in Figure \[fig:VelocityProfile\] show the time evolution of the non-axisymmetric gravitational forces (i.e., spiral perturbation) involved in the damping phase. The component of the non-axisymmetric gravitational force perpendicular to the spiral arm ($\Delta F_{\rm grav,\eta}$), which is stronger than the parallel component ($\Delta F_{\rm grav,\xi}$), is always directed towards the arm ($\Delta F_{\rm grav,\eta} > 0$ for $\eta < 0$, and $\Delta F_{\rm grav,\eta} < 0$ for $\eta > 0$) during the dynamical evolution. The bottom right panels in Figure \[fig:VelocityProfile\] show the time evolution of the net force (i.e., non-axisymmetric gravitational force plus Coriolis force perturbation). The component of the net force perpendicular to the spiral arm ($\Delta F_{\rm grav+cori,\eta}$) evolves in the same manner as $\Delta F_{\rm grav,\eta}$ because there is a strong density gradient along the $\eta$-direction. The parallel component $\Delta F_{\rm grav,\xi}$ is almost zero during the growth and damping phases and does not change. This is because the density gradient along the $\xi$-direction is small and almost unchanged. However, the parallel component of the net force $\Delta F_{\rm grav+cori,\xi}$ changes its $\eta$-dependence from the growing phase ($T_{\rm rot} = 12.1$) to the damping phase ($T_{\rm rot} > 12.3$) according to the change of the sign 