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).
It is possible to show, using basic laws of electrical circuit theory, that the electrical current satisfies the PDE
where the constants and are, for unit length of cable, respectively the resistance, inductance, capacitance and leakage conductance. The voltage also satisfies (5). Special cases of (5) arise in particular situations. For a submarine cable is negligible and frequencies are low so inductive effects can also be neglected. In this case (5) becomes
which is called the submarine equation or telegraph equation . For high frequency alternating currents, again with negligible leakage, (5) can be approximated by
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.