From 075683cdf4588fe16f41d9f7b46b9720b42b2553 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 17 Jul 2024 10:01:43 +0200
Subject: Improve knowledge base

---
 .../concept/korteweg-de-vries-equation/index.md    | 31 ++++++++++++----------
 1 file changed, 17 insertions(+), 14 deletions(-)

(limited to 'source/know/concept/korteweg-de-vries-equation/index.md')

diff --git a/source/know/concept/korteweg-de-vries-equation/index.md b/source/know/concept/korteweg-de-vries-equation/index.md
index 2857e23..e8035d1 100644
--- a/source/know/concept/korteweg-de-vries-equation/index.md
+++ b/source/know/concept/korteweg-de-vries-equation/index.md
@@ -162,11 +162,11 @@ $$\begin{aligned}
     = q_0 - \frac{g}{q_0} \Big( \eta(x, t) + \alpha + \gamma(x, t) \Big)
 \end{aligned}$$
 
-Where $$\alpha$$ is a constant parameter
-(which we will use to handle velocity discrepancies
-between the linear and nonlinear theories).
+Where $$\alpha$$ is a constant parameter,
+which we will use to handle velocity discrepancies
+between the linear and nonlinear theories.
 The correction represented by $$\gamma$$ is much smaller,
-i.e. $$\eta \sim \alpha \gg \gamma$$.
+i.e. $$\eta \gg \alpha \gg \gamma$$.
 We insert this ansatz into the above equations, yielding:
 
 $$\begin{aligned}
@@ -265,14 +265,15 @@ $$\begin{aligned}
     \equiv \frac{h^3}{3} - \frac{h T}{g \rho}
 \end{aligned}$$
 
-What about $$\alpha$$?
+But what about $$\alpha$$?
 Looking at the ansatz for $$f$$, we see that
-the body of water is already assumed to be moving at $$q_0$$,
-minus $$g \alpha / q_0$$, so by varying $$\alpha$$
-we are modifying the water's velocity.
-The term in the KdV equation simply corrects for our chosen value of $$\alpha$$.
-It has no deeper meaning than that: for any value of $$\alpha$$,
-the full range of KdV solutions can still be obtained.
+the body of water is assumed to be moving at $$q_0 - g \alpha / q_0$$,
+and $$q_0$$ is set to $$\pm \sqrt{g h}$$ by almost all authors,
+so $$\alpha$$ controls the velocity of our reference frame.
+Nonlinear waves do not travel at the same speed as linear waves,
+so we can choose $$\alpha$$ to make the wave stationary
+without breaking the $$q_0$$ "tradition".
+That term in the KdV equation simply corrects for our chosen value of $$\alpha$$.
 
 
 
@@ -383,14 +384,16 @@ These are the final scale parameter values,
 leading to the desired dimensionless form:
 
 $$\begin{aligned}
-    0
-    &= \tilde{\eta}_{\tilde{t}} - 6 \tilde{\eta} \tilde{\eta}_{\tilde{x}} + \tilde{\eta}_{\tilde{x} \tilde{x} \tilde{x}}
+    \boxed{
+        0
+        = \tilde{\eta}_{\tilde{t}} - 6 \tilde{\eta} \tilde{\eta}_{\tilde{x}} + \tilde{\eta}_{\tilde{x} \tilde{x} \tilde{x}}
+    }
 \end{aligned}$$
 
 Recall that $$\alpha$$ sets the background fluid velocity,
 and $$v_c$$ controls the coordinate system's motion:
 our choice of $$v_c$$ simply cancels out the effect of $$\alpha$$.
-This reveals the point of $$\alpha$$:
+This demonstrates the purpose of $$\alpha$$:
 the KdV equation has solutions moving at various speeds,
 so, for a given $$\eta$$, we can always choose $$\alpha$$ (and hence $$v_c$$)
 such that the wave appears stationary.
-- 
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