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Phase and Group Velocity

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

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

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

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

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

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

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

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

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

Stefan Bilbao 2002-01-22