= Solution
State a time convention before using complex response: take fields proportional to $e^{i(kz-\omega t)}$, with $k=k_r+i\kappa$ and $\kappa\geq0$ for a passive wave entering the material. In a homogeneous Ohmic medium, <Maxwell equations> with $\mathbf J=\sigma\mathbf E$ give
$$
k^2=\mu\epsilon\omega^2+i\mu\sigma\omega
=\mu\omega^2\left(\epsilon+\frac{i\sigma}{\omega}\right).
$$
The <ohmic-conductor complex wavenumber> describes both phase advance and <amplitude> decay. Real constant conductivity is only a low-frequency approximation. In the <Drude model>, $m\dot{\mathbf v}+m\gamma\mathbf v=-e\mathbf E$ gives
$$
\sigma(\omega)=\frac{ne^2}{m(\gamma-i\omega)},\qquad
\epsilon_r(\omega)=1-\frac{\omega_p^2}{\omega(\omega+i\gamma)},\qquad
\omega_p^2=\frac{ne^2}{m\epsilon_0},
$$
with a fixed positive ionic background and no additional bound-electron contribution in this simple model. The <plasma frequency> marks the inertial electron-response scale. More elaborate band structure, interband transitions and nonlocal transport limit this scalar treatment.<Electromagnetism lecture notes by Richard Fitzpatrick>[Dielectric constant of a collisional plasma, equations (1201)–(1205): electron inertia, collisions and dielectric response]{ref}
In the collisionless nonmagnetic limit, the <dispersion relation> is $\omega^2=\omega_p^2+c^2k^2$. For $\omega>\omega_p$,
$$
v_p=\frac\omega k=\frac{c}{\sqrt{1-\omega_p^2/\omega^2}}>c,\qquad
v_g=\frac{d\omega}{dk}=c\sqrt{1-\omega_p^2/\omega^2}<c.
$$
A crest can move faster than $c$ without carrying a new signal faster than <light>. Below cutoff the ideal wave is evanescent, with penetration <length> $c/\sqrt{\omega_p^2-\omega^2}$; it does not propagate as an ordinary real-$k$ bulk wave. Evanescence in the lossless limit is not Joule dissipation. Nonzero collision rate gives a dissipative conductivity and smooths this ideal cutoff picture.
For a good conductor with approximately real $\sigma\gg\omega\epsilon$, the spatial branch becomes
$$
\boxed{k\simeq\frac{1+i}{\delta},\qquad\delta=\sqrt{\frac{2}{\mu\sigma\omega}}.}
$$
A field <amplitude> falls as $e^{-z/\delta}$, while <intensity> falls as $e^{-2z/\delta}$. The <phase velocity> is $\omega\delta$ in this limit. Magnetic and electric <amplitudes> are not related by the <vacuum> ratio: $Z_s=E/H=(1-i)\sqrt{\mu\omega/(2\sigma)}$ with the chosen time convention. Its positive real part is the surface resistance. The local mean heating is $\tfrac12\operatorname{Re}\sigma\,|E|^2$, consistent with electromagnetic <energy> flowing into the metal. <skin depth> is an <amplitude> <length>; interpreting it as an <intensity> <length> gives a factor-of-two error.
A transmission-line mode propagates chiefly through the <dielectric> between conductors, even though finite metal conductivity causes loss at its boundaries. With per-length parameters $R',L',G',C'$, the <telegrapher's equations> give, for the same $e^{-i\omega t}$ convention,
$$
\frac{dV}{dz}=-(R'-i\omega L')I,\qquad
\frac{dI}{dz}=-(G'-i\omega C')V.
$$
Writing a forward wave as $e^{-\Gamma z}$, $\Gamma=\alpha-i\beta$, gives $\Gamma^2=(R'-i\omega L')(G'-i\omega C')$. In the low-loss regime $R'\ll\omega L'$ and $G'\ll\omega C'$, set $Z_0\simeq\sqrt{L'/C'}$ to obtain
$$
\boxed{\alpha\simeq\frac{R'}{2Z_0}+\frac{G'Z_0}{2},\qquad
\beta\simeq\omega\sqrt{L'C'}.}
$$
This <low-loss transmission-line attenuation> separates series conductor loss from <dielectric> leakage loss. Average forward power falls as $e^{-2\alpha z}$. In the normal skin-effect regime, reduced conducting <area> makes the AC conductor resistance increase roughly as $\sqrt\omega$; <dielectric loss> can supply a separate <frequency> dependence. Reflection from a mismatched termination changes delivered power without being, by itself, dissipation in the line.<Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher>[Chapter14, §§14.7–14.9: distributed losses, Ohmic waves and conductor-loss approximation]{ref}
To test loss, compare several <lengths> of otherwise identical line with matched terminations, calibrate connectors and incident/reflected power, and monitor <temperature>. Separate <amplitude> attenuation from phase delay and from standing-wave minima. Bulk penetration, line attenuation and <dielectric> propagation <speed> are related electromagnetic quantities but describe different experimental geometries.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2010/paper-1-conducting-waves.png]
{title=Conducting-wave propagation}
{description=Chosen dispersion and decay models.}
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