# Doubt with general solution of wave equation

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:

$$$$\sigma=x-ct \\ \gamma=x+ct$$$$

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?