Group Velocity

The wave transfers the energy with the group velocity. Whether the group velocity is positive or negative is quite important considering the energy transfer. Using the dispersion relation, equation (3.38), The group velocity

$$v_g(R)=\frac{d \omega}{d k}={\rm sign}(k)\frac{\vert k\vert c_s^2-\pi G \Sigma_0}{\omega-m\Omega}.$$ (3.41)

For a region $R > R_{\rm CR}$, $\omega-m\Omega >0$ or $\nu > 0$. On the other hand, for a region $R < R_{\rm CR}$, $\omega-m\Omega <0$ or $\nu < 0$. Consider first the trailing wave. In the region $R > R_{\rm CR}$, long-waves propagate inwardly to the CR, since $\vert k\vert c_s^2< \pi G \Sigma_0$ for long-waves and $\nu > 0$. Short-waves propagate outwardly from the CR, since $\vert k\vert c_s^2> \pi G \Sigma_0$ for short-waves and $\nu > 0$. In the region $R < R_{\rm CR}$, long-waves propagate outwardly to the CR since $\vert k\vert c_s^2< \pi G \Sigma_0$ and $\nu < 0$. Short-waves propagate inwardly from the CR, since $\vert k\vert c_s^2> \pi G \Sigma_0$ and $\nu < 0$. As a result, it is concluded that the long-wave propagates toward the CR and the short-wave does away from the CR. As for the leading wave, the short-wave propagates toward the CR and the long-wave does away from the CR.

Assuming that the wave packet is made near the Lindblad resonance points, (1) the long-trailing waves propagate toward the co-rotation resonance points; (2) they are reflected by the $Q$-barrier; (3) they change their character to short-waves and propagate away from the co-rotation resonance points; and finally (4) the waves are absorbed at the center or propagate away to the infinity. (1) the short-leading waves propagate toward the co-rotation resonance points; (2) they are reflected by the $Q$-barrier; (3) they change their character to long-waves and propagate away from the co-rotation resonance points; and finally (4) they reach the Lindblad resonance points and the energy may be converted to the long-trailing waves there.

The wave obtains its energy at the resonance points. The density wave transfer the energy to the co-rotation points. Therefore, the density wave theory predicts the galactic stellar disk has spiral density pattern between the inner Lindblad resonance points and the outer Lindblad resonance point if $Q\mbox{\raisebox{0.3ex}{$>$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 1$.