Telegrapher's Equation

(or equation of telegraphy), in mathematics, a partial differential equation that, under certain simplifying assumptions, describes the propagation of a current along a conductor.

The current i and voltage u satisfy the following system of equations:

Here, x is a coordinate reckoned along the conductor, t is time, and C, G, L, and R are the capacitance, conductance, inductance, and resistance, respectively, of the conductor per unit length. If LC ≠ 0, a suitable change of variables leads to the equation

which is one form of the telegrapher’s equation.

Boundary value problems for the telegrapher’s equation are solved by methods developed for the equation of vibration of a string (seeWAVE EQUATION). The telegrapher’s equation reduces to this equation when k = 0. When k ≠ 0, a dispersion phenomenon exists in the process described by the telegrapher’s equation (see, for example, DISPERSION OF SOUND). Operational calculus and special functions are commonly used to solve the telegrapher’s equation.

The telegrapher’s equation was studied by Lord Kelvin in 1855 for the case L = 0. G. Kirchhoff dealt with the general case in 1857. Others who have investigated the equation include O. Heaviside in 1876 and H. Poincaré in 1897. The term “telegrapher’s equation” was suggested by Poincaré (Téquation destélégraphistes).