Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 27 insertions, 27 deletions

diff --git a/source/know/concept/langmuir-waves/index.md b/source/know/concept/langmuir-waves/index.md
index 0b7f2d7..7dc5dbf 100644
--- a/source/know/concept/langmuir-waves/index.md
+++ b/source/know/concept/langmuir-waves/index.md
@@ -13,8 +13,8 @@ layout: "concept"
 In plasma physics, **Langmuir waves** are oscillations in the electron density,
 which may or may not propagate, depending on the temperature.
 
-Assuming no [magnetic field](/know/concept/magnetic-field/) $\vb{B} = 0$,
-no ion motion $\vb{u}_i = 0$ (since $m_i \gg m_e$),
+Assuming no [magnetic field](/know/concept/magnetic-field/) $$\vb{B} = 0$$,
+no ion motion $$\vb{u}_i = 0$$ (since $$m_i \gg m_e$$),
 and therefore no ion-electron momentum transfer,
 the [two-fluid equations](/know/concept/two-fluid-equations/)
 tell us that:
@@ -34,8 +34,8 @@ $$\begin{aligned}
     = q_e (n_e - n_i)
 \end{aligned}$$
 
-We split $n_e$, $\vb{u}_e$ and $\vb{E}$ into a base component
-(subscript $0$) and a perturbation (subscript $1$):
+We split $$n_e$$, $$\vb{u}_e$$ and $$\vb{E}$$ into a base component
+(subscript $$0$$) and a perturbation (subscript $$1$$):
 
 $$\begin{aligned}
     n_e
@@ -48,8 +48,8 @@ $$\begin{aligned}
     = \vb{E}_0 + \vb{E}_1
 \end{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 $\vb{E}_0$
+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:
 
 $$\begin{aligned}
@@ -65,7 +65,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We insert this decomposistion into the electron continuity equation,
-arguing that $n_{e1} \vb{u}_{e1}$ is small enough to neglect, leading to:
+arguing that $$n_{e1} \vb{u}_{e1}$$ is small enough to neglect, leading to:
 
 $$\begin{aligned}
     0
@@ -78,7 +78,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Likewise, we insert it into Gauss' law,
-and use the plasma's quasi-neutrality $n_i = n_{e0}$ to get:
+and use the plasma's quasi-neutrality $$n_i = n_{e0}$$ to get:
 
 $$\begin{aligned}
     \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big)
@@ -110,7 +110,7 @@ $$\begin{aligned}
     -\! i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1}
 \end{aligned}$$
 
-However, there are three unknowns $n_{e1}$, $\vb{u}_{e1}$ and $\vb{E}_1$,
+However, there are three unknowns $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$,
 so one more equation is needed.
 
 
@@ -118,7 +118,7 @@ so one more equation is needed.
 
 We therefore turn to the electron momentum equation.
 For now, let us assume that the electrons have no thermal motion,
-i.e. the electron temperature $T_e = 0$, so that $p_e = 0$, leaving:
+i.e. the electron temperature $$T_e = 0$$, so that $$p_e = 0$$, leaving:
 
 $$\begin{aligned}
     m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
@@ -126,8 +126,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Inserting the decomposition then gives the following,
-where we neglect $(\vb{u}_{e1} \cdot \nabla) \vb{u}_{e1}$
-because $\vb{u}_{e1}$ is so small by assumption:
+where we neglect $$(\vb{u}_{e1} \cdot \nabla) \vb{u}_{e1}$$
+because $$\vb{u}_{e1}$$ is so small by assumption:
 
 $$\begin{gathered}
     m_e (n_{e0} \!+\! n_{e1}) \Big( \pdv{(\vb{u}_{e0} \!+\! \vb{u}_{e1})}{t}
@@ -147,7 +147,7 @@ $$\begin{aligned}
     -i \omega m_e \vb{u}_{e1} = q_e \vb{E}_1
 \end{aligned}$$
 
-Solving this system of three equations for $\omega^2$
+Solving this system of three equations for $$\omega^2$$
 gives the following dispersion relation:
 
 $$\begin{aligned}
@@ -158,7 +158,7 @@ $$\begin{aligned}
     = \frac{n_{e0} q_e^2}{\varepsilon_0 m_e}
 \end{aligned}$$
 
-This result is known as the **plasma frequency** $\omega_p$,
+This result is known as the **plasma frequency** $$\omega_p$$,
 and describes the frequency of **cold Langmuir waves**,
 otherwise known as **plasma oscillations**:
 
@@ -169,16 +169,16 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Note that this is a dispersion relation $\omega(k) = \omega_p$,
-but that $\omega_p$ does not contain $k$.
+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".
 
 
 ## Warm Langmuir waves
 
-Next, we generalize this result to nonzero $T_e$,
-in which case the pressure $p_e$ is involved:
+Next, we generalize this result to nonzero $$T_e$$,
+in which case the pressure $$p_e$$ is involved:
 
 $$\begin{aligned}
     m_e n_{e0} \pdv{}{\vb{u}{e1}}{t}
@@ -186,7 +186,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 From the two-fluid thermodynamic equation of state,
-we know that $\nabla p_e$ can be written as:
+we know that $$\nabla p_e$$ can be written as:
 
 $$\begin{aligned}
     \nabla p_e
@@ -203,7 +203,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Which once again closes the system of three equations.
-Solving for $\omega^2$ then gives:
+Solving for $$\omega^2$$ then gives:
 
 $$\begin{aligned}
     \omega^2
@@ -213,8 +213,8 @@ $$\begin{aligned}
     &= \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)
 \end{aligned}$$
 
-Recognizing the first term as the plasma frequency $\omega_p^2$,
-we therefore arrive at the **Bohm-Gross dispersion relation** $\omega(\vb{k})$
+Recognizing the first term as the plasma frequency $$\omega_p^2$$,
+we therefore arrive at the **Bohm-Gross dispersion relation** $$\omega(\vb{k})$$
 for **warm Langmuir waves**:
 
 $$\begin{aligned}
@@ -225,16 +225,16 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This expression is typically quoted for 1D oscillations,
-in which case $\gamma = 3$ and $k = |\vb{k}|$:
+in which case $$\gamma = 3$$ and $$k = |\vb{k}|$$:
 
 $$\begin{aligned}
     \omega^2
     = \omega_p^2 + \frac{3 k_B T_e}{m_e} k^2
 \end{aligned}$$
 
-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:
+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:
 
 $$\begin{aligned}
     v_g
@@ -244,7 +244,7 @@ $$\begin{aligned}
 
 This is the root-mean-square velocity of the
 [Maxwell-Boltzmann speed distribution](/know/concept/maxwell-boltzmann-distribution/),
-meaning that information travels at the thermal velocity for large $k$.
+meaning that information travels at the thermal velocity for large $$k$$.