### 3 Transmission line equations

In a long electrical cable or a telephone wire both the current and voltage depend upon position along the wire as well as the time (see Figure 6).

Figure 6

It is possible to show, using basic laws of electrical circuit theory, that the electrical current $i\left(x,t\right)$ satisfies the PDE

$\frac{{\partial }^{2}i}{\partial {x}^{2}}=LC\frac{{\partial }^{2}i}{\partial {t}^{2}}+\left(RC+GL\right)\frac{\partial i}{\partial t}+RGi$ (5)

where the constants $R,L,C$ and $G$ are, for unit length of cable, respectively the resistance, inductance, capacitance and leakage conductance. The voltage $v\left(x,t\right)$ also satisfies (5). Special cases of (5) arise in particular situations. For a submarine cable $G$ is negligible and frequencies are low so inductive effects can also be neglected. In this case (5) becomes

$\frac{{\partial }^{2}i}{\partial {x}^{2}}=RC\frac{\partial i}{\partial t}$ (6)

which is called the submarine equation or telegraph equation . For high frequency alternating currents, again with negligible leakage, (5) can be approximated by

$\frac{{\partial }^{2}i}{\partial {x}^{2}}=LC\frac{{\partial }^{2}i}{\partial {t}^{2}}$ (7)

which is called the high frequency line equation .

What PDEs, already discussed, have the same form as equations (6) or (7)?

(6) has the same form as the one-dimensional heat conduction equation.

(7) has the same form as the one-dimensional wave equation.