Hi i have a doubt with the deduction of general solution of wave equation.
Let $u_{tt}-c^2u_{xx}=0\tag 1$
The wave equation.
We know the wave equation have two family of characteristic
$x-ct=k_1\,\,\,\, , k_1\in \mathbb{R}\tag 2$
$x+ct=k_2\,\,\,\, , k_2\in \mathbb{R}\tag 3$
Then, we need make the change:
$\begin{equation} \sigma=x-ct \\ \gamma=x+ct \end{equation}$
Then $\bar{u}(\sigma,\gamma)=\bar{u}(x-ct,x+ct)=u(x,t)$
With the change of variable, we have the wave equation is reduced to
$$\frac{\partial^2 \bar{u}}{\partial \sigma \partial \gamma}=0 \tag 4$$
This implies
$\int\frac{\partial^2{\bar{u}}}{\partial\sigma\partial\gamma}d\sigma=g(\gamma)\implies \int\frac{\partial{\bar{u}}}{\partial\gamma}=\int g(\gamma)d\gamma$
Then
$\bar{u}(\sigma,\gamma)=G(\gamma)+F(\sigma)$
Where $G(\gamma)=\int g(\gamma)d\gamma$
Returning the change we have:
$u(x,t)=\bar{u}(x-ct,x+ct)=F(x-ct)+G(x+ct)$
I don't see the step $(4)$ Can someone explain me that step?