We know that the classical, one dimensional wave equation
\begin{equation*}
u_{tt}= c^2 u_{xx}
\end{equation*}
with initial conditions
\begin{equation*}
u(x,0)= f(x)\ , \qquad u_t(x,0)= g(x)
\end{equation*}
where $c>0$ is constant, have solutions given by D'Alambert formula:
\begin{equation*} u(x, t)= \frac{1}{2}\big(f(x+ ct)+ f(x- ct) \big)+ \frac{1}{2c} \int_{x-ct}^{x+ct} g(s)\,ds
\end{equation*}
In the classical sense, that is, under some differentiability conditions of $f$ and $g$ (what are they ? $C^1,\ C^2$ ?) , $u$ is twice differentiable both with respect to $x$ and $t$ and satisfies the wave equation.
The same formula suggests that there might be other solutions as well, in a 'weak' sense, for example :
$$u(x,t)= F(x-ct) + G(x+ct)$$
can be verified to describe physical wave functions even when $F, G$ are discontinuous.
$F$ describes a right traveling wave, and $G$ a left traveling wave.
It seems like in this case, $u$ satisfies the wave equation as a distribution (?)
I am looking for a good reference on this matter.
How is D'alembert formula generalized and how it is explained with distribution theory ?
This is already answered in another post
Show that D'Alembert solution gives distributional solution to the wave-equation
It is indeed a distributional solution under far weaker conditions.