Wave equation in hyperbolic canonical form

Question

The wave equation is given $u_{tt}-u_{xx}=0$. I need to show that the wave equation can be put into canonical form by rotation of the coordinate system by 45 degrees. I solved it by writing that $\xi = x/cos45°$ $\eta = t/cos45°$. In the end, I do not get hyperbolic canonical form, I get the same equation that was given to me which is $u_{\xi\xi}-u_{\eta\eta}=0$. How do I change the variables so that I get a hyperbolic canonical form. Thank you.

Answer 1

Does not look like a rotation. Rotation would be

$$\eta = \phantom{-}x\cos(45^\circ)+t\sin(45^\circ)\\\xi = -x\sin(45^\circ)+t\cos(45^\circ)$$

Try this and maybe it works better.