Consider an effectively infinite transmission line (telephone line). Initially the line is dead, with both potential, \(\Phi\), and current, \(d \Phi / d t\), being equal to zero. At \(t=0\) we impose a signal at the origin of the line:

\[\Phi(0, t)=\Phi_{o} \cos (\omega t)\]

Solve the telegrapher's equation:

\[\frac{\partial^{2} \Phi}{\partial x^{2}}=L C \frac{\partial^{2} \Phi}{\partial t^{2}}+(R C+G L) \frac{\partial \Phi}{\partial t}+R G \Phi\]

for the transient and steady-state responses. Here, \(L\) is the inductance, \(C\) is the capacitance, \(R\) is the resistance, and \(G\) is the impedance of the line.