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I am ignorant of EE, but not the wave equation, so bear with me.

The (scalar) wave equation \$d_{tt} - c^2d_{xx} = 0\$ describes the displacement (and velocity) of some medium through space and time. It works for air, EM fields, and many physical quantities.

I am interested in recreating transport phenomena modeled by the wave equation within a conducting wire, so the voltage (or perhaps something else) at any point of the wire would be described by the wave equation.

My main interest in this question derives from the difficulty of experimentally (rather than theoretically or computationally) modeling more complex wave equations \$d_{tt} - c(x,t)^2d_{xx} = 0\$ with variable wave speed. If there were a straightforward way of recreating variable coefficient wave equations using EE, it would be helpful to me!

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    \$\begingroup\$ If you want to change the propagation velocity along the transmission path you could use a microstrip transmission line and change the width. As the line gets skinnier more field lines go through the air and the effective permittivity decreases. This will slightly increase the propagation velocity. \$\endgroup\$
    – DavidG25
    Jan 9, 2020 at 16:28

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Certainly

To quote wikipedia;

The telegrapher's equations (or just telegraph equations) are a pair of coupled, linear partial differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who developed the transmission line model in the 1880s. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line.

The theory applies to transmission lines of all frequencies including direct current and high-frequency. Originally developed to describe telegraph wires, the theory can also be applied to radio frequency conductors, audio frequency (such as telephone lines), low frequency (such as power lines), and pulses of direct current. It can also be used to electrically model wire radio antennas as truncated single-conductor transmission lines.

To clarify, the "telegrapher's equations" are a set of equations that give you the voltage and/or current, present at any length along a transmission line and at any point in time.

A transmission line can be anything which propagates voltages/ conducts current along its length, a close to ideal transmission line would be a good quality coax cable, but even a loose bundle of wires can in theory be modeled as a transmission line.

The telegrapher's equations used to model transmission lines are not wave equations, but they can be rewritten to be of the form of wave equations, here is a link I found which explains how; https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.ittc.ku.edu/~jstiles/723/handouts/The_Transmission_Line_Wave_Equation.pdf&ved=2ahUKEwj4kc_W9_bmAhXlzcQBHdSCC5cQFjAHegQIBBAC&usg=AOvVaw1VnVhK-_iH9CWsDbR7WLeA&cshid=1578588439100

I Hope this helps.

As a side note; thinking of electricity as 1d (scalar) waves on a wire is actually the correct physical way of thinking of electricity, but rather than just having one wave with one amplitude we have two perpendicular waves with two different amplitude, the voltage (or electric field wave) and the current (or magnetic field wave), the proportion of the electric wave to the magnetic wave is what we call the characteristic impedance, in units of ohms (voltage over current).

Another side note; the thermal noise generated by a "Hot" resistor is actually the 1d equivalent of the Black body radiation generated by a three dimensional "Hot" object

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  • \$\begingroup\$ This is so useful, and absolutely fascinating. I had no idea that the voltage and current mapped directly to E and M fields. I knew about the telegrapher’s equation, but had no idea it could be transformed into wave equations! Thanks for this. \$\endgroup\$
    – Wapiti
    Jan 10, 2020 at 17:51
  • \$\begingroup\$ I'm glad that you got something useful out of it (:. Yes it is beautiful! \$\endgroup\$
    – user173292
    Jan 10, 2020 at 18:58

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