telegraph equation
Both the electric voltage and the satisfy the telegraph equation
(1) |
where is distance, is time and are non-negative constants. The equation is a generalised form of the wave equation.
If the initial conditions are and the boundary conditions , , then the Laplace transform of the solution function is
(2) |
In the special case , the solution is
(3) |
Justification of (2). Transforming the partial differential equation (1) ( may be regarded as a parametre) gives
which due to the initial conditions simplifies to
The solution of this ordinary differential equation is
Using the latter boundary condition, we see that
whence . Thus the former boundary condition implies
So we obtain the equation (2).
Justification of (3). When the discriminant (http://planetmath.org/QuadraticFormula) of the quadratic equation vanishes, the roots (http://planetmath.org/Equation) coincide to , and . Therefore (2) reads
According to the delay theorem, we have
Thus we obtain for the expression of (3).
Title | telegraph equation |
Canonical name | TelegraphEquation |
Date of creation | 2013-03-22 18:03:15 |
Last modified on | 2013-03-22 18:03:15 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 13 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 35L15 |
Classification | msc 35L20 |
Synonym | telegrapher’s equation |
Related topic | HeavisideStepFunction |
Related topic | SecondOrderLinearDifferentialEquation |
Related topic | DelayTheorem |
Related topic | MellinsInverseFormula |
Related topic | TableOfLaplaceTransforms |