群速度

\begin{equation} v_g = \frac{\mathrm{d}{\omega}}{\mathrm{d}{k}} ~. \end{equation}

\begin{equation} k, k+\Delta k, \Delta k, v_1 = \omega/k, v_2 = (\omega+\Delta\omega)/(k+\Delta k)~, \end{equation}

\begin{equation} \Delta v = \frac{\Delta \omega}{k} - \frac{\omega}{k^2}\Delta k~, \end{equation}

\begin{equation} v_g = v_1 + \frac{\Delta \lambda}{\Delta v} = v_1 - \frac{2\pi\Delta k}{k^2\Delta v}~. \end{equation}

\begin{equation} \begin{aligned} &f_1(x,t) = A \cos\left(k_1 x - \omega_1 t + \phi_1\right) ~,\\ &f_2(x,t) = A \cos\left(k_2 x - \omega_2 t + \phi_2\right) ~. \end{aligned} \end{equation}

\begin{equation} \begin{aligned} f_1(x,t) + f_2(x,t) &= 2 A \cos \left(\frac{k_2-k_1}{2}x - \frac{\omega_2-\omega_1}{2}t + \frac{\phi_2 - \phi_1}{2} \right) \\ & \times \cos \left(\frac{k_1+k_2}{2}x - \frac{\omega_{2}+\omega_{1}}{2}t + \frac{\phi_1 + \phi_2}{2} \right) ~. \end{aligned} \end{equation}

\begin{equation} v_g = \frac{\mathrm{d}{\omega}}{\mathrm{d}{k}} ~, \end{equation}