summaryrefslogtreecommitdiff
path: root/source/know/concept/langmuir-waves
diff options
context:
space:
mode:
Diffstat (limited to 'source/know/concept/langmuir-waves')
-rw-r--r--source/know/concept/langmuir-waves/index.md12
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:
$$\begin{aligned}
- m_e n_{e0} \pdv{}{\vb{u}{e1}}{t}
+ m_e n_{e0} \pdv{\vb{u}_{e1}}{t}
= q_e n_{e0} \vb{E}_1 - \nabla p_e
\end{aligned}$$