= Solution
A <transmission line> stores electric and magnetic <energy> along its length, so <voltage> and current depend on both position and time. A distributed circuit uses <inductance> $L'$ and <capacitance> $C'$ per length, together with resistance $R'$ and leakage conductance $G'$. Applying <voltage> and charge balance to a short segment gives the <telegrapher's equations>,
$$
\partial_xV=-R'I-L'\partial_tI,\qquad
\partial_xI=-G'V-C'\partial_tV.
$$
For a lossless line, both variables obey <wave equations> with speed $v=1/\sqrt{L'C'}$ and <characteristic impedance> $Z_0=\sqrt{L'/C'}$. Forward and backward <waves> obey $I^+=V^+/Z_0$ and $I^-=-V^-/Z_0$. Matching a termination $Z_L$ gives the <voltage> <reflection coefficient>
$$
\boxed{\Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0}.}
$$
A matched load absorbs without reflection; an open circuit has $\Gamma_L=1$, and a short has $\Gamma_L=-1$. Reflections can produce <standing waves>. A coaxial line confines the fields between conductors and, for a uniform <dielectric> TEM <wave>, has $v=1/\sqrt{\mu\epsilon}$. Finite loss attenuates <waves>; frequency-dependent losses or material properties can also distort pulses. \b[A long line is a wave-carrying distributed system, so it cannot generally be replaced by one lumped capacitor.]
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