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  "body": "\n## Definition\n\nLet $L \u003e 0$, $C \u003e 0$, $R \\ge 0$ and $G \\ge 0$ be constants: the inductance, capacitance, resistance and conductance per unit length of a line.\nThe *telegrapher's equations* are the system of [[partial differential equations|GC5BDZ6QBC]] for the voltage $V(x, t)$ and the current $I(x, t)$, with $x \\in \\R$ and $t \\in \\R$:\n$$\\frac{\\partial V}{\\partial x} = -L \\frac{\\partial I}{\\partial t} - R I, \\qquad \\frac{\\partial I}{\\partial x} = -C \\frac{\\partial V}{\\partial t} - G V.$$\nIt is a first-order [[hyperbolic system|92C3S03AQZ]].\n\n## Remarks\n\nFor $C^2$ solutions, differentiating the first equation in $x$ and using the second gives the telegraph equation\n$$V_{xx} = L C\\, V_{tt} + (R C + G L)\\, V_t + R G\\, V,$$\nand the same equation holds for $I$.\nExample: 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.\nIndeed $V_x = F'$ and $-L I_t = L c \\sqrt{C/L}\\, F' = F'$, and $I_x = \\sqrt{C/L}\\, F'$ and $-C V_t = C c F' = \\sqrt{C/L}\\, F'$.\nNon-example: with $R \u003e 0$ and $G = 0$, this undamped pair fails the first equation, since the extra term $-R I$ is nonzero when $F \\ne 0$.\nWhen $R = G = 0$ the telegraph equation is the [[wave equation|5EHD2XBQDG]] with speed $1/\\sqrt{LC}$.\n",
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