2R_{\rm sd}$ (Figure \[fig:rotcurve\]c). Thus, substituting $\Gamma \simeq 0.8$ in eq. (7), we obtain $i_{\rm th} \simeq 32^\circ$. This value is consistent with the evolution of the spiral arm with the maximum contrast (hatched region in Figure \[fig:PitchAngleAmplitude\]).

![ Evolution of spiral arm on $i-\bar{\delta}$ plane for $T_{\rm rot} = 12.0 - 12.5$. The hatched region corresponds to the predicted maximum pitch angle around the analyzed region ($Q \simeq 1.4$ and $\Gamma \simeq 0.75-0.85$) due to swing amplification (refer to equation (7)). []{data-label="fig:PitchAngleAmplitude"}](fig7.eps){width=".45\textwidth"}

Damping Phase {#sec:DampingSpiralArm}
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![image](fig8.eps){width=".90\textwidth"}

The swing amplification mechanism can explain certain aspects of the evolution of spiral arms, i.e., the amplification (excitation) of density enhancement. However, the destruction process of the non-steady spirals, as seen in Figure \[fig:snapshotModelMSdamping\] [see also @Fujii+2011; @Wada+2011], cannot be understood only by the swing amplification mechanism. The top right panels in Figure \[fig:VelocityProfile\] show the time evolution of the relative velocities ($\Delta v_\eta$ and $\Delta v_\xi$) in the co-rotating frame of the spiral arm. Upon considering the coordinates $(\eta, \xi)$, i.e., the $\eta$- and $\xi$-axes are perpendicular and parallel to the spiral arm, respectively, the velocity component along each is given by $$\begin{aligned}
 && \Delta v_{\eta} \equiv  v_{R} \cos{i} + (v_{\phi} 