# Telegrapher's equations

Telegraph equation is a general form of the wave equation.

## General

With the constitutive equations can see the Maxwell equations in charge-free regions of space to describe

And

In the case of an insulator and the Maxwell equations reduce to ( vector ) wave equation.

Each of these equations is a special form of the telegraph equation. This is a partial differential equation (for hyperbolic, elliptical, and parabolic in ) and is in the general form

In this form it is an equation. Many other nonlinear partial differential equations of physics includes as special cases ( wave equation, diffusion equation, Helmholtz equation, potential equation ) and it is accordingly generally treatable

## Telegraph equation with a> 0, b > 0; c = 0

The equations are generally of the type:

Substituting for example E or F through H and chooses a = b = εμ and σμ we obtain the wave equation for a lossy dielectric.

Or

## Telegraph equation with a> 0; b = 0; c = 0

The equations are generally of the type:

And bear the name generically wave equation. Substituting for example E or F through H and chooses a = εμ, we obtain the wave equations of electromagnetic waves in lossless space.

Or

If we replace F by u or i, we obtain the wave equation for the propagation of voltage and current waves along lossless lines:

Or

If we replace F by the deflection L of mass particles and a by the inverse of the wave propagation velocity v, we obtain the wave equation of mechanical waves:

## Telegraph equation with a = 0; b> 0; c = 0

The equations are generally of the type:

And bear the name generically heat equation or diffusion equation. If we replace F by E, H or JL and selects b = σμ, we obtain the equations for the flow field in conductors with current displacement:

Or

Or

If we replace F by the temperature T and b by Cρ / λ (C specific heat, ρ density, thermal conductivity λ ), we obtain the partial differential equations for spatio-temporal temperature distributions:

## Source

- Adolf J. Schwab: terminology of the field theory. Practical, clear introduction. Electromagnetic fields, Maxwell's equations, gradient, rotation, divergence, finite elements, finite differences, spare charging method, boundary element method, method of moments, Monte Carlo method. 6 unchanged edition. Springer -Verlag, Berlin and others, 2002, ISBN 3-540-42018-5.

- Electrodynamics
- Electromagnetic Theory
- Partial Differential Equations