Consider a single-phase, two-wire, lossless transmission line terminated by an impedance at the receiving end and a source with Thevenin voltage and impedance at the sending end. The line, with length, has a surge impedance and wave velocity determined by the line's inductance and capacitance.

At the receiving end, the boundary condition states that the voltage equals the product of the receiving-end impedance and current. This relationship is expressed as a function of the incident and reflected voltage waves in the Laplace domain. The receiving-end voltage reflection coefficient characterizes how the wave reflects at the receiving end.

At the transmitting end, the boundary condition involves the difference between the source voltage and the product of the sending-end impedance and the current. This boundary condition leads to an equation for the incident voltage wave in terms of the sending-end reflection coefficient. This coefficient indicates the proportion of the wave reflected towards the source due to impedance mismatch.

The complete solutions for voltage and current along the transmission line are derived using the established boundary conditions and the reflection coefficients. These solutions incorporate both the incident and reflected waves, demonstrating how they combine to form the overall voltage and current at any point on the line. These solutions account for the effects of reflections at both the sending and receiving ends.

The line's characteristic impedance is derived from its inductance and capacitance per unit length, and these parameters also determine the wave velocity. By integrating these boundary conditions and parameters, a comprehensive understanding of the behavior of traveling waves on single-phase lossless transmission lines is obtained.