Generals 2002 I 4B

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The phase velocity is:

\mathbf{v}_{\phi}=\frac{\omega\mathbf{k}}{k^{2}}


The group velocity is:

\mathbf{v}_{g}=\frac{\partial\omega}{\partial\mathbf{k}}


The dispersion relation for cold electrostatic waves is:

k_{x}^{2}S+k_{z}^{2}P=0


Or, since S and P are only functions of ω:

\frac{P}{S}=f\left(\omega\right)=-\frac{k_{x}^{2}}{k_{z}^{2}}


Then, taking the derivative with respect to \mathbf{k}:

f^{\prime}\left(\omega\right)\frac{\partial\omega}{\partial\mathbf{k}}=-\frac{2k_{x}^{2}}{k_{z}^{2}}\left(k_{x}^{-1}\hat{x}-k_{z}^{-1}\hat{z}\right)


So:

\mathbf{v}_{\phi}\cdot\mathbf{v}_{g}=\frac{\omega\mathbf{k}}{k^{2}}\cdot\frac{\partial\omega}{\partial\mathbf{k}}=\frac{\omega}{k^{2}f^{\prime}\left(\omega\right)}\left(-\frac{2k_{x}^{2}}{k_{z}^{2}}\right)\left(1-1\right)=0
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