We present an analytical calculation of the transient response of ideal (i.e. lossless) transmission lines. The calculation presented considers a length of transmission line connected to a signal generator with output impedance $Z_\mathrm{g}$ and terminated with a load impedance $Z_\mathrm{L}$. The approach taken is to analyze a circuit model of the system in the complex-frequency or $s$-domain and then apply an inverse Laplace transform to recover the time-domain response. We consider both rectangular pulses and voltage steps (i.e. the Heaviside function) applied to the input of the transmission line. Initially, we assume that $Z_\mathrm{g}$ and $Z_\mathrm{L}$ are purely real/resistive. At the end of the paper, we demonstrate how the calculations can be generalized to consider reactive impedances.