From 1d700ab734aa9b6711eb31796beb25cb7659d8e0 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 20 Dec 2022 20:11:25 +0100
Subject: More improvements to knowledge base

---
 source/know/concept/ion-sound-wave/index.md | 34 ++++++++++++++---------------
 1 file changed, 16 insertions(+), 18 deletions(-)

(limited to 'source/know/concept/ion-sound-wave/index.md')

diff --git a/source/know/concept/ion-sound-wave/index.md b/source/know/concept/ion-sound-wave/index.md
index 8749f1a..6a9dcff 100644
--- a/source/know/concept/ion-sound-wave/index.md
+++ b/source/know/concept/ion-sound-wave/index.md
@@ -49,7 +49,7 @@ $$\begin{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 $$\phi_0$$
-by definition satisfy:
+are assumed to satisfy:
 
 $$\begin{aligned}
     \pdv{n_{i0}}{t} = 0
@@ -63,11 +63,7 @@ $$\begin{aligned}
     \phi_0 = 0
 \end{aligned}$$
 
-Inserting this decomposition into the momentum equations
-yields new equations.
-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:
+Inserting this decomposition into the momentum equations yields new equations:
 
 $$\begin{aligned}
     m_i (n_{i0} \!+\! n_{i1}) \frac{\mathrm{D} (\vb{u}_{i0} \!+\! \vb{u}_{i1})}{\mathrm{D} t}
@@ -77,17 +73,19 @@ $$\begin{aligned}
     &= - q_e (n_{e0} \!+\! n_{e1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_e k_B T_e \nabla (n_{e0} \!+\! n_{e1})
 \end{aligned}$$
 
-Using the defined properties of the equilibrium components
-$$n_{i0}$$, $$n_{e0}$$, $$\vb{u}_{i0}$$ and $$\phi_0$$,
-and neglecting all products of perturbations for being small,
-this reduces to:
+Using the assumed properties of $$n_{i0}$$, $$n_{e0}$$, $$\vb{u}_{i0}$$ and $$\phi_0$$,
+and discarding products of perturbations for being too small,
+we arrive at the below equations.
+Our choice $$\vb{u}_{i0} = 0$$ lets us linearize
+the [material derivative](/know/concept/material-derivative/)
+$$\mathrm{D}/\mathrm{D} t = \ipdv{}{t}$$ for the ions:
 
 $$\begin{aligned}
     m_i n_{i0} \pdv{\vb{u}_{i1}}{t}
-    &= - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1}
+    &\approx - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1}
     \\
     0
-    &= - q_e n_{e0} \nabla \phi_1 - \gamma_e k_B T_e \nabla n_{e1}
+    &\approx - q_e n_{e0} \nabla \phi_1 - \gamma_e k_B T_e \nabla n_{e1}
 \end{aligned}$$
 
 Because we are interested in linear waves,
@@ -123,7 +121,7 @@ to get a relation between $$n_{e1}$$ and $$n_{e0}$$:
 $$\begin{aligned}
      i \vb{k} \gamma_e k_B T_e n_{e1}
      = - i \vb{k} q_e n_{e0} \phi_1
-     \quad \implies \quad
+     \qquad \implies \qquad
      n_{e1}
      = - \frac{q_e \phi_1}{\gamma_e k_B T_e} n_{e0}
 \end{aligned}$$
@@ -159,13 +157,13 @@ $$\begin{aligned}
     \approx \pdv{n_{i1}}{t} + n_{i0} \nabla \cdot \vb{u}_{i1}
 \end{aligned}$$
 
-Then we insert our plane-wave ansatz,
+Into which we insert our plane-wave ansatz,
 and substitute $$n_{i0} = n_0$$ as before, yielding:
 
 $$\begin{aligned}
     0
     = - i \omega n_{i1} + i n_{i0} \vb{k} \cdot \vb{u}_{i1}
-    \quad \implies \quad
+    \qquad \implies \qquad
     \vb{k} \cdot \vb{u}_{i1}
     = \omega \frac{n_{i1}}{n_{i0}}
     = \omega \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}}
@@ -187,9 +185,9 @@ $$\begin{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,
 according to [Gauss' law](/know/concept/maxwells-equations/).
-Nevertheless, authors often ignore this fact,
+Nevertheless, some authors tend to ignore this fact,
 thereby making the so-called **plasma approximation**.
-We will not, and therefore turn to Gauss' law:
+We will not, and thus turn to Gauss' law:
 
 $$\begin{aligned}
     \varepsilon_0 \nabla \cdot \vb{E}
@@ -244,7 +242,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Curiously, unlike a neutral gas,
+Curiously, unlike in a neutral gas,
 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 $$T_e \gg T_i$$,
-- 
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