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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?

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