$ and $n=x-ct$.
But how was it decided that this was the transformation required.How do we know that the transformation should be selected so that $\epsilon =x+ct$ and $n=x-ct$.

Answer 1

You suspect the solutions are travelling waves. So you try rewriting it in a coordinate system where the travelling waves are stationary. So you try $x+dt$ and $x-et$ as the new coordinate system, and quickly figure out that $d=e=c$ works very nicely.

Incidentally, this method is used for many different wave equations. E.g. when looking for solutions of the KdV equation (which is supposed to describe water waves in shallow water), the first thing you try is to rewrite the equation in a different coordinate system $x-ct$, and then solve for $c$. In the case of the KdV equation, you will find that $c$ depends upon the amplitude of the wave.

http://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation