Home exercise 6.2.3: Dispersion in a good conductor

${\beta }^2=\omega \mu \sigma /2$, so $\omega =2{\beta }^2/\mu \sigma$, a parabola in the $\omega$–$\beta$ plane, and

$v_g>v_p$: the conductor is anomalously dispersive. On the parabola the tangent is always twice as steep as the secant.

$v_p=\omega /\beta =\omega \delta$. Copper at $1\mathrm{MHz}$: $\delta =66.1\mu \mathrm{m}$, $v_p=2\pi \times 10^6\times 66.1\times 10^{-6}=415m/s$ and $v_g=830m/s$. Seawater at $10\mathrm{kHz}$ ($\sigma /\omega \epsilon =8.9\times 10^4$, a good conductor): $\delta =1/\sqrt{\pi \times 10^4\times 4\pi \times 10^{-7}\times 4}=2.52\mathrm{m}$, $v_p=1.58\times 10^{5}m/s$ and $v_g=3.16\times 10^{5}m/s$.

$v_p=\omega \delta =\sqrt{2\omega /\mu \sigma }$ grows as $\sqrt{f}$, and so does $v_g$. For copper both stay far below $c$. These speeds are those of a wave inside the metal, where it survives only a skin depth or two: a signal on a copper cable travels in the dielectric between the conductors, at $c/\sqrt{{\epsilon }_r}$. In a lossy medium the group velocity also loses its meaning as the speed of energy once the pulse is damped within one wavelength.