Phase and Group Velocity

In the constant-coefficient case, where $r=g=0$, the dispersion relation, defined in (3.10), will be

$$\chi_{TL}(\omega, \beta) = -\omega^{2}lc+\beta^{2} = 0$$

in terms of real frequencies $\omega$ and wavenumbers $\beta$, and has solutions

$$\omega = \pm\frac{\beta}{\sqrt{lc}}$$

The phase and group velocities, from (3.12) are then

$$\gamma^{p}_{TL} = \gamma_{TL}^{g} = \pm\frac{1}{\sqrt{lc}}$$ (3.54)

and if $l$ and $c$ are functions of $x$, the maximal group velocity will be

$$\gamma_{TL, max}^{g} = \frac{1}{\sqrt{(lc)_{min}}}$$ (3.55)

where $(lc)_{min} = \min_{x\in\mathcal{D}}(lc)$.