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Diffstat (limited to 'source/know/concept/langmuir-waves')
| -rw-r--r-- | source/know/concept/langmuir-waves/index.md | 12 |
1 files changed, 7 insertions, 5 deletions
diff --git a/source/know/concept/langmuir-waves/index.md b/source/know/concept/langmuir-waves/index.md index be47567..2dbce8f 100644 --- a/source/know/concept/langmuir-waves/index.md +++ b/source/know/concept/langmuir-waves/index.md @@ -22,7 +22,7 @@ tell us that: $$\begin{aligned} m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} = q_e n_e \vb{E} - \nabla p_e - \qquad \quad + \qquad \qquad \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) = 0 \end{aligned}$$ @@ -50,7 +50,7 @@ $$\begin{aligned} Where the perturbations $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$ are very small, and the equilibrium components $$n_{e0}$$, $$\vb{u}_{e0}$$ and $$\vb{E}_0$$ -by definition satisfy: +are assumed to satisfy: $$\begin{aligned} \pdv{n_{e0}}{t} = 0 @@ -64,7 +64,7 @@ $$\begin{aligned} \vb{E}_0 = 0 \end{aligned}$$ -We insert this decomposistion into the electron continuity equation, +We insert this decomposition into the electron continuity equation, arguing that $$n_{e1} \vb{u}_{e1}$$ is small enough to neglect, leading to: $$\begin{aligned} @@ -114,6 +114,7 @@ However, there are three unknowns $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$, so one more equation is needed. + ## Cold Langmuir waves We therefore turn to the electron momentum equation. @@ -172,7 +173,8 @@ $$\begin{aligned} Note that this is a dispersion relation $$\omega(k) = \omega_p$$, but that $$\omega_p$$ does not contain $$k$$. This means that cold Langmuir waves do not propagate: -the oscillation is "stationary". +the oscillation is stationary. + ## Warm Langmuir waves @@ -181,7 +183,7 @@ Next, we generalize this result to nonzero $$T_e$$, in which case the pressure $$p_e$$ is involved: $$\begin{aligned} - m_e n_{e0} \pdv{}{\vb{u}{e1}}{t} + m_e n_{e0} \pdv{\vb{u}_{e1}}{t} = q_e n_{e0} \vb{E}_1 - \nabla p_e \end{aligned}$$ |