Maximum Group Velocity

For the constant coefficient problem, the characteristic polynomial relating frequencies $\omega$ to spatial wavenumber $\Vert$$\beta$$\Vert _{2} = \sqrt{\beta_{x}^{2}+\beta_{y}^{2}}$, from (3.10), will be

$$\left(\omega^{4}-\frac{\omega^{2}}{\rho}\left(\frac{12\kappa^{2} ... ...th$\beta$}\Vert _{2}^{2}-\frac{12 G\kappa^{2}}{\rho h^{2}}\right)\omega^{2} = 0$$

The first factor, which is similar in form to that which defines the Timoshenko system, from (5.19), has four roots, $\omega_{1\pm}$, $\omega_{2\pm}$ which behave as

$$\lim_{\Vert\mbox{{\scriptsize\boldmath$\beta$}}\Vert _{2}\rightar... ...2\pm}= \pm\Vert\mbox{\boldmath$\beta$}\Vert _{2}\sqrt{\frac{G\kappa^{2}}{\rho}}$$

and the second factor has a pair of roots $\omega_{3\pm}$, which have the limiting behavior

$$\lim_{\Vert\mbox{{\scriptsize\boldmath$\beta$}}\Vert _{2}\rightar... ...}\omega_{3\pm} = \pm\Vert\mbox{\boldmath$\beta$}\Vert _{2}\sqrt{\frac{G}{\rho}}$$

All phase and group velocities are thus bounded. For the varying-coefficient problem, the maximum global group velocity will be

$$\gamma_{M, max}^{g} = \max_{{\bf x}\in\mathcal{D}}\sqrt{\frac{E}{\rho(1-\nu^{2})}}$$