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

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It is possible to show, using basic laws of electrical circuit theory, that the electrical current [maths rendering] satisfies the PDE

[maths rendering] (5)

where the constants [maths rendering] and [maths rendering] are, for unit length of cable, respectively the resistance, inductance, capacitance and leakage conductance. The voltage [maths rendering] also satisfies (5). Special cases of (5) arise in particular situations. For a submarine cable [maths rendering] is negligible and frequencies are low so inductive effects can also be neglected. In this case (5) becomes

[maths rendering] (6)

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

[maths rendering] (7)

which is called the high frequency line equation .

Task!

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.