The arm merges with with a neighboring weak spiral arm ($T_{\rm rot} = 12.5-12.7$), and the peak density contrast again becomes $\delta \sim 1$ when $T_{\rm rot} = 12.8$. In the following sections, we discuss the growth phase (Figure \[fig:snapshotModelMSgrowth\]) and the damping phase (Figure \[fig:snapshotModelMSdamping\] ) of the spiral separately.

![image](fig5.eps){width=".8\textwidth"}

![image](fig6.eps){width=".8\textwidth"}

Growth Phase {#sec:AmplifySpiralArm}
------------

In the growth phase of the spiral arm ($T_{\rm rot} \simeq 12.0-12.2$), the pitch angle of the arm reduces due to the differential rotation of the disk. This is clearly observed in the right hand panels of Figures \[fig:snapshotModelMSgrowth\] and \[fig:snapshotModelMSdamping\]; the two peaks at $2R_{\rm sd}+1.5$ kpc and $2R_{\rm sd}-1.5$ kpc show increasing separation in terms of $\phi$. More quantitatively, Figure \[fig:PitchAngleAmplitude\] shows the evolution of the spiral arm along the pitch angle - density contrast ($i-\bar{\delta}$) plane. Here, we calculated the pitch angle ($i$) of the spiral arms using the relation $$\begin{aligned}
 \tan i = \frac{1}{R}\frac{\Delta R}{\Delta\phi}, \end{aligned}$$ where $R=2 R_{\rm sd}$ (= 8.6 kpc), $\Delta R = 3$ kpc, and $\Delta \phi$ is the azimuthal angle difference between the contrast peaks at $R=2 R_{\rm sd} - 1.5$ kpc (dot-dashed lines) and $R=2 R_{\rm sd} + 1.5$ kpc (dotted lines) shown 