The following steps gives mathematical analysis of velocity modulation for two cavity Klystron.

Due to potential difference $V_0$ between anode and cathode, the electrons from a high current density beam with velocity $ϑ_0$

$ϑ_0=\sqrt{2e V_0/M}$

Where $\text{e = charge an electron} \\ \text{M = Mass of electron}$

The time taken by beam to cross cavity gap ‘d’ is the transit time and transit angle through gap ‘d’ is,

Transit time $= t_2 – t_1 \\ t_g=\dfrac{d}{v_0}$

Transit angle = wtg -------- (1)

The input given to buncher cavity is the RF input.

Average RF input in the gap of buncher cavity is

$V_{av}=\dfrac{1}{t_g \int^{t_2}_{t_1}V_1 \sin(wt)dt}=\dfrac{V_1}{wtg} (-\cos⁡ wt)_{t_1}^{t_2} \\ V_{av}=\dfrac{V_1}{wtg} \bigg[\sin⁡ \bigg(\dfrac{wt_1+wt_2}{2}\bigg).\sin⁡\bigg(\dfrac{wt_1.wt_2}{2}\bigg) \bigg] ………….\text{by trignometric} \\ V_{avg}=\dfrac{V_1}{wtg}\bigg[\sin\bigg(wt_1+\dfrac{\theta_g}{2}\bigg).\sin \bigg(\dfrac{wtg}{2}\bigg)\bigg].....................\text{(from(1))}$

Let $β_1=\dfrac{\sin⁡ \dfrac{θ_g}{2}}{\dfrac{θ_g}{2}}$=buncher cavity beam coupling coefficient

$V_{avg}=V_1.\beta_1 \sin \bigg(wt_1+\dfrac{\theta_g}{2}\bigg) ...................(2)$

When the electron passes through the buncher cavity their velocity either increases or decreases depending on positive or negative cycle of KF ZIP. Let Vavg = velocity of electron at mid of gap

$\dfrac{v_{avg}}{V_0} =\sqrt{\dfrac{V_0+V_av}{V_0}} \\ V_{avg}=V_0 \sqrt{1+\dfrac{V_1 β_1}{V_0} \sin⁡\bigg(wt_1+\dfrac{θ_g}{2}} \bigg) ………(from (2))$

Let $M=\dfrac{V_1 β_1}{V_0}$ =Dept of modulation

$V_{avg}=V_0\sqrt{1+M \sin \bigg(wt_1+\dfrac{\theta_g}{2}}\bigg)$