1 files changed, 29 insertions, 29 deletions
diff --git a/source/know/concept/ion-sound-wave/index.md b/source/know/concept/ion-sound-wave/index.md
index 622605c..cb86c04 100644
--- a/source/know/concept/ion-sound-wave/index.md
+++ b/source/know/concept/ion-sound-wave/index.md
@@ -16,9 +16,9 @@ at lower temperatures and pressures
 than would be possible in a neutral gas.
 
 We start from the [two-fluid model's](/know/concept/two-fluid-equations/) momentum equations,
-rewriting the [electric field](/know/concept/electric-field/) $\vb{E} = - \nabla \phi$
-and the pressure gradient $\nabla p = \gamma k_B T \nabla n$,
-and arguing that $m_e \approx 0$ because $m_e \ll m_i$:
+rewriting the [electric field](/know/concept/electric-field/) $$\vb{E} = - \nabla \phi$$
+and the pressure gradient $$\nabla p = \gamma k_B T \nabla n$$,
+and arguing that $$m_e \approx 0$$ because $$m_e \ll m_i$$:
 
 $$\begin{aligned}
     m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
@@ -29,9 +29,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Note that we neglect ion-electron collisions,
-and allow for separate values of $\gamma$.
-We split $n_i$, $n_e$, $\vb{u}_i$ and $\phi$ into an equilibrium
-(subscript $0$) and a perturbation (subscript $1$):
+and allow for separate values of $$\gamma$$.
+We split $$n_i$$, $$n_e$$, $$\vb{u}_i$$ and $$\phi$$ into an equilibrium
+(subscript $$0$$) and a perturbation (subscript $$1$$):
 
 $$\begin{aligned}
     n_i
@@ -47,8 +47,8 @@ $$\begin{aligned}
     = \phi_0 + \phi_1
 \end{aligned}$$
 
-Where the perturbations $n_{i1}$, $n_{e1}$, $\vb{u}_{i1}$ and $\phi_1$ are tiny,
-and the equilibrium components $n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$
+Where the perturbations $$n_{i1}$$, $$n_{e1}$$, $$\vb{u}_{i1}$$ and $$\phi_1$$ are tiny,
+and the equilibrium components $$n_{i0}$$, $$n_{e0}$$, $$\vb{u}_{i0}$$ and $$\phi_0$$
 by definition satisfy:
 
 $$\begin{aligned}
@@ -65,9 +65,9 @@ $$\begin{aligned}
 
 Inserting this decomposition into the momentum equations
 yields new equations.
-Note that we will implicitly use $\vb{u}_{i0} = 0$
+Note that we will implicitly use $$\vb{u}_{i0} = 0$$
 to pretend that the [material derivative](/know/concept/material-derivative/)
-$\mathrm{D}/\mathrm{D} t$ is linear:
+$$\mathrm{D}/\mathrm{D} t$$ is linear:
 
 $$\begin{aligned}
     m_i (n_{i0} \!+\! n_{i1}) \frac{\mathrm{D} (\vb{u}_{i0} \!+\! \vb{u}_{i1})}{\mathrm{D} t}
@@ -78,7 +78,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Using the defined properties of the equilibrium components
-$n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$,
+$$n_{i0}$$, $$n_{e0}$$, $$\vb{u}_{i0}$$ and $$\phi_0$$,
 and neglecting all products of perturbations for being small,
 this reduces to:
 
@@ -118,7 +118,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The electron equation can easily be rearranged
-to get a relation between $n_{e1}$ and $n_{e0}$:
+to get a relation between $$n_{e1}$$ and $$n_{e0}$$:
 
 $$\begin{aligned}
      i \vb{k} \gamma_e k_B T_e n_{e1}
@@ -131,12 +131,12 @@ $$\begin{aligned}
 Due to their low mass, the electrons' heat conductivity
 can be regarded as infinite compared to the ions'.
 In that case, all electron gas compression is isothermal,
-meaning it obeys the ideal gas law $p_e = n_e k_B T_e$, so that $\gamma_e = 1$.
+meaning it obeys the ideal gas law $$p_e = n_e k_B T_e$$, so that $$\gamma_e = 1$$.
 Note that this yields the first-order term of a Taylor expansion
 of the [Boltzmann relation](/know/concept/boltzmann-relation/).
 
-At equilibrium, quasi-neutrality demands that $n_{i0} = n_{e0} = n_0$,
-so we can rearrange the above relation to $n_0 = - k_B T_e n_{e1} / (q_e \phi_1)$,
+At equilibrium, quasi-neutrality demands that $$n_{i0} = n_{e0} = n_0$$,
+so we can rearrange the above relation to $$n_0 = - k_B T_e n_{e1} / (q_e \phi_1)$$,
 which we insert into the ion equation to get:
 
 $$\begin{gathered}
@@ -148,8 +148,8 @@ $$\begin{gathered}
     = T_e n_{e1} |\vb{k}|^2 - \gamma_i T_i n_{i1} |\vb{k}|^2
 \end{gathered}$$
 
-Where we have taken the dot product with $\vb{k}$,
-and used that $q_i / q_e = -1$.
+Where we have taken the dot product with $$\vb{k}$$,
+and used that $$q_i / q_e = -1$$.
 In order to simplify this equation,
 we turn to the two-fluid ion continuity relation:
 
@@ -160,7 +160,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Then we insert our plane-wave ansatz,
-and substitute $n_{i0} = n_0$ as before, yielding:
+and substitute $$n_{i0} = n_0$$ as before, yielding:
 
 $$\begin{aligned}
     0
@@ -172,7 +172,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Substituting this in the ion momentum equation
-leads us to a dispersion relation $\omega(\vb{k})$:
+leads us to a dispersion relation $$\omega(\vb{k})$$:
 
 $$\begin{gathered}
     \omega^2 m_i \frac{T_e n_{e1}}{q_e \phi_1} \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}}
@@ -184,8 +184,8 @@ $$\begin{gathered}
     = \frac{|\vb{k}|^2}{m_i} \Big( k_B T_e \frac{n_{e1}}{n_{i1}} - \gamma_i k_B T_i \Big)
 \end{gathered}$$
 
-Finally, we would like to find an expression for $n_{e1} / n_{i1}$.
-It cannot be $1$, because then $\phi_1$ could not be nonzero,
+Finally, we would like to find an expression for $$n_{e1} / n_{i1}$$.
+It cannot be $$1$$, because then $$\phi_1$$ could not be nonzero,
 according to [Gauss' law](/know/concept/maxwells-equations/).
 Nevertheless, authors often ignore this fact,
 thereby making the so-called **plasma approximation**.
@@ -199,8 +199,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 One final time, we insert our plane-wave ansatz,
-and use our Boltzmann-like relation between $n_{e1}$ and $n_{e0}$
-to substitute $\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0})$:
+and use our Boltzmann-like relation between $$n_{e1}$$ and $$n_{e0}$$
+to substitute $$\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0})$$:
 
 $$\begin{gathered}
     q_e (n_{e1} - n_{i1})
@@ -213,7 +213,7 @@ $$\begin{gathered}
     = n_{e1} \big( 1 + |\vb{k}|^2 \lambda_{De}^2 \big)
 \end{gathered}$$
 
-Where $\lambda_{De}$ is the electron [Debye length](/know/concept/debye-length/).
+Where $$\lambda_{De}$$ is the electron [Debye length](/know/concept/debye-length/).
 We thus reach the following dispersion relation,
 which governs **ion sound waves** or **ion acoustic waves**:
 
@@ -224,7 +224,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The aforementioned plasma approximation is valid if $|\vb{k}| \lambda_{De} \ll 1$,
+The aforementioned plasma approximation is valid if $$|\vb{k}| \lambda_{De} \ll 1$$,
 which is often reasonable,
 in which case this dispersion relation reduces to:
 
@@ -233,7 +233,7 @@ $$\begin{aligned}
     = \frac{|\vb{k}|^2}{m_i} \bigg( k_B T_e + \gamma_i k_B T_i \bigg)
 \end{aligned}$$
 
-The phase velocity $v_s$ of these waves,
+The phase velocity $$v_s$$ of these waves,
 i.e. the speed of sound, is then given by:
 
 $$\begin{aligned}
@@ -245,11 +245,11 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Curiously, unlike a neutral gas,
-this velocity is nonzero even if $T_i = 0$,
+this velocity is nonzero even if $$T_i = 0$$,
 meaning that the waves still exist then.
-In fact, usually the electron temperature $T_e$ dominates $T_e \gg T_i$,
+In fact, usually the electron temperature $$T_e$$ dominates $$T_e \gg T_i$$,
 even though the main feature of these waves
-is that they involve ion density fluctuations $n_{i1}$.
+is that they involve ion density fluctuations $$n_{i1}$$.