Given the wave equation and $u_{yw}=0$, conclude $u(x,t)=\phi(x-\alpha t)+\psi(x+\alpha t)$.

Question

I know the wave equation is $\alpha^2u_{xx}=u_{tt}$. I am given the result from a previous problem, $u_{yw}(x,t)=0$ where we let $y=x-\alpha t$ and $w=x+\alpha t$.

For this, $\phi$ and $\psi$ are arbitrary functions.

My initial thought is to start with $u_{yw}=0$, integrate both sides with respect to $w$, then again with respect to $y$, but I am unsure how the math itself works out.

Any help is greatly appreciated!