One Dimension Wave Equation (Calculus of several variables) I can't find any way to get to that expression. Honestly I don't how to start this. Any recommendations?      
Consider the One Dimension Wave Equation
$$
\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}
$$
where $u:=u(x, t) .$ If now we make a change of variable $\xi=x+c t, \eta=x-c t,$ show that the wave equation can be written as $$\frac{\partial^{2} u}{\partial \eta \partial \xi}=0$$
 A: The solution is quite simple: Try to substitude $\xi$ and $\eta$ in
$\frac{\partial^2 u}{\partial \eta \partial \xi}=0$,
with the given relations and try to recover your first equation. The rest is up to you. It is done in one second.
Regards
A: If you want to approach this from the other direction and take the brute force approach, note that by the chain rule,

*

*$\frac{\partial u}{\partial t} = \frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial t}+\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial t}$,

*$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial x}+\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x}$.

Since $\xi=x+c t, \eta=x-c t$,
$
\frac{\partial \eta}{\partial t}=-c, \frac{\partial \eta}{\partial x}=1,\\
\frac{\partial \xi}{\partial t}=c, \frac{\partial \xi}{\partial x}=1.
$
Replacing these in the first two equations:

*

*$\frac{\partial u}{\partial t} = -c\frac{\partial u}{\partial \eta}+c\frac{\partial u}{\partial \xi}$,

*$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial \eta}+\frac{\partial u}{\partial \xi}$.

Now, take the derivatives again,

*

*$\frac{\partial^2 u}{\partial t^2} = -c\frac{\partial^2 u}{\partial \eta \partial t}+c\frac{\partial^2 u}{\partial \xi \partial t}$,

*$\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial \eta \partial x}+\frac{\partial^2 u}{\partial \xi \partial x},$
and use the chain rule again:

*

*$\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial^2 \eta}-c^2\frac{\partial^2 u}{\partial \eta \partial \xi}-c^2\frac{\partial^2 u}{\partial \xi \partial \eta}+c^2\frac{\partial^2 u}{\partial^2 \xi }$,

*$\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial^2 \eta} +\frac{\partial^2 u}{\partial \eta \partial \xi} +\frac{\partial^2 u}{\partial \xi \partial \eta} +\frac{\partial^2 u}{\partial^2 \xi}$.

Finally, replace these in
$$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$$
to get
$$ c^2\frac{\partial^2 u}{\partial^2 \eta}-c^2\frac{\partial^2 u}{\partial \eta \partial \xi}-c^2\frac{\partial^2 u}{\partial \xi \partial \eta}+c^2\frac{\partial^2 u}{\partial^2 \xi } = \frac{\partial^2 u}{\partial x^2} = c^2\frac{\partial^2 u}{\partial^2 \eta} +c^2\frac{\partial^2 u}{\partial \eta \partial \xi} +c^2\frac{\partial^2 u}{\partial \xi \partial \eta} +c^2\frac{\partial^2 u}{\partial^2 \xi},$$ which simplifies to
$$\frac{\partial^2 u}{\partial \xi \partial \eta}=0$$
after cancellations.
