in Figures \[fig:snapshotModelMSgrowth\] and \[fig:snapshotModelMSdamping\]. Here, we evaluated the positions of the contrast peaks at each radius by visual inspection. The arm density contrast, $\bar{\delta}$, is calculated by averaging the corresponding density contrast over the radial range in Figures \[fig:snapshotModelMSgrowth\] and \[fig:snapshotModelMSdamping\]. As the pitch angle of the spiral arm decreases from $i \simeq 40^\circ$ ($T_{\rm rot} =12.0$) to $i \simeq 32^\circ$ ($T_{\rm rot} =12.20$), the density contrast increases to a maximum, and subsequently, it decreases with increase in the pitch angle. Thus, the spiral arm has a maximum amplitude when $i \sim 32^\circ$.

This amplification process associated with the galactic shear motion is known as the swing amplification [@Toomre1981; @GoldreichLynden-Bell1965; @JulianToomre1966; @ToomreKalnajs1991] [^3]. @Fuchs2001a calculated grid models of swing amplification process by varying the shear rate $\Gamma$ for $Q=1.4$. In his work, he derived a fitting empirical equation for swing amplification (eq.98 in his paper) as the following: $$\begin{aligned}
  \tan i_{\rm th} 
    = 1.932 - 5.186 \left(\frac{1}{2}\Gamma \right) + 4.704 \left(\frac{1}{2}\Gamma \right)^2,\end{aligned}$$ where $i_{\rm th}$ is the pitch angle at which the spiral arm reaches the maximum amplitude. We adopt this equation to evaluate the predicted pitch angle $i_{\rm th}$. In our model, $\Gamma \simeq 0.8$ around $R = 