一般認為快速磁聲波(fast magnetosonic waves)在靠近赤道平面上垂直於周遭磁場的方向上傳遞。而這磁聲波的不穩定度事件目前被視為可能由離子速度環狀分佈所激發。為了去研究這個事件的特性，常使用動力學線性色散方程式(kinetic linear dispersion relation)去討論波函數的頻率波長關係(ω=ω_r+ω_i i v.s. k ⃑)，此處將頻率劃分為實部及虛部來討論。在此，採用了無環、單環和雙環三種質子垂直速度環分佈函數，分別由Maxwellian函數加上0個、1個和3個漂移bi-Maxwellian函數組成。在平行方向上的波長為0的情況下我們利用圖解和數值方法研究波色散關係，方法包含contour plots和Newton’s method來找出每個分佈波函數的色散關係。我們從推導的色散關係可以發現在雙環分佈的情況下找到比單環分佈還多一個不穩定度峰值，並且發現雙環分佈情況下的波長範圍來得更大。此外還有更多關於波極化的詳細分析，如相位差、電場發展和相速。最後從相速的範圍對應到分佈函數對速度的正梯度的關係找出不同不穩定波的極化電場種類。

Fast magnetosonic waves are considered to propagate in the direction perpendicular to the ambient magnetic field (B_0 ) ⃑ near the equatorial plane (k ⃑⊥(B_0 ) ⃑, where k ⃑ is a wave vector). The instabilities of these waves are recently regarded as being excited by the ion velocity ring distributions. In order to study the properties of such events, kinetic linear dispersion relations are used to discuss the relations between the frequency ω=ω_r+ω_i i and the wave number k ⃑, where the real and imaginary parts of the frequency are indicated by ω_r and ω_i. Three types of ring distributions in perpendicular velocity f_p (v) are adopted: zero-ring, one-ring, and two-ring distribution function. These are generated by superposing 0, 1, and 3 drift bi-Maxwellian distribution functions on a thermal Maxwellian distribution, respectively. Under the limit k_∥=0, we apply both graphical and numerical methods to the dispersion relation, including contour plots and Newton’s method, to obtain the approximate solutions of ω vs. k for each distribution function. From the derived dispersion relations, we find instability peaks (i.e., ω_i peaks) for the two-ring distribution with an extra peak than the one-ring distribution function. Further detail analysis on the wave polarization is implemented, including the phase difference α, the evolution of the electric field, and the phase velocity ω/k corresponding to the positive gradient of the proton velocity distribution functions (〖∂f〗_p (v))/∂v>0 for the unstable waves, different kinds of the polarization with the electric field evolution due to the range of ω/k are found.

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