Telegrapher's equations

Remarks

For $C^2$ solutions, differentiating the first equation in $x$ and using the second gives the telegraph equation $$V_{xx} = L C\, V_{tt} + (R C + G L)\, V_t + R G\, V,$$ and the same equation holds for $I$. Example: with $R = G = 0$, for any $F \in C^1(\R)$ the pair $V = F(x - ct)$, $I = \sqrt{C/L}\, F(x - ct)$ with $c = 1/\sqrt{LC}$ is a solution. Indeed $V_x = F'$ and $-L I_t = L c \sqrt{C/L}\, F' = F'$, and $I_x = \sqrt{C/L}\, F'$ and $-C V_t = C c F' = \sqrt{C/L}\, F'$. Non-example: with $R > 0$ and $G = 0$, this undamped pair fails the first equation, since the extra term $-R I$ is nonzero when $F \ne 0$. When $R = G = 0$ the telegraph equation is the wave equation with speed $1/\sqrt{LC}$.