He developed the transmission line theory (also known as the "telegrapher's equations").

The circuit in the top figure implements the solutions of the telegrapher's equations.

The solutions of the telegrapher's equations can be inserted directly into a circuit as components.

This model is now known as the telegrapher's equation and the distributed elements are called the primary line constants.

It also implements the solutions of the telegrapher's equations.

If and are not neglected, the Telegrapher's equations become:

For a lossless transmission line, the second order steady-state Telegrapher's equations are:

(Interestingly, Kirchhoff derived the telegrapher's equations in 1857 without using displacement current.

The Telegrapher's Equations are developed in similar forms in the following references:

See Telegrapher's equation.

Telegrapher's Equations.

The telegrapher's equations, like all other equations describing electrical phenomena, result from Maxwell's equations.

In 1885 Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the telegrapher's equations.

Applying the transmission line model based on the telegrapher's equations, the general expression for the characteristic impedance of a transmission line is:

(Concept of inhomogeneous waves propagation - Show the importance of the telegrapher's equation with Heaviside's condition.)

The primary coefficients being the physical properties of the line; R,C,L and G, from which the secondary coefficients may be derived using the telegrapher's equation.

John Greaton Wöhlbier, ""Fundamental Equation" and "Transforming the Telegrapher's Equations".

The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time.

For a transmission line, the Heaviside condition of the telegrapher's equation tells us that the wavenumber must be proportional to frequency for the transmission of the wave to be undistorted in the time domain.

The first Transatlantic telegraph cable suffered from severe signal integrity problems, and analysis of the problems yielded many of the mathematical tools still used today to analyze signal integrity problems, such as the telegrapher's equations.

By the 1890s, Oliver Heaviside had produced the modern general form of the telegrapher's equations which included the effects of inductance and which were essential to extending the theory of transmission lines to higher frequencies required for high-speed data and voice.

Heaviside and Mihajlo Idvorski Pupin in later decades understood that the bandwidth of a cable is hindered by an imbalance between capacitive and inductive reactance, which causes a severe dispersion and hence a signal distortion; see Telegrapher's equations.