I am trying to understand my lecture notes.
We have $$\frac{\partial^2 u}{\partial t^2}=a^2\frac{\partial^2 u}{\partial x^2}.$$
For $a>0$, substitute: $\xi=x-at, \eta=x+at.$ Then
$u(x,t)=v(x-at,x+at)=v(\xi,\eta), \\\frac{\partial^2v}{\partial \xi \partial \eta}=0, \\ v(\xi,\eta)=\psi_1(\xi)+\psi_2(\eta).$
Then this notation is used:
$S(\xi)=\frac{\partial v}{\partial\xi}, R(\eta)=\frac{\partial v}{\partial \eta}.$ I don't understand why not write $S(\eta)=\frac{\partial v}{\partial\eta}$, but so it is in my notes. $s(t,x)=S(x-at), r(t,x)=R(x+at).$ I do not understand why we need this chain of different notations.
From this, a conclusion is made that the original equation is equivalent to this system:
$$ \begin{equation} \begin{cases} \frac{\partial r}{\partial t}+ a\frac{\partial r}{\partial x}=0\\ \frac{\partial s}{\partial t}-a\frac{\partial s}{\partial x}=0 \end{cases}\, \end{equation}. $$
I do not understand how we came to this conclusion. I see that $\frac{\partial^2 u}{\partial t^2}=a^2\frac{\partial^2 u}{\partial x^2}$ can be written as $\left(\frac{\partial }{\partial t }+ a\frac{\partial }{\partial x}\right) \left(\frac{\partial }{\partial t}-a \frac{\partial }{\partial x}\right)u=0$, but this has been of no help to me.
I would be very grateful if someone could give a detailed explanation. Thank you.