Change of variables to go from $u_{tt}-c^2u_{xx}=0$ to $u_{\epsilon n}=0$.

Question

Consider the following PDE $u_{tt}-c^2u_{xx}=0$, $c=1.$ Notice that with the following change of variables $$x=\epsilon+n \\t=\epsilon-n\\\implies \epsilon=\frac{(x+t)}2,n=\frac{(x-t)}2$$ we get $u_{\epsilon n}=0$.

Exercise Use an analogous change of variables to go from $u_{tt}-c^2u_{xx}=0$ to $u_{\epsilon n}=0$.

My attempt:

I've thinking about consider $x=\frac{(\epsilon+n)}{2}$ and $t=\frac{(\epsilon-n)}{2}$. This implies: $\epsilon=x+t$ and $n=x-t$.

Will this change of variables work?

Is there an easier change of variables to work with? please tell me!

Thank you in advance for your help.

Answer 1

In this case, try $x = c(\epsilon + n)$ and $t = \epsilon - n$. This is an exercise of chain rule. For instance, \begin{align*} u_\epsilon & = u_x\frac{\partial x}{\partial\epsilon} + u_t\frac{\partial t}{\partial\epsilon} \\ & = cu_x + u_t. \end{align*} I shall let you finish the computation.