Consider the wave equation: $\alpha^2 u_{xx}=u_{tt}$. I am told to let $y=x-\alpha t$ and $w=x+\alpha t$ and use chain rule but I'm not confident in my attempt.
I got $x=\frac{y+w}{2}$ and $t=\frac{w-y}{2\alpha}$. Found $u_{yw}(x,t)=\frac{\alpha^2u_{xx}-u_{tt}}{4\alpha^2}$, then used then wave equation to conclude $u_{yw}=\frac{0}{4\alpha^2}=0$.
Any help is greatly appreciated!