The wave equation as a system of first-order PDE's
Not exactly what you asked for, but maybe a sensible alternative (I hope so).
Employing Operator Calculus we indeed have sort of a splitting into first order equations:
$$
u_{xx}-u_{tt} =
\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial t^2} =
\left[ \left(\frac{\partial}{\partial x}\right)^2 -
\left(\frac{\partial}{\partial t}\right)^2 \right] u = 0
\quad \Longleftrightarrow \\
\left(\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right)
\left(\frac{\partial}{\partial x} - \frac{\partial}{\partial t}\right) u = 0
\\ \left(\frac{\partial}{\partial x} - \frac{\partial}{\partial t}\right)
\left(\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right) u = 0
$$
Let $u(x,t) = f(x-t)$ , then:
$$
\left(\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right) f(x-t)
= \frac{\partial f}{\partial (x-t)}\frac{\partial (x-t)}{\partial x}
+ \frac{\partial f}{\partial (x-t)}\frac{\partial (x-t)}{\partial t} = \\
= \frac{\partial f}{\partial (x-t)}(+1) + \frac{\partial f}{\partial (x-t)}(-1) = 0
$$
Let $u(x,t) = g(x+t)$ , then:
$$
\left(\frac{\partial}{\partial x} - \frac{\partial}{\partial t}\right) g(x+t)
= \frac{\partial g}{\partial (x+t)}\frac{\partial (x+t)}{\partial x}
- \frac{\partial g}{\partial (x+t)}\frac{\partial (x+t)}{\partial t} = \\
= \frac{\partial g}{\partial (x+t)}(+1) - \frac{\partial g}{\partial (x+t)}(+1) = 0
$$
It is concluded that the general solution is given by any linear combination
of $f$ and $g$ as :
$$
u(x,t) = \lambda\,f(x-t) + \mu\,g(x+t)
$$
Update. Writing the wave equation as a system of first-order partial differential equations, using the above:
$$
\left(\frac{\partial}{\partial x} - \frac{\partial}{\partial t}\right) u = v \quad ; \quad
\left(\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right) v = 0 \\
\left(\frac{\partial}{\partial x} + \frac{\partial}{\partial t}\right) u = v \quad ; \quad
\left(\frac{\partial}{\partial x} - \frac{\partial}{\partial t}\right) v = 0
$$
Or:
$$
\frac{\partial u}{\partial x} - \frac{\partial u}{\partial t} = v \quad ; \quad
\frac{\partial v}{\partial x} + \frac{\partial v}{\partial t} = 0 \\
\frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = v \quad ; \quad
\frac{\partial v}{\partial x} - \frac{\partial v}{\partial t} = 0
$$