D'Alembert's Solution to wave equation

[pierce15]

Hello,

How does the change of variables $\alpha = x + at , \quad \beta = x - at$ change the differential equation

$$ a^2 \frac{ \partial ^2 y}{ \partial x^2 } = \frac{ \partial ^2 y} {\partial t ^2} $$

to

$$ \frac{ \partial ^2 y}{\partial \alpha \partial \beta } = 0$$

? I'm having a hard time following the proofs on wolfram alpha, etc

 

[pierce15]

I think I've got it, although I don't think this is a legitimate proof:

$$ \frac{ \partial \alpha}{\partial t} = a, \quad \frac{\partial \beta}{\partial t} = -a \implies \frac{1}{-a^2} \frac{ \partial \alpha \partial \beta}{\partial t^2 } = 1 $$

By the same logic,

$$ \frac{ \partial \alpha \partial \beta}{ \partial x^2} = 1$$

From the wave equation:

$$ a^2 \frac{ \partial ^2 y}{\partial x^2} \frac{\partial x^2}{\partial \alpha \partial \beta } = \frac{ \partial ^2 y}{\partial t^2} \frac{\partial t^2}{\partial \alpha \partial \beta} (-a^2) $$

$$ \implies \frac{ \partial ^2 y}{\partial \alpha \partial \beta } = 0 $$

Is that OK? It's cool to just treat everything like a fraction?