Improve knowledge base

5 files changed, 264 insertions, 58 deletions

diff --git a/source/know/concept/boussinesq-wave-theory/index.md b/source/know/concept/boussinesq-wave-theory/index.md
index 31228ba..ad2fe4c 100644
--- a/source/know/concept/boussinesq-wave-theory/index.md
+++ b/source/know/concept/boussinesq-wave-theory/index.md
@@ -20,7 +20,7 @@ which were not predicted by the linear theories existing at the time.
 ## Boundary conditions
 
 Consider the [Euler equations](/know/concept/euler-equations/)
-for an incompressible fluid with negligible viscosity:
+for an incompressible fluid with negligible [viscosity](/know/concept/viscosity/):
 
 $$\begin{aligned}
     \va{g} - \frac{\nabla p}{\rho}
@@ -441,6 +441,168 @@ which are both on the order of $$h / \lambda$$.
 
 
 
+## Dimensionless form
+
+Let us non-dimensionalize the equation by introducing
+dimensionless quantities $$\tilde{\eta}$$, $$\tilde{t}$$ and $$\tilde{x}$$:
+
+$$\begin{aligned}
+    \tilde{\eta}(\tilde{x}, \tilde{t})
+    = \frac{\eta(x, t)}{\eta_c}
+    \qquad \qquad
+    \tilde{t}
+    = \frac{t}{t_c}
+    \qquad \qquad
+    \tilde{x}
+    = \frac{x}{x_c}
+\end{aligned}$$
+
+Where $$\eta_c$$, $$t_c$$ and $$x_c$$ are unspecified scale parameters.
+We rewrite the Boussinesq equation with these quantities
+by using the chain rule of differentiation,
+and divide by $$\eta_c / t_c^2$$:
+
+$$\begin{aligned}
+    0
+    &= \tilde{\eta}_{\tilde{t} \tilde{t}}
+    - \frac{g h t_c^2}{x_c^2} \tilde{\eta}_{\tilde{x} \tilde{x}}
+    - \pdvn{2}{}{x} \bigg( \frac{3 g \eta_c t_c^2}{2 x_c^2} \tilde{\eta}^2 + \frac{g h^3 t_c^2}{3 x_c^4} \tilde{\eta}_{\tilde{x} \tilde{x}} \bigg)
+\end{aligned}$$
+
+Now we must choose values for $$\eta_c$$, $$t_c$$ and $$x_c$$
+such that the prefactors become simple constants.
+Conventionally it is demanded that:
+
+$$\begin{aligned}
+    \frac{g h t_c^2}{x_c^2}
+    = 1
+    \qquad \qquad
+    \frac{3 g \eta_c t_c^2}{2 x_c^2}
+    = 3
+    \qquad \qquad
+    \frac{g h^3 t_c^2}{3 x_c^4}
+    = 1
+\end{aligned}$$
+
+Solving this system of equations yields the following values for the scale parameters:
+
+$$\begin{aligned}
+    \eta_c
+    = 2 h
+    \qquad \qquad
+    t_c
+    = \sqrt{\frac{h}{3 g}}
+    \qquad \qquad
+    x_c
+    = \frac{h}{\sqrt{3}}
+\end{aligned}$$
+
+And the Boussinesq equation is reduced to its standard dimensionless form:
+
+$$\begin{aligned}
+    \boxed{
+        \pdvn{2}{\tilde{\eta}}{\tilde{t}} - \pdvn{2}{\tilde{\eta}}{\tilde{x}}
+        - \pdvn{2}{}{x} \bigg( 3 \tilde{\eta}^2 + \pdvn{2}{\tilde{\eta}}{\tilde{x}} \bigg)
+        = 0
+    }
+\end{aligned}$$
+
+Many authors flip the sign of $$\tilde{\eta}_{\tilde{x}\tilde{x}\tilde{x}\tilde{x}}$$
+to get the so-called "good" Boussinesq equation
+(as opposed to the "bad" one above).
+For fluid surface waves, this is unphysical,
+but it makes the problem more well-behaved mathematically;
+the details are beyond the scope of this article.
+
+
+
+## Soliton solution
+
+Let us make an ansatz for $$\tilde{\eta}$$ that describes a wave
+with a fixed shape propagating in the positive $$\tilde{x}$$-direction
+at dimensionless velocity $$v$$:
+
+$$\begin{aligned}
+    \tilde{\eta}(\tilde{x}, \tilde{t})
+    = \phi(\xi)
+    \qquad
+    \xi
+    \equiv \tilde{x} - v \tilde{t}
+    \qquad \implies \qquad
+    \pdv{}{\tilde{t}}
+    = -v \pdv{}{\xi}
+    \qquad
+    \pdv{}{\tilde{x}}
+    = \pdv{}{\xi}
+\end{aligned}$$
+
+With this, the Boussinesq equation becomes a nonlinear ordinary differential equation:
+
+$$\begin{aligned}
+    0
+    &= (v^2 - 1) \phi_{\xi \xi} - \pdvn{2}{}{\xi} (3 \phi^2 + \phi_{\xi \xi})
+\end{aligned}$$
+
+We abbreviate $$w \equiv v^2 \!-\! 1$$ and integrate twice,
+introducing integration constants $$A$$ and $$B$$:
+
+$$\begin{aligned}
+    w \phi - 3 \phi^2 - \phi_{\xi \xi}
+    = A \xi + B
+\end{aligned}$$
+
+We restrict ourselves to localized solutions
+by demanding that $$\phi \to 0$$ for $$\xi \to \pm \infty$$.
+This implies that also $$\phi_\xi \to 0$$ and $$\phi_{\xi \xi} \to 0$$,
+meaning that we must set $$A = B = 0$$ to satisfy the equation at infinity.
+The remaining terms are multiplied by $$\phi_\xi$$ to give:
+
+$$\begin{aligned}
+    0
+    &= w \phi \phi_\xi - 3 \phi^2 \phi_\xi - \phi_{\xi \xi} \phi_\xi
+    = \frac{1}{2} \pdv{}{\xi} \Big( w \phi^2 - 2 \phi^3 - (\phi_\xi)^2 \Big)
+\end{aligned}$$
+
+Integrating (and dropping the integration constant due to localization) yields:
+
+$$\begin{aligned}
+    (\phi_\xi)^2
+    = \phi^2 (w - 2 \phi)
+\end{aligned}$$
+
+Because $$\phi_\xi$$ is real, we need the right-hand side to be positive,
+so $$w > 2 \phi$$; for $$\phi \to 0$$, this means that $$w > 0$$.
+This equation is similar to the one encountered when solving
+the [Korteweg-de Vries equation](/know/concept/korteweg-de-vries-equation/)
+and is integrated in the same way; refer there for details.
+The result is:
+
+$$\begin{aligned}
+    \boxed{
+        \tilde{\eta}(\tilde{x}, \tilde{t})
+        = \frac{w}{2} \sech^2\!\bigg( \frac{\sqrt{w}}{2} \big( \tilde{x} - \sqrt{1 \!+\! w} \: \tilde{t} - \tilde{x}_0 \big) \bigg)
+    }
+\end{aligned}$$
+
+This is known as a **soliton**.
+Reintroducing units by replacing $$\tilde{\eta} = \eta / \eta_c$$ etc.
+leads to:
+
+$$\begin{aligned}
+    \boxed{
+        \eta(x, t)
+        = w h \sech^2\!\bigg( \frac{\sqrt{3 w}}{2 h} \Big( x - \sqrt{(1 \!+\! w) g h} \: t - x_0 \Big) \bigg)
+    }
+\end{aligned}$$
+
+Note that Boussinesq's original calculation had $$(1 \!+\! w/2)$$ instead of $$\sqrt{1 \!+\! w}$$;
+the former is simply a first-order approximation of the latter.
+Recall that $$\sqrt{g h}$$ is the phase velocity of Lagrange's linear theory:
+this shows that nonlinear waves are faster,
+and speed up with amplitude.
+
+
+
 ## References
 1.  J. Boussinesq,
     [Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond](http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16416&Deb=63&Fin=116&E=PDF),
@@ -451,3 +613,6 @@ which are both on the order of $$h / \lambda$$.
 3.  D. Dutykh, F. Dias,
     [Dissipative Boussinesq equations](https://doi.org/10.1016/j.crme.2007.08.003),
     2007, Elsevier.
+4.  M.B. Almatrafi, A.R. Alharbi, C. Tunç,
+    [Constructions of the soliton solutions to the good Boussinesq equation](doi.org/10.1186/s13662-020-03089-8),
+    2020, Springer.
diff --git a/source/know/concept/euler-equations/index.md b/source/know/concept/euler-equations/index.md
index 3730ea3..2654d2b 100644
--- a/source/know/concept/euler-equations/index.md
+++ b/source/know/concept/euler-equations/index.md
@@ -11,14 +11,13 @@ layout: "concept"
 
 The **Euler equations** are a system of partial differential equations
 that govern the movement of **ideal fluids**,
-i.e. fluids without viscosity.
-There exist several forms, depending on
-the surrounding assumptions about the fluid.
+i.e. fluids without [viscosity](/know/concept/viscosity/).
 
 
-## Incompressible fluid
 
-In a fluid moving according to the velocity vield $$\va{v}(\va{r}, t)$$,
+## Incompressible fluids
+
+In a fluid moving according to the velocity field $$\va{v}(\va{r}, t)$$,
 the acceleration felt by a particle is given by
 the **material acceleration field** $$\va{w}(\va{r}, t)$$,
 which is the [material derivative](/know/concept/material-derivative/) of $$\va{v}$$:
@@ -33,14 +32,15 @@ This infinitesimal particle obeys Newton's second law,
 which can be written as follows:
 
 $$\begin{aligned}
-    \va{w} \dd{m}
+    \va{w} m
     = \va{w} \rho \dd{V}
     = \va{f^*} \dd{V}
 \end{aligned}$$
 
-Where $$\dd{m}$$ and $$\dd{V}$$ are the particle's mass volume,
-and $$\rho$$ is the fluid density, which we assume, in this case, to be constant in space and time.
-Then the **effective force density** $$\va{f^*}$$ represents the net force-per-particle.
+Where $$m$$ and $$\dd{V}$$ are the particle's mass and volume,
+and $$\rho$$ is the fluid density, which we assume
+to be constant in space and time in this case.
+Now, the **effective force density** $$\va{f^*}$$ represents the net force-per-particle.
 By dividing the law by $$\dd{V}$$, we find:
 
 $$\begin{aligned}
@@ -51,13 +51,13 @@ $$\begin{aligned}
 Next, we want to find another expression for $$\va{f^*}$$.
 We know that the overall force $$\va{F}$$ on an arbitrary volume $$V$$ of the fluid
 is the sum of the gravity body force $$\va{F}_g$$,
-and the pressure contact force $$\va{F}_p$$ on the enclosing surface $$S$$.
+and the pressure contact force $$\va{F}_p$$ on the enclosing surface $$\partial V$$.
 Using the divergence theorem, we then find:
 
 $$\begin{aligned}
     \va{F}
     = \va{F}_g + \va{F}_p
-    = \int_V \rho \va{g} \dd{V} - \oint_S p \dd{\va{S}}
+    = \int_V \rho \va{g} \dd{V} - \oint_{\partial V} p \dd{\va{S}}
     = \int_V (\rho \va{g} - \nabla p) \dd{V}
     = \int_V \va{f^*} \dd{V}
 \end{aligned}$$
@@ -76,31 +76,28 @@ Dividing this by $$\rho$$,
 we get the first of the system of Euler equations:
 
 $$\begin{aligned}
-    \boxed{
-        \va{w}
-        = \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
-        = \va{g} - \frac{\nabla p}{\rho}
-    }
+    \va{w}
+    = \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
+    = \va{g} - \frac{\nabla p}{\rho}
 \end{aligned}$$
 
-The last ingredient is **incompressibility**:
+The last ingredient is incompressibility:
 the same volume must simultaneously
-be flowing in and out of an arbitrary enclosure $$S$$.
+be flowing in and out of an arbitrary enclosure $$\partial V$$.
 Then, by the divergence theorem:
 
 $$\begin{aligned}
     0
-    = \oint_S \va{v} \cdot \dd{\va{S}}
+    = \oint_{\partial V} \va{v} \cdot \dd{\va{S}}
     = \int_V \nabla \cdot \va{v} \dd{V}
 \end{aligned}$$
 
-Since $$S$$ and $$V$$ are arbitrary,
-the integrand must vanish by itself everywhere:
+Since $$V$$ is arbitrary,
+the integrand must vanish by itself,
+leading to the **continuity relation**:
 
 $$\begin{aligned}
-    \boxed{
-        \nabla \cdot \va{v} = 0
-    }
+    \nabla \cdot \va{v} = 0
 \end{aligned}$$
 
 Combining this with the equation for $$\va{w}$$,
@@ -118,62 +115,104 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The above form is straightforward to generalize to incompressible fluids
-with non-uniform spatial densities $$\rho(\va{r}, t)$$.
-In other words, these fluids are "lumpy" (variable density),
-but the size of their lumps does not change (incompressibility).
 
-To update the equations, we demand conservation of mass:
+
+## Compressible fluids
+
+If the fluid is compressible,
+the condition $$\nabla \cdot \va{v} = 0$$ no longer holds,
+so to update the equations we demand that mass is conserved:
 the mass evolution of a volume $$V$$
-is equal to the mass flow through its boundary $$S$$.
+is equal to the mass flow through its boundary $$\partial V$$.
 Applying the divergence theorem again:
 
 $$\begin{aligned}
     0
-    = \dv{}{t}\int_V \rho \dd{V} + \oint_S \rho \va{v} \cdot \dd{\va{S}}
+    = \dv{}{t}\int_V \rho \dd{V} + \oint_{\partial V} \rho \va{v} \cdot \dd{\va{S}}
     = \int_V \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) \dd{V}
 \end{aligned}$$
 
 Since $$V$$ is arbitrary, the integrand must be zero.
-This leads to the following **continuity equation**,
-to which we apply a vector identity:
+The new **continuity equation** is therefore:
 
 $$\begin{aligned}
     0
     = \dv{\rho}{t} + \nabla \cdot (\rho \va{v})
-    = \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho + \rho (\nabla \cdot \va{v})
+    = \dv{\rho}{t} + \va{v} \cdot \nabla \rho + \rho \nabla \cdot \va{v}
+    = \frac{\mathrm{D} \rho}{\mathrm{D} t} + \rho \nabla \cdot \va{v}
 \end{aligned}$$
 
-Thanks to incompressibility, the last term disappears,
-leaving us with a material derivative:
+When the fluid gets compressed in a certain location, thermodynamics
+states that the pressure, temperature and/or entropy must increase there.
+For simplicity, let us assume an *isothermal* and *isentropic* fluid,
+such that only $$p$$ is affected by compression, and the
+[fundamental thermodynamic relation](/know/concept/fundamental-thermodynamic-relation/)
+reduces to $$\dd{E} = - p \dd{V}$$.
+
+Then the pressure is given by a thermodynamic equation of state $$p(\rho, T)$$,
+which depends on the system being studied
+(e.g. the ideal gas law $$p = \rho R T$$).
+However, the quantity in control of the dynamics
+is not $$p$$, but the internal energy $$E$$.
+Dividing the fundamental thermodynamic relation by $$m \: \mathrm{D}t$$,
+where $$m$$ is the mass of $$\dd{V}$$:
 
 $$\begin{aligned}
-    \boxed{
-        0
-        = \frac{\mathrm{D} \rho}{\mathrm{D} t}
-        = \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho
-    }
+    \frac{\mathrm{D} e}{\mathrm{D} t}
+    = - p \frac{\mathrm{D} v}{\mathrm{D} t}
 \end{aligned}$$
 
-Putting everything together, Euler's system of equations
-now takes the following form:
+With $$e$$ and $$v$$ the specific (i.e. per unit mass)
+internal energy and volume.
+Using that $$\rho = 1 / v$$,
+and substituting the above continuity relation:
+
+$$\begin{aligned}
+    \frac{\mathrm{D} e}{\mathrm{D} t}
+    = - p \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{1}{\rho} \Big)
+    = \frac{p}{\rho^2} \frac{\mathrm{D} \rho}{\mathrm{D} t}
+    = - \frac{p}{\rho} \nabla \cdot \va{v}
+\end{aligned}$$
+
+It makes sense to see a factor $$-\nabla \cdot \va{v}$$ here:
+an incoming flow increases $$e$$.
+This gives us the time-evolution of $$e$$ due to compression,
+but its initial value is another equation of state $$e(\rho, T)$$.
+
+Putting it all together,
+Euler's system of equations now takes the following form:
 
 $$\begin{aligned}
     \boxed{
         \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
         = \va{g} - \frac{\nabla p}{\rho}
-        \qquad
-        \nabla \cdot \va{v}
-        = 0
-        \qquad
+        \qquad \quad
         \frac{\mathrm{D} \rho}{\mathrm{D} t}
-        = 0
+        = - \rho \nabla \cdot \va{v}
+        \qquad \quad
+        \frac{\mathrm{D} e}{\mathrm{D} t}
+        = - \frac{p}{\rho} \nabla \cdot \va{v}
     }
 \end{aligned}$$
 
-Usually, however, when discussing incompressible fluids,
-$$\rho$$ is assumed to be spatially uniform,
-in which case the latter equation is trivially satisfied.
+What happens if the fluid is actually incompressible,
+so $$\nabla \cdot \va{v} = 0$$ holds again? Clearly:
+
+$$\begin{aligned}
+    \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
+    = \va{g} - \frac{\nabla p}{\rho}
+    \qquad \quad
+    \frac{\mathrm{D} \rho}{\mathrm{D} t}
+    = 0
+    \qquad \quad
+    \frac{\mathrm{D} e}{\mathrm{D} t}
+    = 0
+\end{aligned}$$
+
+So $$e$$ is constant, which is in fact equivalent to saying that $$\nabla \cdot \va{v} = 0$$.
+The equation for $$\rho$$ enforces conservation of mass
+for inhomogeneous fluids, i.e. fluids that are "lumpy",
+but where the size of the lumps is conserved by incompressibility.
 
 
 
diff --git a/source/know/concept/korteweg-de-vries-equation/index.md b/source/know/concept/korteweg-de-vries-equation/index.md
index 4a050fe..2857e23 100644
--- a/source/know/concept/korteweg-de-vries-equation/index.md
+++ b/source/know/concept/korteweg-de-vries-equation/index.md
@@ -10,8 +10,8 @@ layout: "concept"
 
 The **Korteweg-de Vries (KdV) equation** is
 a nonlinear 1+1D partial differential equation
-derived to describe water waves.
-It is usually given in its dimensionless form, namely:
+that was originally derived to describe water waves.
+It is usually given in its dimensionless form:
 
 $$\begin{aligned}
     \boxed{
diff --git a/source/know/concept/optical-bloch-equations/index.md b/source/know/concept/optical-bloch-equations/index.md
index fe74b7e..d663c9e 100644
--- a/source/know/concept/optical-bloch-equations/index.md
+++ b/source/know/concept/optical-bloch-equations/index.md
@@ -193,7 +193,10 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Putting everything together,
-we arrive at the **optical Bloch equations** governing $$\hat{\rho}$$:
+we arrive at the **optical Bloch equations** governing $$\hat{\rho}$$,
+which are the basis of the
+[Maxwell-Bloch equations](/know/concept/maxwell-bloch-equations/)
+and by extension all laser theory:
 
 $$\begin{aligned}
     \boxed{
diff --git a/source/know/concept/reynolds-number/index.md b/source/know/concept/reynolds-number/index.md
index 9ae4f4b..4236617 100644
--- a/source/know/concept/reynolds-number/index.md
+++ b/source/know/concept/reynolds-number/index.md
@@ -77,15 +77,14 @@ $$\begin{aligned}
 
 If we choose $$U$$ and $$L$$ appropriately for a given system,
 the Reynolds number allows us to predict the general trends.
-It can be regarded as the inverse of an "effective viscosity":
+It can be regarded as the inverse of an "effective [viscosity](/know/concept/viscosity/)":
 when $$\mathrm{Re}$$ is large, viscosity only has a minor role,
 but when $$\mathrm{Re}$$ is small, it dominates the dynamics.
 
 Another way is thus to see the Reynolds number
 as the characteristic ratio between the advective term
 (see [material derivative](/know/concept/material-derivative/))
-to the [viscosity](/know/concept/viscosity/) term,
-since $$\va{v} \sim U$$:
+to the viscosity term, since $$\va{v} \sim U$$:
 
 $$\begin{aligned}
      \mathrm{Re}