= Solution
A <transmission line> must be treated as a distributed circuit when its propagation time is significant on the <signal> timescale. Let $R',L',G',C'$ denote resistance, <inductance>, leakage conductance and <capacitance> per unit length. Applying <Kirchhoff's voltage law> and <Kirchhoff's current law> to a short length $dx$ gives the <telegrapher's equations>
$$
\frac{\partial V}{\partial x}=-R'I-L'\frac{\partial I}{\partial t},\qquad
\frac{\partial I}{\partial x}=-G'V-C'\frac{\partial V}{\partial t}.
$$
For a lossless line, $R'=G'=0$. Differentiating again gives <wave equations> with
$$
\boxed{v=\frac1{\sqrt{L'C'}},\qquad Z_0=\sqrt{\frac{L'}{C'}}.}
$$
The travelling solutions can be written $V=V_+(t-x/v)+V_-(t+x/v)$ and $I=[V_+(t-x/v)-V_-(t+x/v)]/Z_0$. The minus sign for the backward wave is essential to <power> transport.
At a load $Z_L$, continuity of voltage and current gives the voltage <reflection coefficient>
$$
\boxed{\Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0}.}
$$
A matched termination $Z_L=Z_0$ absorbs the incident <signal> without reflection. An open circuit has $\Gamma_L=+1$ and a short circuit has $\Gamma_L=-1$. Superposed incident and reflected waves produce <standing waves>. A sinusoidal lossless line of length $\ell$ transforms the termination into
$$
Z_{\mathrm{in}}=Z_0\frac{Z_L+iZ_0\tan(k\ell)}{Z_0+iZ_L\tan(k\ell)}
$$
for the $e^{i\omega t}$ convention. At a quarter <wavelength>, $Z_{\mathrm{in}}=Z_0^2/Z_L$, allowing impedance transformation. A coaxial line supports a transverse electromagnetic mode whose ideal $L',C'$ are calculated in A4. Its <energy> travels in the fields between conductors; the <characteristic impedance> is not the wire's DC resistance. Losses introduce attenuation and make the <characteristic impedance> and propagation constant complex.
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