General solution of transmissionline equations
Equations of a transmission line:
Initial conditions:
General solutions in the Laplace transform 's' domain:
General solutions in the time domain:
where u is the propagation speed and Z_{0} is the characteristic impedance:

We will derive the general solution of transmission line equations.
Equations of a transmission line are:
The Laplace transform of the voltage v(x,t) and current i(x,t) are:
and the Laplace transform of its time derivative is:
Therefore, the Laplace transform of the equations are:
Here we assume the initial conditions at t=0 as:
, then we obtain the wave equation.:
Its general solutions are:
Using the propagation speed u and the characteristic impedance Z_{0} instead of L and C:
Multiplying exp(Ńs) in the 's' domain is equivalent to shifting in the time domain by Ń.
Therefore the general solutions in the time domain are
where the v_{a}/ v_{b} are a forward/ backward moving waves at the propagation speed u.