Given the wave equation:
$$ \frac{\partial^2 f}{\partial t^2} = c^2 \frac{\partial^2 f}{\partial x^2} $$
I change variables in this way:
$$ \xi = x+ct \\ \eta=x-ct $$
And the differential operators transform:
$$ \frac{\partial }{\partial x} = \frac{\partial }{\partial \xi} + \frac{\partial }{\partial \eta} \\ \frac{\partial }{\partial t} = c^2 ( \frac{\partial }{\partial \xi} – \frac{\partial }{\partial \eta} ) $$
I know this has something to do with the chain rule, but I'm not able to understand.
How do I verify that these equalities are correct?
How do I get to them?
Thank you.
Best Answer
Define $F(\xi,\eta)=f(t,x)$. Then, with some abuse of notation, $$ \frac{\partial}{\partial x}f(t,x)=\frac{\partial}{\partial x}F(\xi,\eta)=\frac{\partial F}{\partial \xi}\frac{\partial\xi}{\partial x}+\frac{\partial F}{\partial \eta}\frac{\partial\eta}{\partial x}=\frac{\partial F}{\partial \xi}+\frac{\partial F}{\partial \eta}. $$ This is the meaning of $$ \frac{\partial}{\partial x}=\frac{\partial }{\partial \xi}+\frac{\partial }{\partial \eta}. $$ A similar computation applies to the other derivative.