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Diffstat (limited to 'source/know/concept/langmuir-waves/index.md')
| -rw-r--r-- | source/know/concept/langmuir-waves/index.md | 39 |
1 files changed, 21 insertions, 18 deletions
diff --git a/source/know/concept/langmuir-waves/index.md b/source/know/concept/langmuir-waves/index.md index 2dbce8f..736ef71 100644 --- a/source/know/concept/langmuir-waves/index.md +++ b/source/know/concept/langmuir-waves/index.md @@ -54,13 +54,13 @@ are assumed to satisfy: $$\begin{aligned} \pdv{n_{e0}}{t} = 0 - \qquad + \qquad \quad \pdv{\vb{u}_{e0}}{t} = 0 - \qquad + \qquad \quad \nabla n_{e0} = 0 - \qquad + \qquad \quad \vb{u}_{e0} = 0 - \qquad + \qquad \quad \vb{E}_0 = 0 \end{aligned}$$ @@ -73,8 +73,7 @@ $$\begin{aligned} \\ &= \pdv{n_{e1}}{t} + \nabla \cdot \Big( n_{e0} \vb{u}_{e1} + n_{e1} \vb{u}_{e1} \Big) \\ - &\approx \pdv{n_{e1}}{t} + \nabla \cdot (n_{e0} \vb{u}_{e1}) - = \pdv{n_{e1}}{t} + n_{e0} \nabla \cdot \vb{u}_{e1} + &\approx \pdv{n_{e1}}{t} + n_{e0} \nabla \cdot \vb{u}_{e1} \end{aligned}$$ Likewise, we insert it into Gauss' law, @@ -83,7 +82,7 @@ and use the plasma's quasi-neutrality $$n_i = n_{e0}$$ to get: $$\begin{aligned} \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big) = q_e (n_{e0} + n_{e1} - n_i) - \quad \implies \quad + \qquad \implies \qquad \varepsilon_0 \nabla \cdot \vb{E}_1 = q_e n_{e1} \end{aligned}$$ @@ -106,12 +105,13 @@ Inserting this into the continuity equation and Gauss' law yields, respectively: $$\begin{aligned} - i \omega n_{e1} = - i n_{e0} \vb{k} \cdot \vb{u}_{e1} - \qquad \quad + \qquad \qquad -\! i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1} \end{aligned}$$ +These form a system of equations to be solved. However, there are three unknowns $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$, -so one more equation is needed. +so one more equation is needed before we can do so. @@ -180,7 +180,8 @@ the oscillation is stationary. ## Warm Langmuir waves Next, we generalize this result to nonzero $$T_e$$, -in which case the pressure $$p_e$$ is involved: +in which case the pressure $$p_e$$ is involved, +so the electron momentum equation is given by: $$\begin{aligned} m_e n_{e0} \pdv{\vb{u}_{e1}}{t} @@ -198,10 +199,11 @@ $$\begin{aligned} \end{aligned}$$ With this, insertion of our plane-wave ansatz -into the electron equation results in: +into the momentum equation results in: $$\begin{aligned} - -i \omega m_e n_{e0} \vb{u}_{e1} = q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} + -i \omega m_e n_{e0} \vb{u}_{e1} + = q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} \end{aligned}$$ Which once again closes the system of three equations. @@ -209,7 +211,8 @@ Solving for $$\omega^2$$ then gives: $$\begin{aligned} \omega^2 - = \frac{\omega n_{e0}}{n_{e1}} \vb{k} \cdot \vb{u}_{e1} + &= \frac{\omega n_{e0}}{n_{e1}} \vb{k} \cdot \vb{u}_{e1} + \\ &= \frac{i \omega n_{e0}}{\omega n_{e0} m_e n_{e1}} \vb{k} \cdot \Big( q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} \Big) \\ &= \frac{n_{e0} q_e^2}{\varepsilon_0 m_e} - \frac{i \omega}{\omega m_e n_{e1}} i \gamma k_B T_e n_{e1} \big(\vb{k} \cdot \vb{k}\big) @@ -235,13 +238,13 @@ $$\begin{aligned} \end{aligned}$$ Unlike for $$T_e = 0$$, these "warm" waves do propagate, -carrying information at group velocity $$v_g$$, -which, in the limit of large $$k$$, is given by: +because $$k$$ appears in the dispersion relation. +They carry information at group velocity $$v_g = \ipdv{w}{k}$$, +which in the limit of large $$k$$ becomes: $$\begin{aligned} - v_g - = \pdv{\omega}{k} - \to \sqrt{\frac{3 k_B T_e}{m_e}} + \lim_{k \to \infty} v_g + = \sqrt{\frac{3 k_B T_e}{m_e}} \end{aligned}$$ This is the root-mean-square velocity of the |