In 1858 a long cable was lied onto the see floor for transatlantic communication, from UK to Terranova in America. But the signal was very messy, incomprehensible, so they cranked up the voltage and, PUFF!, they burnt it, years of work gone in an instant. The problem was that the various spectral component of the signal did not have the same velocity, so the signal was distorted and broadened in time: Lord Kelvin (Thomson) proposed a solution:"More Voltage, Longer Signals!" It had been taking days for sending one message, and the cable was now burnt. Raw power this times was not the solution, even tho I agree it usually is the solution. A man without any degree or academic position, Mr. Heaviside, a true self made scientist, proposed (1876) a mathematical solution that was purely theoretical: adding inductance (another electrical component) along the way to make all velocities, for all frequencies equal. He was insulted by the engineers working on the problem for his lack of practical sense, they said "How can adding an obstacle along the path make the signal clearer, you are stupid Heaviside!"1. The solution he gave was the right one tho, and it stands to this day. Only in 1899 a Serbian-American scientist, Mr. Pupin proposed this stuff to the AT&T labs and they patented it. From 1899 underwater cable communication was tens of times faster, cheaper and more reliable. So sometimes is not important what you say, but how many people can hear you.
Here there is a small text concerning Wave Propagation in electric Lines. This does not suffice to explain the Heaviside trick because it regards lossless lines. Loss do exist tho, but they can easily taken into account by just adding a series resistance and a parallel resistance to the lossless line. But theoretical framework explained is what is most important: how to schematize a transmission line.
This derivation works because of an approximation: the Transverse Electric Magnetic approximation, that states that E_z and H_z along the wire are 0. This is obviously false for example for E_z or the electron could not move in the wire, but because the conductance rho is very high it is E_z is practically 0. The important thing to know is that the assumption made to derive the telegraphic equation can be justified starting from Maxwell equation in a medium. In the future I may give here a justification.
Thanks for the attention, Aug. 2026, Rome, Tommaso Virgilii.
1: These were not the exact words they used but this was the concept. Look it up: Victorian times were a disgrace for humanity...