telegraph equation
Both the electric voltage and the current in a double conductor 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 differential equation (1) 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 of the quadratic equation
vanishes, the roots coincide to
, and
. Therefore (2) reads
According to the delay theorem, we have
wnere
is Heaviside step function. Thus we obtain for
the expression of (3).
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As of this snapshot date, this entry was owned by pahio.