Both the electric voltage and the satisfy the telegraph equation
where is distance, is time and are non-negative constants. The equation is a generalised form of the wave equation.
In the special case , the solution is
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).
|Date of creation||2013-03-22 18:03:15|
|Last modified on||2013-03-22 18:03:15|
|Last modified by||pahio (2872)|