commit 4f4ab27f2fff4eeb493417d60e4cd49fce763e07
parent 8ce911ece70e6d388d9cd789443ae126f7eddedd
Author: miksa <milutin@popovic.xyz>
Date: Sun, 5 Jun 2022 13:38:57 +0200
done with basics
Diffstat:
7 files changed, 1295 insertions(+), 955 deletions(-)
diff --git a/app_pde/basics_fluids.tex b/app_pde/basics_fluids.tex
@@ -1,955 +0,0 @@
-\include{./preamble.tex}
-
-\usepackage{amsmath}
-\numberwithin{equation}{section}
-
-\begin{document}
-
-\maketitle
-\tableofcontents
-
-\section{Governing Equations of Fluid Dynamics}
-We first start of with a fluid with a density
-\begin{align}
- \rho(\mathbf{x}, t),
-\end{align}
-in three dimensional Cartesian coordinates $\mathbf{x} = (x, y, z)$ at time
-$t$. For water-wave applications, we should note that we take
-$\rho=\text{constant}$, but we will go into this fact later. The fluid moves
-in time and space with a velocity field
-\begin{align}
- \mathbf{u}(\mathbf{x}, t) = (u, v, w).
-\end{align}
-Additionally it is also described by its pressure
-\begin{align}
- P(\mathbf{x}, t),
-\end{align}
-generally depending on time and position. When thinking of e.g. water the
-pressure increases the deeper we go, that is with decreasing or increasing $z$
-direction (depending how we set up our system $z$ pointing up or down
-respectively).
-
-The general assumption in fluid dynamics is the \textbf{Continuum
-Hypothesis}, which assumes continuity of $\textbf{u}, \rho$ and $P$ in
-$\mathbf{x}$ and $t$. In other words, we premise that the velocity field,
-density and pressure are ''nice enough`` functions of position and time, such
-that we can do all the differential operations we desire in the framework of
-differential analysis.
-\subsection{Mass Conservation}
-Our aim is to derive a model of the fluid and its dynamics, with respect to
-time and position, in the most general way. This is usually done thinking
-of the density of a given fluid, which is a unit mass per unit volume,
-intrinsically an integral representation to derive these equations suggests
-by itself.
-
-Let us now thing of an arbitrary fluid. Within this fluid we define a fixed
-volume $V$ relative to a chosen inertial frame and bound it by a surface $S$
-within the fluid, such that the fluid motion $\mathbf{u}(\mathbf{x}, t)$ may
-cross the surface $S$. The fluid density is given by $\rho(\mathbf{x}, t)$,
-thereby the mass of the fluid in the defined Volume $V$ is an integral
-expression
-\begin{align}
- m = \int_V \rho(\mathbf{x}, t) dV.
-\end{align}
-The figure bellow \ref{fig:volume}, expresses the above described picture.
-\begin{figure}[H]
- \centering
- \begin{tikzpicture}[>=latex,scale=1, xscale=1, opacity=.8]
-% second sphere
- \begin{scope}[rotate=10, xscale=3, yscale=2, shift={(2.3,-0.2)}]
- \coordinate (O) at (0,0);
- \shade[ball color=gray!10!] (0,0) coordinate(Hp) circle (1) ;
-
- \draw[thick] (O) circle (1);
- \draw[rotate=5] (O) ellipse (1cm and 0.66cm);
- \draw[rotate=90] (O) ellipse (1cm and 0.33cm);
-\node[circle, fill=black, inner sep=1pt] at (0.15, 0.25) {} ; \draw[-latex, thick] (0.15, 0.25) -- (1, 1) ;
- \node[right] at (1, 1) {$\mathbf{u}(\mathbf{x}, t)$};
-
- \node[] at (O) {$V$};
- \node[] at (0.55, -0.25) {$\rho(\mathbf{x}, t)$};
-
- \draw[-] (0.76, -0.66) -- (1.2, -0.7);
- \node[right] at (1.2, -0.7) {$S$};
-
- \draw[-latex, thick] (-0.25, -0.65) -- (-1, -1);
- \node[left] at (-1, -1) {$\mathbf{n}$};
-
- \end{scope}
-
-% axis
- \end{tikzpicture}
- \caption{Volume bounded by a surface in a fluid with density and momentum,
- with a surface normal vector $\mathbf{n}$ \label{fig:volume}}
-\end{figure}
-
-Since we want to figure out the fluid's dynamics, we can consider the rate
-of change in the completely arbitrary $V$. The rate of change of mass needs to
-disappear, i.e. it is equal to zero since we cannot lose mass. Matter (mass) is
-neither created nor destroyed anywhere in the fluid, leading us to
-\begin{align}
- \frac{d}{dt}\left( \int_V \rho(\mathbf{x}, t)\ dV \right) = 0.
-\end{align}
-\textbf{NOT SURE HERE YET!!!!!!!!!!!, CHECK LEIBINZ FORMULA}
-To get more information we simply ''differentiate under the integral
-sign``, also known as the Leibniz Rule of Integration, see appendix
-\ref{appendix:leibniz}, the integral equation representing the rate of change
-of mass reads
-\begin{align}\label{eq:mass balance}
- \frac{dm}{dt} = \int_V \frac{\partial \rho(\mathbf{x}, t)}{\partial t}\ dV
- +\int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u}\cdot\mathbf{n}\ dS
- = 0.
-\end{align}
-\textbf{----------------------}
-The above equation in \ref{eq:mass balance} is an underlying equation, describing that the rate of
-change of mass in V is brought about, only by the rate of mass flowing into
-V across S, and thus the mass does not change.
-
-For the second integral in \ref{eq:mass balance} we utilize the Gaussian
-integration law to acquire an integral over the volume
-\begin{align}
- \int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u} \cdot \mathbf{n} \ dS =
- \int_V \nabla (\rho \mathbf{u})\ dV.
-\end{align}
-Thereby we can put everything inside the volume integral
-\begin{align}
- \frac{d m}{dt} = \int_V \left(\partial_t \rho + \nabla(\rho \mathbf{u}) \right) \ dV = 0.
-\end{align}
-Everything under the integral sign needs to be zero, thus we obtain
-the \textbf{Equation of Mass Conservation} or in the general sense also
-called the \textbf{Continuity Equation}
-\begin{align}\label{eq:continuity}
- \partial_t \rho + \nabla(\rho \mathbf{u}) = 0
-\end{align}
-
-In light of the results of the equation of mass conservation
-in \ref{eq:continuity}, an product rule gives
-\begin{align}
- \partial_t \rho + (\nabla \rho)\mathbf{u} + \rho(\nabla \mathbf{u}),
-\end{align}
-for notational purposes, we define the \textbf{material/convective derivative}
-as follows
-\begin{align}
- \frac{D}{Dt} = \partial_t + \mathbf{u}\nabla.
-\end{align}
-With the material derivative the equation of mass conservation reads
-\begin{align}
- \frac{D\rho}{Dt} + \rho \nabla\mathbf{u} = 0
-\end{align}
-We may undertake the first case separation, initiating $\rho = \text{cosnt.}$
-called \textbf{incompressible flow} causes the material derivative of $\rho$ to
-be zero, and thereby
-\begin{align}
- \frac{D\rho}{Dt} = 0 \quad \Rightarrow \quad \nabla \mathbf{u} = 0,
-\end{align}
-following that the divergence of the velocity field is zero, in this case
-$\mathbf{u}$ is called \textbf{solenoidal}.
-\subsection{Euler's Equation of Motion}
-Additional consideration we undertake is the assumption of an
-\textbf{inviscid} fluid, that is we set viscosity to zero. Otherwise we would
-get a viscous contribution under the integral which results in the
-Navier-Stokes equation. In this regard we apply Newton's second law to our
-fluid in terms of infinitesimal pieces $\delta V$ of the fluid. The
-acceleration divides into two terms, a \textbf{body force} given by gravity
-of earth in the $z$ coordinate $\mathbf{F} = (0, 0, -g)$ and a
-\textbf{local/short-rage force} described by the stress tensor in the fluid.
-In the inviscid case we the local force retains the pressure $P$, producing a
-normal force, with respect to the surface, acting onto any infinitesimal
-element in the fluid. The integral formulation of the force would be
-\begin{align}
- \int_V \rho \mathbf{F}\ dV - \int_S P\mathbf{n}\ dV.
-\end{align}
-Now applying the Gaussian rule of integration on the second integral over the
-surface, the resulting force in per unit volume is
-\begin{align}
- \int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV.
-\end{align}
-The acceleration of the fluid particles is given by $\frac{D\mathbf{u}}{Dt}$,
-and thus the total force per unit volume on the other hand is
-\begin{align}
- \int_V \rho \frac{D\mathbf{u}}{Dt}\ dV =
- \int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV.
-\end{align}
-Newton's Second Law for a fluid in an Volume is essentially saying that the
-rate of change of momentum of the fluid in the fixed volume $V$, which is the particle
-acceleration is the resulting force acting on V together with the rate of
-flow of momentum across the surface $S$ into the volume $V$. Hence we arrive
-at the \textbf{Euler's Equation(s) of Motion}
-\begin{align}
- \frac{D\mathbf{u}}{Dt} = \left(\frac{\partial \mathbf{u}}{\partial t}
- (\mathbf{u}\nabla)\mathbf{u}\right) =
- -\frac{1}{\rho}\nabla P + \mathbf{F}.
-\end{align}
-As a side note we have mentioned that there is another contribution if the
-fluid is viscid. Indeed there is a tangential force due to the velocity
-gradient, which into introduces the additional term
-\begin{align}
- \mu \nabla^2 \mathbf{u}, \qquad
- \mu = \text{viscosity of the Fluid}.
-\end{align}
-Thereby the equations become
-\begin{align}
- \rho\frac{D\mathbf{u}}{Dt}
- = -\nabla P + \rho \mathbf{F} + \mu \nabla^2 \mathbf{u}.
-\end{align}
-
-For now we have separated two simplifications, that define an
-\textbf{idealized/perfect fluid}
-\begin{enumerate}
- \item \textbf{incompressible} $\qquad \mu=0$
- \item \textbf{inviscid} $\quad \rho = \text{const.},\ \nabla \mathbf{u}=
- 0$
-\end{enumerate}
-\subsection{Vorticity and irrotational Flow}
-The curl of the velocity field $\mathbf{\omega} = \nabla \times \mathbf{u}$
-of a fluid (i.e. the vorticity), describes a spinning motion of the fluid
-near a position $\mathbf{x}$ at time $t$. The vorticity is an important
-property of a fluid, flows or regions of flows where $\mathbf{\omega}=0$ are
-\textbf{irrotational}, and thus can be modeled and analyzed following well
-known routine methods. Even though real flows are rarely irrotational
-anywhere (!), in water wave theory wave problems, from the classical aspect
-of vorticity have a minor contribution. Hence we can assume irrotational flow
-modeling water waves. To arrive at the vorticity in the equations of motions
-derived in the last section we resort to a differential identity derived in appendix
-\ref{appendix:diff identity}, which gives for the material derivative
-\begin{align}
- \frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t}
- \nabla(\frac{1}{2}\mathbf{u}\mathbf{u)}
- - \left( \mathbf{u}\times (\nabla \times \mathbf{u} \right).
-\end{align}
-Thus the equations of motion become
-\begin{align}
- \frac{\partial \mathbf{u}}{\partial t} + \nabla\left(
- \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega \right)
- = \mathbf{u} \times \mathbf{\omega},
-\end{align}
-where $\Omega$ is the force potential per
-unite mass given by $\mathbf{F} = -\nabla \Omega$.
-
-At this point we may differentiate between \textbf{stead and unsteady flow}.
-For \textbf{Steady Flow} we assume that $\mathbf{u}, P$ and $\Omega$ are time
-independent, thus we get
-\begin{align}
- \nabla\left( \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega
- \right) = \mathbf{u} \times \mathbf{\omega}.
-\end{align}
-It is general knowledge that the gradient of a function $\nabla f$ is
-perpendicular the level sets of $f(\mathbf{x})$, where $f(\mathbf{x}) =
-\text{const.}$. Thus $\mathbf{u} \times \mathbf{\omega}$ is orthogonal to
-the surfaces where
-\begin{align} \label{eq:bernoulli}
- \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega =
- \text{const.},
-\end{align}
-The above equation is called \textbf{Bernoulli's Equation}.
-
-Secondly \textbf{Unsteady Flow} but irrotational (+ incompressible), first of
-all gives us the condition for the existence of a velocity potential $\phi$
-in the sense
-\begin{align}
- \mathbf{\omega} = \nabla \times \mathbf{u} = 0 \quad \Rightarrow \quad
- \mathbf{u} = \nabla \phi,
-\end{align}
-where $\phi$ needs to satisfy the Laplace equation
-\begin{align}
- \Delta \phi = 0.
-\end{align}
-According to the Theorem of Schwartz we may exchange $\frac{\partial
-}{\partial t}$ and $\nabla$, giving us an expression for the material
-derivative
-\begin{align}
- \nabla\left( \frac{\partial \phi}{\partial t} +\frac{1}{2}
- \mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega \right) = 0
-\end{align}
-Thus the expression differentiated by the $\nabla$ operator is an arbitrary
-function $f(\mathbf{x}, t)$, writing
-\begin{align}
- \frac{\partial \phi}{\partial t} +\frac{1}{2}
- \mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega = f(\mathbf{x}, t).
-\end{align}
-The function $f(\mathbf{x}, t)$ can be removed by gauge transformation of
-$\phi \rightarrow \phi + \int f(\mathbf{x}, t)\ dt$, never the less this is
-not further discussed and left to the reader in the reference.
-\subsection{Boundary Conditions for water waves}
-The boundary conditions for water-wave problems vary, generally on the
-simplification we undertake. At the surface, called the free surface as in
-free from the velocity conditions, we have the atmospheric stress on the
-fluid. The stress component would again have a viscid component, this however
-is only relevant when modeling surface wind, in this review we model the
-fluid as unaffectedly and within reason as inviscid. The atmosphere employs
-only a pressure on the surface, this pressure is taken to be the atmospheric
-pressure, dependent on time and point in space. Thereby any surface tension
-effects can also include a scenario at a curved surface (e.g. wave), giving
-rise to the pressure difference across the surface. A more precise
-description would use Thermodynamics to derive boundary conditions coupling
-water surface and the air above it, yet the density component of air
-compared to that of water makes our ansatz viable. The described conditions
-are called the \textbf{dynamic conditions}
-
-An additional condition revolves around the fluid particles on the moving
-surface, called the \textbf{kinematic condition}. This condition bounds
-the vertical velocity component on the surface.
-
-The logical step now is to define boundary conditions on the bod of the
-fluid, i.e. the bottom. If the viscid case bottom is impermeable, we a no
-slip condition to all fluid particles $\mathbf{u}_\text{bottom}= 0$. If we
-assume that the fluid is inviscid then the bottom becomes a surface of the
-fluid in the sense that the fluid particles in contact with the bed move in
-the surface, we more or less mirror the kinematic condition of the surface.
-For many problems the condition is going to vary, in most cases the bottom
-will be rigid and fixed not necessarily horizontal. This condition is simply
-called the \textbf{bottom condition}.
-\subsubsection{Kinematic Condition}
-Obtaining the free surface is the primary objective in the theory of modeling
-water waves, represented by
-\begin{align}
- z = h(\mathbf{x}_\perp, t),
-\end{align}
-where $\mathbf{x}_\perp = (x, y)$ in Cartesian, or $\mathbf{x}_\perp = (r,
-\theta)$ in cylindrical coordinates. A surfaces that moves with the fluid,
-always contains the same fluid particles, described as
-\begin{align}
- \frac{D}{Dt}\left(z - h(\mathbf{x}_\perp, t \right) = 0.
-\end{align}
-Upon expanding the derivative we get
-\begin{align}
- \frac{Dz}{Dt} - \frac{Dh}{Dt}
- &= \frac{\partial z}{\partial t}+
- (\mathbf{u}\nabla)z - \frac{\partial h}{\partial t} -(\mathbf{u}\nabla)\\
- &= w - \left(h_t - (\mathbf{u}_\perp \nabla_\perp) h\right) = 0,
-\end{align}
-where the subscript $\perp$ describes the components with regard to
-$\mathbf{x}_\perp$. The \textbf{kinematic condition} reads
-\begin{align}
- w = h_t - (\mathbf{u}_\perp \nabla_\perp) h \qquad \text{on}\;\;
- z=h(\mathbf{u}_\perp, t).
-\end{align}
-
-\subsubsection{Dynamic Condition}
-As described in the prescript of this section, the case of an inviscid fluid,
-requires that only the pressure $P$ needs to be described on the free surface
-$z = h(\mathbf{x}_\perp, t)$. Assuming incompressible, irrotational,
-unsteady flow and setting $P=P_a$ for atmospheric pressure and $\Omega =
-g\cdot z$ for the force per unit mass potential the equations of motion are
-\begin{align}
- \frac{\partial \phi}{\partial t} +\frac{1}{2}\mathbf{u}\mathbf{u}
- + P_\frac{a}{\rho}+gh = f(t) \qquad \text{on}\;\; on z=h.
-\end{align}
-Somewhere $\|\mathbf{x}_\perp\| \rightarrow \infty$ the fluid reaches
-equilibrium and is thereby stationary, thereby has no motion and the pressure
-is $P=P_a$ and the surface is a constant $h = h_0$ $f(t)$ is
-\begin{align}
- f(t) = \frac{P_a}{\rho}+gh_0.
-\end{align}
-The simplest description for the \textbf{dynamic condition} may be written as
-\begin{align}
- \frac{\partial \phi}{\partial t}
- +\frac{1}{2}\mathbf{u}\mathbf{u}+g(h-h_0) = 0 \qquad \text{on}\;\; z=h.
-\end{align}
-
-Regarding the pressure difference on a curved surface, we may expand the
-dynamic condition by introducing the pressure difference known as the
-\textbf{Young-Laplace Equation}
-\begin{align}
- \Delta P = \frac{\Gamma}{R},
-\end{align}
-where $\Gamma>0$ is the coefficient of surface tension and $\frac{1}{R}$ is
-the curvature representing an implicit function, in our case the implicit
-function is $z - h(\mathbf{x}_\perp, t)$ for fixed time. The curvature in
-Cartesian coordinates takes the form
-\begin{align}
- \frac{1}{R} = \frac{(1+h_y^2)h_{x x}+(1+h_y^2)h_{yy} -
- 2h_xh_yh_{xy}}{\left( h_x^2+h_y^2+1 \right)^{\frac{3}{2}} },
-\end{align}
-the derivation is precisely described in \ref{appendix:curvature}
-
-
-
-\subsubsection{The Bottom Condition}
-The representation for the bottom is
-\begin{align}
- z = b(\mathbf{x}_\perp, t),
-\end{align}
-where the fluid surface needs to satisfy
-\begin{align}
- \frac{D}{Dt} \left(z - b(\mathbf{x}_\perp) \right) = 0.
-\end{align}
-Hence we arrive at the bottom boundary conditions
-\begin{align}
- w = b_t + (\mathbf{u}_\perp \nabla_\perp)b \qquad \text{on}\;\; z=b ,
-\end{align}
-where $b(\mathbf{x}_\perp, t)$ is already known for most water wave
-problems. If we consider a stationary bottom then the time derivative
-vanishes, leaving us with the following condition
-\begin{align}
- w = (\mathbf{u}_\perp \nabla_\perp)b \qquad \text{on}\;\; z=b
-\end{align}
-
-
-\subsubsection{Integrated Mass Condition}
-In this section we want to combine the kinematics of both the free and the
-bottom surface with the mass conservation equation on the perpendicular
-components
-\begin{align}
- \nabla \mathbf{u} = \nabla_\perp \mathbf{u}_\perp + w_z = 0 .
-\end{align}
-Integrating the above expression from bottom to surface, i.e. from
-$z=b(\mathbf{x}_\perp,t)$ to $z = h (\mathbf{x},t)$ gives
-\begin{align}
- \int_b^h \nabla_\perp \mathbf{u}_\perp\ dz + w\bigg|_{z=b}^{z=h} = 0,
-\end{align}
-where we insert the conditions on the free surface and on the bottom surface
-\begin{align}
- w &= h_t + (\mathbf{u}_{\perp \text{s}} \nabla_\perp) h \quad
- \text{on}\;\; z = h\\
- w &= b_t + (\mathbf{u}_{\perp \text{b}} \nabla_\perp) h \quad
- \text{on}\;\; z =b,
-\end{align}
-with the subscript $s$ and $b$ indicating the evaluation of a quantity
-on the free surface and the bottom surface respectively. Inserting the
-boundary conditions we get
-\begin{align}
- \int_b^h \nabla_\perp \mathbf{u}_\perp
- + h_t + (\mathbf{u}_{\perp \text{s}} \nabla_\perp) h
- - b_t - (\mathbf{u}_{\perp \text{b}} \nabla_\perp) b= 0.
-\end{align}
-To simplify the equation we resort again to the Leibniz Rule of Integration
-\begin{align}
- \int_b^h \nabla_\perp\mathbf{u}_\perp =
- \nabla_\perp \int_b^h \mathbf{u}_\perp\ dz - (\mathbf{u}_{\perp \text{s}}
- \nabla_\perp)h - (\mathbf{u}_{\perp \text{b}})b.
-\end{align}
-As a consequence the \textbf{Integrated Mass Condition} is given by
-\begin{align}
- \nabla_\perp \int_b^h \mathbf{u}_\perp\ dz + \underbrace{h_t -
- b_t}_{=d_t} = 0.
-\end{align}
-\subsection{Energy Equation}
-To derive the energy equation we start off with Euler's Equation of Motion
-\begin{align}
- \mathbf{u} _t + \nabla
- (\frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}+\Omega) = \mathbf{u}\times
- \mathbf{w},
-\end{align}
-multiplying the equation with $\mathbf{u}$ we get
-\begin{align}
- &\mathbf{u}\mathbf{u} _t \label{eq:energy1} \\
- &+(\mathbf{u}\nabla)(\frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}+\Omega)\label{eq:energy2}\\
- &= \mathbf{u}(\mathbf{u}\times
- \mathbf{w})\label{eq:energy3}.
-\end{align}
-The first equation given in \ref{eq:energy1} can we rewritten using inverse
-product rule of differentiation
-\begin{align}
- \mathbf{u}\frac{\partial \mathbf{u}}{\partial t}
- &= \frac{\partial
- }{\partial t} (\mathbf{u}\mathbf{u}) - \frac{\partial \mathbf{u}}{\partial t}
- \mathbf{u} \\
- &= \frac{\partial
- }{\partial t} (\mathbf{u}\mathbf{u}) - \mathbf{u}\frac{\partial
- \mathbf{u}}{\partial t}\\
- \Rightarrow\quad & \mathbf{u} \frac{\partial \mathbf{u}}{\partial t} =
- \frac{1}{2}\frac{\partial }{\partial t} (\mathbf{u}\mathbf{u}).
-\end{align}
-Then we may add
-\begin{align}
- \left(\frac{1}{2} \mathbf{u}\mathbf{u}+\frac{P}{\rho} +\Omega \right)
- \underbrace{(\nabla u)}_{=0} = 0,
-\end{align}
-to above not changing anything. Thereby getting
-\begin{align}
- \frac{\partial }{\partial t} (\frac{1}{2}\mathbf{u}\mathbf{u})
- +(\mathbf{u}\nabla \mathbf{u})\left(
- \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho} \right)
- +\left( \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho} + \Omega \right)
- (\nabla \mathbf{u}) = 0.
-\end{align}
-Applying the product rule we can simplify
-\begin{align}
- \frac{\partial }{\partial t} \left(\frac{1}{2}\mathbf{u}\mathbf{u}\right)
- +\nabla \left(\mathbf{u}\left(\mathbf{u}(
- \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}\right) \right) = 0,
-\end{align}
-additionally adding $\frac{\partial \Omega}{\partial t} =0$ leads us to
-\begin{align}
- \underbrace{\frac{\partial }{\partial t} \left(\frac{1}{2}\mathbf{u}\mathbf{u}
- +\Omega\right)}_{\text{change of total energy density}}
- +\underbrace{\nabla \left(\mathbf{u}\left(\mathbf{u}(
- \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}\right)
-\right)}_{\text{energy flow of the velocity field}} = 0.\label{eq:energy}
-\end{align}
-This is called the \textbf{energy equation} and is a general result for a
-inviscid and incompressible fluids, which we can apply to study water waves.
-We start off with replacing $\nabla = \nabla_\perp + \frac{\partial }{\partial
-z} $ and $\Omega = g z$ and multiplying by $\rho$, then our energy equation
-in \ref{eq:energy} becomes
-\begin{align}
- \frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
- g z\right) + \nabla_\perp\left( \mathbf{u}_\perp\left(
- \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right) \right)
- \frac{\partial}{\partial z} \left( w\left(
- \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho g z \right) \right) = 0.
-\end{align}
-Integrating from bottom to top, i.e. from bed to free surface gets us to
-\begin{align}
- &\int_b^h\frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
- g z\right)\ dz \label{eq:e-int1}\\
- &+ \int_b^h \nabla_\perp\left( \mathbf{u}_\perp\left(
- \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right) \right)\
- dz\label{eq:e-int2}\\
- &+ \left(\frac{\partial}{\partial z} \left( w\left(
- \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho g z \right)
-\right)\right)\Bigg|_b^h \label{eq:e-int3}
- = 0.
-\end{align}
-For equation \ref{eq:e-int1} we use Leibniz Rule of Integration, leaving us
-with
-\begin{align}
- \int_b^h\frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
- g z\right)\ dz
- &= \frac{\partial }{\partial t} \int_b^h
- \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho gz \ dz\\
- &+ \left( \frac{1}{2}\rho \mathbf{u}_s \mathbf{u}_s + \rho g h \right)
- h_t\\
- &- \left( \frac{1}{2}\rho \mathbf{u}_b \mathbf{u}_b + \rho g b \right)
- b_t
-\end{align}
-For equation \ref{eq:e-int2} we again take note of the Leibniz Rule of
-Integration, getting
-\begin{align}
- \int_b^h \nabla_\perp\left( \mathbf{u}_\perp\left(
- \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right) \right)\
- dz
- &= \nabla_\perp \int_b^h \mathbf{u}_\perp\left(
- \frac{1}{2}\rho\mathbf{u}\mathbf{u} + P + \rho g z \right) \ dz\\
- &- \left( \frac{1}{2}\rho \mathbf{u}_s\mathbf{u}_s + P + \rho g h \right)
- \left( \mathbf{u}_{\perp s} \nabla_\perp \right) h\\
- &+\left( \frac{1}{2}\rho \mathbf{u}_b\mathbf{u}_b + P + \rho g b \right)
- \left( \mathbf{u}_{\perp b} \nabla_\perp \right) b
-\end{align}
-Thereby transforming our equation into
-\begin{align}
- \frac{\partial }{\partial t} \underbrace{\int_b^h \frac{1}{2}\rho
- \mathbf{u}\mathbf{u}+\rho g z\ dz}_{=:\mathcal{E}}
- + \nabla_\perp&\underbrace{\int_b^h
- \mathbf{u}_\perp\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho g z
-\right)\ dz}_{:=\mathcal{F}}
-+ \underbrace{P_s h_t - P_b b_t}_{:=\mathcal{P}} = 0\\
-\nonumber\\
- &\frac{\partial \mathcal{E}}{\partial t}
- + \nabla_\perp \mathcal{F} + \mathcal{P} = 0,
-\end{align}
-where $\mathcal{E}$ represents the energy in the flow per unit horizontal
-area, since we are integrating from bed to free surface. Where $\mathcal{F}$
-is the horizontal energy flux vector and lastly $\mathcal{P} = P_s h_t -
-P_b b_t$ is the net energy input due to the pressure forces doing work on the
-upper and lower boundaries, i.e. bottom and free surface of the fluid.
-Assuming stationary rigid bottom condition and constant surface pressure, we
-can set $P_s=0$, such that $\mathcal{P} =0$ leaving us with the equation
-\begin{align}
- \frac{\partial \mathcal{E}}{\partial t}
- + \nabla_\perp \mathcal{F} = 0.
-\end{align}
-We note that the assumption $P_s=0$ is only possible if the coefficient of
-surface tension is set to 0, which usually is not the case.
-\section{Dimensional Analysis}
-Our derived model of fluid dynamics yields formal connections between
-physical quantities. These quantities bear units, e.g. the velocity of fluid
-particles $\mathbf{u}$ has the ``SI'' unites of $\frac{m}{s}$, meters per
-second. The idea is the make use of these scales and formulate a model, where
-the quantities are nondimensionalized, i.e. to get rid of physical units by
-scaling each quantity appropriately. The appropriate length scales are that
-of the typical water depth $h_0$ and the typical wavelength $\lambda$ of a
-surface wave.
-
-\subsection{Nondimensionalisation}
-In summary we use these adaptations
-\begin{itemize}
- \item $h_0$ for the typical water depth
- \item $\lambda$ for the typical wavelength
- \item $\frac{\lambda}{\sqrt{g h_0}}$ time scale of wave propagation
- \item $\sqrt{g h_0}$ velocity scale of waves in $(x, y)$
- \item $\frac{h_0 \sqrt{g h_0} }{\lambda}$ velocity scale in the $z$
- direction.
-\end{itemize}
-$(x, z, t)$, then
-\begin{align}
- u = \psi _z, \qquad w = - \psi_x;
-\end{align}
-and the scale of $\psi$ must be $h_0\sqrt{g h_0}$. Additionally we write the
-boundary condition on the free surface as follows
-\begin{align}
- h = h_0 + a \eta (\mathbf{x}_\perp, t) = z,
-\end{align}
-where $a$ is the typical amplitude and $\eta$ nondimensional function. All in
-all we have the following scaling for the physical quantities of our context
-\begin{align}
- &x \rightarrow\ \lambda x, \quad u \rightarrow \sqrt{gh_0} u, \\
- &y \rightarrow\ \lambda y, \quad v \rightarrow \sqrt{gh_0} v, \qquad
- t\rightarrow \frac{\lambda}{\sqrt{gh_0}}t,\\
- &z \rightarrow\ h_0 z, \quad w \rightarrow
- \frac{h_0\sqrt{gh_0}}{\lambda} w.
-\end{align}
-with
-\begin{align}
- h = h_0 + a \eta, \qquad b \rightarrow h_0 b.
-\end{align}
-The pressure is also rewritten into
-\begin{align}
- P = P_a + \rho g(h_0 -z) + \rho g h_0 p,
-\end{align}
-where $P_a$ is the atmospheric pressure, the term $h_0-z$ represent the
-hydrostatic pressure distribution, i.e. pressure at depth and the term with the pressure
-variable $p$ measures the deviation from the hydrostatic pressure
-distribution. Indeed $p\neq 0 $ for wave propagation. Now we can perform a
-rescaling of the Euler's Equation of Motion, we introduce the notation
-\begin{align}
- &t = \frac{\lambda}{\sqrt{gh_0}}\tau,\quad x = \lambda \xi,\quad u =
- \sqrt{gh_0} \tilde{u}\\
- &y = \lambda \chi,\quad v = \sqrt{gh_0} \tilde{v}\\
- &z = h_0 \zeta, \quad w = \frac{h_0\sqrt{gh_0} }{\lambda}\tilde{w}.
-\end{align}
-We start off with the $x$ coordinate, substitute and apply the chain rule
-leading us to
-\begin{align}
- \frac{Du}{Dt}
- &= \frac{\partial u}{\partial t} +u \frac{\partial
- u}{\partial x} \\
- &= \sqrt{gh_{0}}\frac{\partial \tilde{u}}{\partial \tau} \frac{\partial
- \tau}{\partial t} +gh_0 \tilde{u} \frac{\partial \tilde{u}}{\partial \xi}
- \frac{\partial \xi}{\partial x} \\
- &= \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau}
- \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right),
-\end{align}
-on the other hand
-\begin{align}
- \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau}
- \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right)
- &=-\frac{1}{\rho}\frac{1}{\lambda}\frac{\partial P}{\partial x} \\
- &=-\frac{ g h_0 }{\lambda}\rho \frac{\partial p}{\partial \xi}.
-\end{align}
-Thereby the rescaling evolves to
-\begin{align}
- \frac{D \tilde{u}}{D\tau} = -\frac{\partial p}{\partial \xi}.
-\end{align}
-Because of the same scaling in $y$ we get the same result as in $x$, that is
-\begin{align}
- \frac{D \tilde{v}}{D\tau} = -\frac{\partial p}{\partial \chi}.
-\end{align}
-In the $z$ coordinate we have
-\begin{align}
- \frac{Dw}{Dt}
- &= \frac{\partial w}{\partial t} +w \frac{\partial
- w}{\partial \zeta} \\
- &= \frac{h_0\sqrt{gh_0}}{\lambda} \frac{\sqrt{gh_0}}{\lambda}
- \frac{\partial \tilde{w}}{\partial \tau} + \frac{1}{h_0}
- \frac{h_0\sqrt{gh_0} }{\lambda} \frac{h_0\sqrt{gh_0}}{\lambda}
- \tilde{w}\frac{\partial \tilde{v}}{\partial \zeta}\\
- &= \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
- + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right) .
-\end{align}
-On the other side we have
-\begin{align}
- \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
- + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right)
- &=
- -\frac{1}{h_0\rho} \frac{\partial P}{\partial z} +g \\
- &=-\frac{1}{h_0\rho}(-\rho gh_0 \frac{\partial \zeta}{\partial \zeta}
- \rho gh_0
- \frac{\partial p}{\partial \zeta} ) + g \\
- &= -g \frac{\partial p}{\partial z}.
-\end{align}
-In total for the $z$ direction we get
-\begin{align}
- \underbrace{\left( \frac{h_0}{\lambda} \right)^2}_{=: \delta^2}
- \frac{Dw}{Dt} = -\frac{\partial p}{\partial z},
-\end{align}
-where $\delta$ is the \textbf{long wavelength} or \textbf{shallowness}
-parameter, a very important constant for developing model hierarchies. For
-clarity we resubstitute for $x, y, z, t, u, v$ and $w$, and for completeness
-the we display the equations again, which are
-\begin{align}\label{eq:nondim-motion}
- \frac{Du}{Dt} = - \frac{\partial p}{\partial x}&, \quad
- \frac{Dv}{Dt} = - \frac{\partial p}{\partial y}, \quad
- \delta^2\frac{Dw}{Dt} = - \frac{\partial p}{\partial z}, \\
- &\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}
- +\frac{\partial w}{\partial z} = 0.
-\end{align}
-We can now turn our attention to the boundary conditions, on both free
-surface $z=h$ and the bottom $z=b$ we have $z \Rightarrow h_0 z$ and thereby
-\begin{align}
- z = 1+
- \underbrace{\frac{a}{h_0}}_{:=\varepsilon}\eta(\mathbf{x}_\perp,t) \quad
- \text{and}\quad z= b,
-\end{align}
-where we arrive at our second very important parameter $\varepsilon$ called
-the \textbf{amplitude} parameter. As for the kinematic condition, we
-substitute the free surface $z=h = 1+\varepsilon \eta$ and get
-\begin{align}
- \frac{Dz}{Dt} = \varepsilon\left(\eta_t + (\mathbf{u}_\perp
- \nabla_\perp)\eta\right) \qquad \text{on}\;\; z= 1+\varepsilon \eta.
-\end{align}
-Respectively the bottom condition is not changed
-\begin{align}
- w = b_t + (\mathbf{u}_\perp \nabla_\perp) b \quad \text{on}\;\; z= b.
-\end{align}
-The general dynamic condition for $h = h(x, y, t)$ yields a rescaling of the
-curvature in terms of
-\begin{align}
- \frac{1}{R}
- &= \frac{(1+h_y^2)h_{x x} + (1+h_x^2)h_yy - 2h_xh_yh_{xy}
- }{\left(h_x^2+h_y^2 +1 \right)^{\frac{3}{2}} } \\
- &= -\frac{\varepsilon h_0}{\lambda^2} \frac{(
- 1+\varepsilon^2\delta^2\eta_y^2 )\eta_{x x}+
- (1+\varepsilon^2\delta^2\eta_x^2)\eta_{yy} -
- 2\varepsilon^2\delta^2\eta_x\eta_y\eta_{xy}}{\left(
- 1+\varepsilon^2\delta^2\eta_x^2+\varepsilon^2\delta^2\eta_y^2
- \right)^{\frac{3}{2}} },
-\end{align}
-together with the pressure difference
-\begin{align}
- \Delta P = \rho g h_0(p - \varepsilon \eta) = \frac{\Gamma}{R},
-\end{align}
-leaving us ultimately with the dynamic condition
-\begin{align}
- p-\varepsilon\eta= \varepsilon\left( \frac{\Gamma}{\rho g\lambda^2}
- \right) \left(\frac{\lambda^2}{\varepsilon h_0}\frac{1}{R}\right),
-\end{align}
-where $W_e = \frac{\Gamma}{\rho g h_0^2}$ is the \textbf{Weber number}. This
-dimensionless parameter can be considered as a measure of the fluid's inertia
-compered to its surface tension, which satisfies the relation
-\begin{align}
- \delta^2 W_e = \frac{\Gamma}{\rho g \lambda^2}.
-\end{align}
-\subsection{Scaling of Variables}
-Admits a simple observation of the governing equations in the last chapter we
-notice that $w$ and $p$ on the free surface $z = 1 + \varepsilon\eta$ are
-directly proportional to $\varepsilon$. Hence we want to ''scale this way``
-by introducing the following transformation
-\begin{align}
- p \rightarrow \varepsilon p, \quad w \rightarrow \varepsilon w, \quad
- \mathbf{u}_\perp \rightarrow \varepsilon \mathbf{u}_\perp.
-\end{align}
-Because of this scaling our material derivative changes slightly to
-\begin{align}\label{eq:mod-material}
- \frac{D}{Dt} = \frac{\partial }{\partial t} + \varepsilon\left(u
- \frac{\partial }{\partial x} + v \frac{\partial }{\partial y} + w
- \frac{\partial }{\partial z} \right)
-\end{align}
-A simple recalculation yields the rescaled, nondimensionalized Euler's
-Equation of motion are the same as in equations \ref{eq:nondim-motion} with
-the modified material derivative from \ref{eq:mod-material}, and the boundary
-conditions are
-\begin{align}
- p &= \eta - \frac{\delta^2\varepsilon h_0}{\lambda^2} \frac{W_e}{R}\\
- w &= \frac{1}{\varepsilon}\eta_t + (\mathbf{u}_\perp \nabla_\perp)\eta
- \quad \text{on}\;\; z = 1+\varepsilon\eta\\
- w &=\frac{1}{\varepsilon}b_t + (\mathbf{u}_\perp \nabla_\perp)b \quad
- \text{on}\;\; z=b
-\end{align}
-\subsection{Model Hierarchies}
-As we have derived a model of fluid dynamics, with small parameters
-$\varepsilon$ and $\delta$, we can conduct a series of classifications and
-perform asymptotic analysis on them. The main hierarchies important in this
-review are derived from the following problem classifications
-\begin{itemize}
- \item $\varepsilon\rightarrow 0$: linearized problem, small amplitude
- \item $\delta\rightarrow 0$: shallow Water, long-wave
- \item$\delta \rightarrow 0;\; \varepsilon~1$: shallow Water, large
- amplitude
- \item $\delta\ll 1;\; \varepsilon~\delta$: shallow water, medium
- amplitude
- \item $\delta\ll 1;\; \varepsilon~\delta^2$: shallow water, small
- amplitude
- \item $\delta \gg 1;\; \varepsilon\delta\ll 1$: deep water, small
- steepness.
-\end{itemize}
-
-\section{The Solitary Wave and The KdV Equation}
-The solitary wave is a wave of translation, it is stable and can travel long
-distances additionally the speed depends on the size of the wave. An
-interesting feature is that two solitary waves do not merge together to form
-one solitary wave, rather the small wave is overtaken by a larger one. If a
-solitary wave is too big for the depth it splits into two, a big and a small
-one. Solitary waves arise in the region $\varepsilon=O(\delta^2)$.
-
-
-\subsection{Solitary Wave}
-To describe
-a solitary wave we begin with Euler's Equation of Motion, where we assume
-there is no surface tension we set $W_e = 0$ and additionally assume
-irrotational flow $\mathbf{\omega}=\nabla \times \mathbf{u} = 0$. This means
-that there exists a velocity potential $\phi(\mathbf{x},t)$ given
-by$\mathbf{u} = \nabla \phi$ satisfying the Laplace equation. In regard of a
-solitary wave being a plane wave, we rotate our coordinate system such that
-the propagation is in the $x$-direction and a stationary \& fixed bottom
-$b=0$. Ultimately leaving us with the following model
-\begin{align}\label{eq:soliton}
-\begin{drcases}
- & \phi_{zz} + \delta \phi_{x x } = 0,\\
- &\text{with the boundary conditions}\\
- &\begin{drcases}
- &\phi_z = \delta^2 (\eta_t + \varepsilon \phi_x \eta_x) \\
- &\phi_t + \eta + \frac{1}{2}\varepsilon\left( \frac{1}{\delta^2}\phi^2_z
- + \phi_x^2\right) =0
- \end{drcases}\quad \text{on}\;\; z = 1+\varepsilon\eta,\\
- &\text{and}\\
- & \phi_z =0 \quad \text{on}\;\; z = b = 0.
-\end{drcases}
-\end{align}
-Since the model arises $\varepsilon = O(\delta^2)$, for convince we set
-$\varepsilon=1$. The fact of the matter is we are seeking a traveling wave
-solution, thereby we can go into the coordinate system of the traveling wave,
-one in the variable $\xi = x - ct$ for a from left to right traveling wave,
-where $c$ is the nondimensional speed of the wave. Our goal is to find the
-solution for the velocity potential $\phi(\xi, z)$ and the wave profile
-$\eta(\xi)$. The chain rule gives us
-\begin{align}
- \frac{\partial }{\partial x} &= \frac{\partial \xi}{\partial x}
- \frac{\partial }{\partial \xi} = \frac{\partial }{\partial \xi}, \\
- \frac{\partial }{\partial t} &= \frac{\partial \xi}{\partial t}
- \frac{\partial }{\partial \xi} = -c\frac{\partial }{\partial \xi}.
-\end{align}
-Together with the equations in \ref{eq:soliton} we obtain
-\begin{align}\label{eq:soliton-xi}
- \begin{drcases}
- & \phi_{zz} + \delta \phi_{\xi\xi} = 0,\\
- &\text{with the boundary conditions}\\
- &\begin{drcases}
- &\phi_z = \delta^2 (\phi_\xi -c)\eta_\xi \\
- &-c\phi_\xi + \eta + \frac{1}{2}\varepsilon\left( \frac{1}{\delta^2}\phi^2_z
- + \phi_\xi^2\right) =0
- \end{drcases}\quad \text{on}\;\; z = 1+\eta,\\
- &\text{and}\\
- & \phi_z =0 \quad \text{on}\;\; z = b = 0.
- \end{drcases}
-\end{align}
-\subsubsection{Exponential Decay}
-We would like to analyze if the equation in \ref{eq:soliton-xi} gives viable a
-solution that decays exponentially, we make the ansatz
-\begin{align}
- \eta \simeq a e^{-\alpha |\psi|},\quad \phi \simeq \psi(z)e^{-\alpha
- |\xi|}, \qquad \mid \xi \mid \rightarrow \infty,
-\end{align}
-where $\alpha>0$ is the exponent. The equations in \ref{eq:soliton-xi}
-transforms to
-\begin{align}
- \psi'' + \alpha^2 \delta^2\psi = 0.
-\end{align}
-The above equation is a standard well known ordinary differential equation
-reading
-\begin{align}
- \psi = A \cos(\alpha\delta z),
-\end{align}
-where $A$ is the integration constant. On the free surface $z\simeq 1$ gives
-\begin{align}
- &-cA\alpha\sin(\alpha\delta) = ca\alpha,\label{eq:sol1}\\
- &cA\alpha \cos(\alpha\delta) = -a \label{eq:sol2}.
-\end{align}
-Dividing equation \ref{eq:sol1} with equation \ref{eq:sol2} gives
-\begin{align} \label{eq:soliton-dispersion}
- c^2 = \frac{\tan\left(\alpha\delta \right) }{\alpha\delta}.
-\end{align}
-We conclude that the solution for such a wave exists provided that the
-dispersion relation on the wave propagation speed holds, thereby all solitary
-waves exhibit exponential decay in their tail and satisfy the dispersion
-relation in equation \ref{eq:soliton-dispersion}.
-\subsubsection{Asymptotic Analysis}
-The underlining equations in \ref{eq:soliton} extend from $-\infty$ to
-$\infty$, so the length scale is much greater than any finite depth of
-water. Therefore the classification $\delta \rightarrow 0$ is appropriate for
-a solitary wave, this however goes with the assumption
-$\varepsilon\rightarrow 0$ otherwise we cannot make an appropriate expansion.
-Let us look at the main equation
-\begin{align}\label{eq:sol-laplace}
- \phi_{zz} + \delta \phi_{x x} = 0.
-\end{align}
-For small $\delta$ we conduct the $\delta^2 = O(\varepsilon)$ standard ansatz
-in asymptotic analysis
-\begin{align}
- \phi_{\delta}(x, t, z) \simeq \sum_{n=0}^{\infty} \delta^{2n}\phi_n(x, t,
- z).
-\end{align}
-Substituting $\phi_\delta$ into equation \ref{eq:sol-laplace} we get
-\begin{align}
- \delta^{2\cdot 0}\left( \phi_{0zz} \right) + \delta^{2\cdot 1}\left(
- \phi_{1zz}+\phi_{0 x x} \right) + \delta^{2\cdot 2}\left( \phi_{2zz}+
- \phi_{1 x x} \right) + O(\delta^{2\cdot 3}) = 0.
-\end{align}
-We start off with $O(\delta^{2\cdot0}) $, which gives us an arbitrary function
-$\phi_{0} = \theta(x, t)$. Next we may generalize the results for all
-$O(\delta^{2\cdot n})$ in the means of
-\begin{align}
- \phi_{n+1zz} = -\phi_{nx x}\qquad \forall n\in \mathbb{N} .
-\end{align}
-Therefore leaving us for $\phi_1$ and $\phi_2$ with
-\begin{align}
- &\phi_1 = -\frac{1}{2} z^2 \theta_0(x,t) + \theta(x, t),\\
- \Rightarrow& \phi_2 =
- \frac{1}{24}z^4\theta_0(x,t)-\frac{1}{2}z^2\theta_1(x,t) + \theta_2(x,t).
-\end{align}
-The boundary condition on the bottom comes around to be
-\begin{align}
- \phi_{nz} =0 \quad \text{on}\;\; z=0.
-\end{align}
-The free surface boundary condition $z= 1+\varepsilon\eta$ n evolves more calculation, we consider
-only terms up the order of $\delta^2$, initializing with
-\begin{align}
- &\phi_z = \delta^2(\eta_t + \varepsilon\phi_x \eta_x)\\
- \Leftrightarrow &\frac{1}{\delta}\phi_z = \eta_t + \varepsilon\phi_x
- \eta_x,
-\end{align}
-substituting $\phi_\delta$ into the above proceeds to be
-\begin{align}
- \frac{1}{\delta^2}\underbrace{\phi_{0z}}_{=0} + \phi_{1z}+ \delta^2\phi_{zz}
- O(\delta^{2\cdot 2})
- &= -z\theta_{x x} + \delta^2\left( \frac{1}{6}z^3\theta_{0 x x x x} - z
- \theta_{0x x} \right) + O(\delta^{2\cdot 2})\\
- &=-(1+\varepsilon\eta)\theta_{0 x x} + \delta^2\left(
- \frac{1}{6}(1+\varepsilon\eta)^3\theta_{0 x x} -
-(1+\varepsilon\eta)\theta_{0 x x} \right) +O(\delta^{2\cdot 2})\\
- &= \eta_t + \varepsilon\eta_x \left( \theta_{0x}
- \delta^2(\theta_{1x}-\frac{1}{2}( 1+ \varepsilon\eta)^2 \theta_{0x x x}
-\right).
-\end{align}
-The second condition is
-\begin{align}
- \phi_t + \eta + \frac{1}{2}\varepsilon \left( \frac{1}{\delta}\phi^2_z
- \phi_x^2\right) = 0,
-\end{align}
-becomes after substitution
-\begin{align}
- \theta_{0t}+ \delta^2\left( -\frac{1}{2}(1+\varepsilon\eta)^2\theta_{0 x xt}
- + \theta_{1t}\right) + \eta + O(\delta^{2\cdot 2})
- &=-\frac{1}{2}\delta^2\varepsilon(-(1+\varepsilon\eta)\theta_{0 x x
- })^2\\
- &-\frac{1}{2}\left( \theta_{0 x} + \delta^2\left( \theta_{1x} -
- \frac{1}{2}(1+\varepsilon\eta)^2\theta_{0 x x x x} \right) \right) ^2
-\end{align}
-The simplest case is $\varepsilon,\delta \rightarrow 0$.
-
-
-
-
-
-
-
-
-
-
-
-
-\newpage
-\include{appendix.tex}
-
-
-
-\nocite{johnson_1997}
-\nocite{vallis_2017}
-\nocite{constantin_tsunami}
-\nocite{rupert_2009}
-\nocite{mathe-physik}
-
-\printbibliography
-
-\end{document}
diff --git a/app_pde/build/basics_fluids.pdf b/app_pde/build/basics_fluids.pdf
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@@ -0,0 +1,544 @@
+\section{Governing Equations of Fluid Dynamics}
+We first start of with a fluid with a density
+\begin{align}
+ \rho(\mathbf{x}, t),
+\end{align}
+in three dimensional Cartesian coordinates $\mathbf{x} = (x, y, z)$ at time
+$t$. For water-wave applications, we should note that we take
+$\rho=\text{constant}$, but we will go into this fact later. The fluid moves
+in time and space with a velocity field
+\begin{align}
+ \mathbf{u}(\mathbf{x}, t) = (u, v, w).
+\end{align}
+Additionally it is also described by its pressure
+\begin{align}
+ P(\mathbf{x}, t),
+\end{align}
+generally depending on time and position. When thinking of e.g. water the
+pressure increases the deeper we go, that is with decreasing or increasing $z$
+direction (depending how we set up our system $z$ pointing up or down
+respectively).
+
+The general assumption in fluid dynamics is the \textbf{Continuum
+Hypothesis}, which assumes continuity of $\textbf{u}, \rho$ and $P$ in
+$\mathbf{x}$ and $t$. In other words, we premise that the velocity field,
+density and pressure are ''nice enough`` functions of position and time, such
+that we can do all the differential operations we desire in the framework of
+differential analysis.
+\subsection{Mass Conservation}
+Our aim is to derive a model of the fluid and its dynamics, with respect to
+time and position, in the most general way. This is usually done thinking
+of the density of a given fluid, which is a unit mass per unit volume,
+intrinsically an integral representation to derive these equations suggests
+by itself.
+
+Let us now thing of an arbitrary fluid. Within this fluid we define a fixed
+volume $V$ relative to a chosen inertial frame and bound it by a surface $S$
+within the fluid, such that the fluid motion $\mathbf{u}(\mathbf{x}, t)$ may
+cross the surface $S$. The fluid density is given by $\rho(\mathbf{x}, t)$,
+thereby the mass of the fluid in the defined Volume $V$ is an integral
+expression
+\begin{align}
+ m = \int_V \rho(\mathbf{x}, t) dV.
+\end{align}
+The figure bellow \ref{fig:volume}, expresses the above described picture.
+\begin{figure}[H]
+ \centering
+ \begin{tikzpicture}[>=latex,scale=1, xscale=1, opacity=.8]
+% second sphere
+ \begin{scope}[rotate=10, xscale=3, yscale=2, shift={(2.3,-0.2)}]
+ \coordinate (O) at (0,0);
+ \shade[ball color=gray!10!] (0,0) coordinate(Hp) circle (1) ;
+
+ \draw[thick] (O) circle (1);
+ \draw[rotate=5] (O) ellipse (1cm and 0.66cm);
+ \draw[rotate=90] (O) ellipse (1cm and 0.33cm);
+\node[circle, fill=black, inner sep=1pt] at (0.15, 0.25) {} ; \draw[-latex, thick] (0.15, 0.25) -- (1, 1) ;
+ \node[right] at (1, 1) {$\mathbf{u}(\mathbf{x}, t)$};
+
+ \node[] at (O) {$V$};
+ \node[] at (0.55, -0.25) {$\rho(\mathbf{x}, t)$};
+
+ \draw[-] (0.76, -0.66) -- (1.2, -0.7);
+ \node[right] at (1.2, -0.7) {$S$};
+
+ \draw[-latex, thick] (-0.25, -0.65) -- (-1, -1);
+ \node[left] at (-1, -1) {$\mathbf{n}$};
+
+ \end{scope}
+
+% axis
+ \end{tikzpicture}
+ \caption{Volume bounded by a surface in a fluid with density and momentum,
+ with a surface normal vector $\mathbf{n}$ \label{fig:volume}}
+\end{figure}
+
+Since we want to figure out the fluid's dynamics, we can consider the rate
+of change in the completely arbitrary $V$. The rate of change of mass needs to
+disappear, i.e. it is equal to zero since we cannot lose mass. Matter (mass) is
+neither created nor destroyed anywhere in the fluid, leading us to
+\begin{align}
+ \frac{d}{dt}\left( \int_V \rho(\mathbf{x}, t)\ dV \right) = 0.
+\end{align}
+\textbf{NOT SURE HERE YET!!!!!!!!!!!, CHECK LEIBINZ FORMULA}
+To get more information we simply ''differentiate under the integral
+sign``, also known as the Leibniz Rule of Integration, see appendix
+\ref{appendix:leibniz}, the integral equation representing the rate of change
+of mass reads
+\begin{align}\label{eq:mass balance}
+ \frac{dm}{dt} = \int_V \frac{\partial \rho(\mathbf{x}, t)}{\partial t}\ dV
+ +\int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u}\cdot\mathbf{n}\ dS
+ = 0.
+\end{align}
+\textbf{----------------------}
+The above equation in \ref{eq:mass balance} is an underlying equation, describing that the rate of
+change of mass in V is brought about, only by the rate of mass flowing into
+V across S, and thus the mass does not change.
+
+For the second integral in \ref{eq:mass balance} we utilize the Gaussian
+integration law to acquire an integral over the volume
+\begin{align}
+ \int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u} \cdot \mathbf{n} \ dS =
+ \int_V \nabla (\rho \mathbf{u})\ dV.
+\end{align}
+Thereby we can put everything inside the volume integral
+\begin{align}
+ \frac{d m}{dt} = \int_V \left(\partial_t \rho + \nabla(\rho \mathbf{u}) \right) \ dV = 0.
+\end{align}
+Everything under the integral sign needs to be zero, thus we obtain
+the \textbf{Equation of Mass Conservation} or in the general sense also
+called the \textbf{Continuity Equation}
+\begin{align}\label{eq:continuity}
+ \partial_t \rho + \nabla(\rho \mathbf{u}) = 0
+\end{align}
+
+In light of the results of the equation of mass conservation
+in \ref{eq:continuity}, an product rule gives
+\begin{align}
+ \partial_t \rho + (\nabla \rho)\mathbf{u} + \rho(\nabla \mathbf{u}),
+\end{align}
+for notational purposes, we define the \textbf{material/convective derivative}
+as follows
+\begin{align}
+ \frac{D}{Dt} = \partial_t + \mathbf{u}\nabla.
+\end{align}
+With the material derivative the equation of mass conservation reads
+\begin{align}
+ \frac{D\rho}{Dt} + \rho \nabla\mathbf{u} = 0
+\end{align}
+We may undertake the first case separation, initiating $\rho = \text{cosnt.}$
+called \textbf{incompressible flow} causes the material derivative of $\rho$ to
+be zero, and thereby
+\begin{align}
+ \frac{D\rho}{Dt} = 0 \quad \Rightarrow \quad \nabla \mathbf{u} = 0,
+\end{align}
+following that the divergence of the velocity field is zero, in this case
+$\mathbf{u}$ is called \textbf{solenoidal}.
+\subsection{Euler's Equation of Motion}
+Additional consideration we undertake is the assumption of an
+\textbf{inviscid} fluid, that is we set viscosity to zero. Otherwise we would
+get a viscous contribution under the integral which results in the
+Navier-Stokes equation. In this regard we apply Newton's second law to our
+fluid in terms of infinitesimal pieces $\delta V$ of the fluid. The
+acceleration divides into two terms, a \textbf{body force} given by gravity
+of earth in the $z$ coordinate $\mathbf{F} = (0, 0, -g)$ and a
+\textbf{local/short-rage force} described by the stress tensor in the fluid.
+In the inviscid case we the local force retains the pressure $P$, producing a
+normal force, with respect to the surface, acting onto any infinitesimal
+element in the fluid. The integral formulation of the force would be
+\begin{align}
+ \int_V \rho \mathbf{F}\ dV - \int_S P\mathbf{n}\ dV.
+\end{align}
+Now applying the Gaussian rule of integration on the second integral over the
+surface, the resulting force in per unit volume is
+\begin{align}
+ \int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV.
+\end{align}
+The acceleration of the fluid particles is given by $\frac{D\mathbf{u}}{Dt}$,
+and thus the total force per unit volume on the other hand is
+\begin{align}
+ \int_V \rho \frac{D\mathbf{u}}{Dt}\ dV =
+ \int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV.
+\end{align}
+Newton's Second Law for a fluid in an Volume is essentially saying that the
+rate of change of momentum of the fluid in the fixed volume $V$, which is the particle
+acceleration is the resulting force acting on V together with the rate of
+flow of momentum across the surface $S$ into the volume $V$. Hence we arrive
+at the \textbf{Euler's Equation(s) of Motion}
+\begin{align}
+ \frac{D\mathbf{u}}{Dt} = \left(\frac{\partial \mathbf{u}}{\partial t}
+ (\mathbf{u}\nabla)\mathbf{u}\right) =
+ -\frac{1}{\rho}\nabla P + \mathbf{F}.
+\end{align}
+As a side note we have mentioned that there is another contribution if the
+fluid is viscid. Indeed there is a tangential force due to the velocity
+gradient, which into introduces the additional term
+\begin{align}
+ \mu \nabla^2 \mathbf{u}, \qquad
+ \mu = \text{viscosity of the Fluid}.
+\end{align}
+Thereby the equations become
+\begin{align}
+ \rho\frac{D\mathbf{u}}{Dt}
+ = -\nabla P + \rho \mathbf{F} + \mu \nabla^2 \mathbf{u}.
+\end{align}
+
+For now we have separated two simplifications, that define an
+\textbf{idealized/perfect fluid}
+\begin{enumerate}
+ \item \textbf{incompressible} $\qquad \mu=0$
+ \item \textbf{inviscid} $\quad \rho = \text{const.},\ \nabla \mathbf{u}=
+ 0$
+\end{enumerate}
+\subsection{Vorticity and irrotational Flow}
+The curl of the velocity field $\mathbf{\omega} = \nabla \times \mathbf{u}$
+of a fluid (i.e. the vorticity), describes a spinning motion of the fluid
+near a position $\mathbf{x}$ at time $t$. The vorticity is an important
+property of a fluid, flows or regions of flows where $\mathbf{\omega}=0$ are
+\textbf{irrotational}, and thus can be modeled and analyzed following well
+known routine methods. Even though real flows are rarely irrotational
+anywhere (!), in water wave theory wave problems, from the classical aspect
+of vorticity have a minor contribution. Hence we can assume irrotational flow
+modeling water waves. To arrive at the vorticity in the equations of motions
+derived in the last section we resort to a differential identity derived in appendix
+\ref{appendix:diff identity}, which gives for the material derivative
+\begin{align}
+ \frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t}
+ \nabla(\frac{1}{2}\mathbf{u}\mathbf{u)}
+ - \left( \mathbf{u}\times (\nabla \times \mathbf{u} \right).
+\end{align}
+Thus the equations of motion become
+\begin{align}
+ \frac{\partial \mathbf{u}}{\partial t} + \nabla\left(
+ \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega \right)
+ = \mathbf{u} \times \mathbf{\omega},
+\end{align}
+where $\Omega$ is the force potential per
+unite mass given by $\mathbf{F} = -\nabla \Omega$.
+
+At this point we may differentiate between \textbf{stead and unsteady flow}.
+For \textbf{Steady Flow} we assume that $\mathbf{u}, P$ and $\Omega$ are time
+independent, thus we get
+\begin{align}
+ \nabla\left( \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega
+ \right) = \mathbf{u} \times \mathbf{\omega}.
+\end{align}
+It is general knowledge that the gradient of a function $\nabla f$ is
+perpendicular the level sets of $f(\mathbf{x})$, where $f(\mathbf{x}) =
+\text{const.}$. Thus $\mathbf{u} \times \mathbf{\omega}$ is orthogonal to
+the surfaces where
+\begin{align} \label{eq:bernoulli}
+ \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega =
+ \text{const.},
+\end{align}
+The above equation is called \textbf{Bernoulli's Equation}.
+
+Secondly \textbf{Unsteady Flow} but irrotational (+ incompressible), first of
+all gives us the condition for the existence of a velocity potential $\phi$
+in the sense
+\begin{align}
+ \mathbf{\omega} = \nabla \times \mathbf{u} = 0 \quad \Rightarrow \quad
+ \mathbf{u} = \nabla \phi,
+\end{align}
+where $\phi$ needs to satisfy the Laplace equation
+\begin{align}
+ \Delta \phi = 0.
+\end{align}
+According to the Theorem of Schwartz we may exchange $\frac{\partial
+}{\partial t}$ and $\nabla$, giving us an expression for the material
+derivative
+\begin{align}
+ \nabla\left( \frac{\partial \phi}{\partial t} +\frac{1}{2}
+ \mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega \right) = 0
+\end{align}
+Thus the expression differentiated by the $\nabla$ operator is an arbitrary
+function $f(\mathbf{x}, t)$, writing
+\begin{align}
+ \frac{\partial \phi}{\partial t} +\frac{1}{2}
+ \mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega = f(\mathbf{x}, t).
+\end{align}
+The function $f(\mathbf{x}, t)$ can be removed by gauge transformation of
+$\phi \rightarrow \phi + \int f(\mathbf{x}, t)\ dt$, never the less this is
+not further discussed and left to the reader in the reference.
+\subsection{Boundary Conditions for water waves}
+The boundary conditions for water-wave problems vary, generally on the
+simplification we undertake. At the surface, called the free surface as in
+free from the velocity conditions, we have the atmospheric stress on the
+fluid. The stress component would again have a viscid component, this however
+is only relevant when modeling surface wind, in this review we model the
+fluid as unaffectedly and within reason as inviscid. The atmosphere employs
+only a pressure on the surface, this pressure is taken to be the atmospheric
+pressure, dependent on time and point in space. Thereby any surface tension
+effects can also include a scenario at a curved surface (e.g. wave), giving
+rise to the pressure difference across the surface. A more precise
+description would use Thermodynamics to derive boundary conditions coupling
+water surface and the air above it, yet the density component of air
+compared to that of water makes our ansatz viable. The described conditions
+are called the \textbf{dynamic conditions}
+
+An additional condition revolves around the fluid particles on the moving
+surface, called the \textbf{kinematic condition}. This condition bounds
+the vertical velocity component on the surface.
+
+The logical step now is to define boundary conditions on the bod of the
+fluid, i.e. the bottom. If the viscid case bottom is impermeable, we a no
+slip condition to all fluid particles $\mathbf{u}_\text{bottom}= 0$. If we
+assume that the fluid is inviscid then the bottom becomes a surface of the
+fluid in the sense that the fluid particles in contact with the bed move in
+the surface, we more or less mirror the kinematic condition of the surface.
+For many problems the condition is going to vary, in most cases the bottom
+will be rigid and fixed not necessarily horizontal. This condition is simply
+called the \textbf{bottom condition}.
+\subsubsection{Kinematic Condition}
+Obtaining the free surface is the primary objective in the theory of modeling
+water waves, represented by
+\begin{align}
+ z = h(\mathbf{x}_\perp, t),
+\end{align}
+where $\mathbf{x}_\perp = (x, y)$ in Cartesian, or $\mathbf{x}_\perp = (r,
+\theta)$ in cylindrical coordinates. A surfaces that moves with the fluid,
+always contains the same fluid particles, described as
+\begin{align}
+ \frac{D}{Dt}\left(z - h(\mathbf{x}_\perp, t \right) = 0.
+\end{align}
+Upon expanding the derivative we get
+\begin{align}
+ \frac{Dz}{Dt} - \frac{Dh}{Dt}
+ &= \frac{\partial z}{\partial t}+
+ (\mathbf{u}\nabla)z - \frac{\partial h}{\partial t} -(\mathbf{u}\nabla)\\
+ &= w - \left(h_t - (\mathbf{u}_\perp \nabla_\perp) h\right) = 0,
+\end{align}
+where the subscript $\perp$ describes the components with regard to
+$\mathbf{x}_\perp$. The \textbf{kinematic condition} reads
+\begin{align}
+ w = h_t - (\mathbf{u}_\perp \nabla_\perp) h \qquad \text{on}\;\;
+ z=h(\mathbf{u}_\perp, t).
+\end{align}
+
+\subsubsection{Dynamic Condition}
+As described in the prescript of this section, the case of an inviscid fluid,
+requires that only the pressure $P$ needs to be described on the free surface
+$z = h(\mathbf{x}_\perp, t)$. Assuming incompressible, irrotational,
+unsteady flow and setting $P=P_a$ for atmospheric pressure and $\Omega =
+g\cdot z$ for the force per unit mass potential the equations of motion are
+\begin{align}
+ \frac{\partial \phi}{\partial t} +\frac{1}{2}\mathbf{u}\mathbf{u}
+ + P_\frac{a}{\rho}+gh = f(t) \qquad \text{on}\;\; on z=h.
+\end{align}
+Somewhere $\|\mathbf{x}_\perp\| \rightarrow \infty$ the fluid reaches
+equilibrium and is thereby stationary, thereby has no motion and the pressure
+is $P=P_a$ and the surface is a constant $h = h_0$ $f(t)$ is
+\begin{align}
+ f(t) = \frac{P_a}{\rho}+gh_0.
+\end{align}
+The simplest description for the \textbf{dynamic condition} may be written as
+\begin{align}
+ \frac{\partial \phi}{\partial t}
+ +\frac{1}{2}\mathbf{u}\mathbf{u}+g(h-h_0) = 0 \qquad \text{on}\;\; z=h.
+\end{align}
+
+Regarding the pressure difference on a curved surface, we may expand the
+dynamic condition by introducing the pressure difference known as the
+\textbf{Young-Laplace Equation}
+\begin{align}
+ \Delta P = \frac{\Gamma}{R},
+\end{align}
+where $\Gamma>0$ is the coefficient of surface tension and $\frac{1}{R}$ is
+the curvature representing an implicit function, in our case the implicit
+function is $z - h(\mathbf{x}_\perp, t)$ for fixed time. The curvature in
+Cartesian coordinates takes the form
+\begin{align}
+ \frac{1}{R} = \frac{(1+h_y^2)h_{x x}+(1+h_y^2)h_{yy} -
+ 2h_xh_yh_{xy}}{\left( h_x^2+h_y^2+1 \right)^{\frac{3}{2}} },
+\end{align}
+the derivation is precisely described in \ref{appendix:curvature}
+
+
+
+\subsubsection{The Bottom Condition}
+The representation for the bottom is
+\begin{align}
+ z = b(\mathbf{x}_\perp, t),
+\end{align}
+where the fluid surface needs to satisfy
+\begin{align}
+ \frac{D}{Dt} \left(z - b(\mathbf{x}_\perp) \right) = 0.
+\end{align}
+Hence we arrive at the bottom boundary conditions
+\begin{align}
+ w = b_t + (\mathbf{u}_\perp \nabla_\perp)b \qquad \text{on}\;\; z=b ,
+\end{align}
+where $b(\mathbf{x}_\perp, t)$ is already known for most water wave
+problems. If we consider a stationary bottom then the time derivative
+vanishes, leaving us with the following condition
+\begin{align}
+ w = (\mathbf{u}_\perp \nabla_\perp)b \qquad \text{on}\;\; z=b
+\end{align}
+
+
+\subsubsection{Integrated Mass Condition}
+In this section we want to combine the kinematics of both the free and the
+bottom surface with the mass conservation equation on the perpendicular
+components
+\begin{align}
+ \nabla \mathbf{u} = \nabla_\perp \mathbf{u}_\perp + w_z = 0 .
+\end{align}
+Integrating the above expression from bottom to surface, i.e. from
+$z=b(\mathbf{x}_\perp,t)$ to $z = h (\mathbf{x},t)$ gives
+\begin{align}
+ \int_b^h \nabla_\perp \mathbf{u}_\perp\ dz + w\bigg|_{z=b}^{z=h} = 0,
+\end{align}
+where we insert the conditions on the free surface and on the bottom surface
+\begin{align}
+ w &= h_t + (\mathbf{u}_{\perp \text{s}} \nabla_\perp) h \quad
+ \text{on}\;\; z = h\\
+ w &= b_t + (\mathbf{u}_{\perp \text{b}} \nabla_\perp) h \quad
+ \text{on}\;\; z =b,
+\end{align}
+with the subscript $s$ and $b$ indicating the evaluation of a quantity
+on the free surface and the bottom surface respectively. Inserting the
+boundary conditions we get
+\begin{align}
+ \int_b^h \nabla_\perp \mathbf{u}_\perp
+ + h_t + (\mathbf{u}_{\perp \text{s}} \nabla_\perp) h
+ - b_t - (\mathbf{u}_{\perp \text{b}} \nabla_\perp) b= 0.
+\end{align}
+To simplify the equation we resort again to the Leibniz Rule of Integration
+\begin{align}
+ \int_b^h \nabla_\perp\mathbf{u}_\perp =
+ \nabla_\perp \int_b^h \mathbf{u}_\perp\ dz - (\mathbf{u}_{\perp \text{s}}
+ \nabla_\perp)h - (\mathbf{u}_{\perp \text{b}})b.
+\end{align}
+As a consequence the \textbf{Integrated Mass Condition} is given by
+\begin{align}
+ \nabla_\perp \int_b^h \mathbf{u}_\perp\ dz + \underbrace{h_t -
+ b_t}_{=d_t} = 0.
+\end{align}
+\subsection{Energy Equation}
+To derive the energy equation we start off with Euler's Equation of Motion
+\begin{align}
+ \mathbf{u} _t + \nabla
+ (\frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}+\Omega) = \mathbf{u}\times
+ \mathbf{w},
+\end{align}
+multiplying the equation with $\mathbf{u}$ we get
+\begin{align}
+ &\mathbf{u}\mathbf{u} _t \label{eq:energy1} \\
+ &+(\mathbf{u}\nabla)(\frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}+\Omega)\label{eq:energy2}\\
+ &= \mathbf{u}(\mathbf{u}\times
+ \mathbf{w})\label{eq:energy3}.
+\end{align}
+The first equation given in \ref{eq:energy1} can we rewritten using inverse
+product rule of differentiation
+\begin{align}
+ \mathbf{u}\frac{\partial \mathbf{u}}{\partial t}
+ &= \frac{\partial
+ }{\partial t} (\mathbf{u}\mathbf{u}) - \frac{\partial \mathbf{u}}{\partial t}
+ \mathbf{u} \\
+ &= \frac{\partial
+ }{\partial t} (\mathbf{u}\mathbf{u}) - \mathbf{u}\frac{\partial
+ \mathbf{u}}{\partial t}\\
+ \Rightarrow\quad & \mathbf{u} \frac{\partial \mathbf{u}}{\partial t} =
+ \frac{1}{2}\frac{\partial }{\partial t} (\mathbf{u}\mathbf{u}).
+\end{align}
+Then we may add
+\begin{align}
+ \left(\frac{1}{2} \mathbf{u}\mathbf{u}+\frac{P}{\rho} +\Omega \right)
+ \underbrace{(\nabla u)}_{=0} = 0,
+\end{align}
+to above not changing anything. Thereby getting
+\begin{align}
+ \frac{\partial }{\partial t} (\frac{1}{2}\mathbf{u}\mathbf{u})
+ +(\mathbf{u}\nabla \mathbf{u})\left(
+ \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho} \right)
+ +\left( \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho} + \Omega \right)
+ (\nabla \mathbf{u}) = 0.
+\end{align}
+Applying the product rule we can simplify
+\begin{align}
+ \frac{\partial }{\partial t} \left(\frac{1}{2}\mathbf{u}\mathbf{u}\right)
+ +\nabla \left(\mathbf{u}\left(\mathbf{u}(
+ \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}\right) \right) = 0,
+\end{align}
+additionally adding $\frac{\partial \Omega}{\partial t} =0$ leads us to
+\begin{align}
+ \underbrace{\frac{\partial }{\partial t} \left(\frac{1}{2}\mathbf{u}\mathbf{u}
+ +\Omega\right)}_{\text{change of total energy density}}
+ +\underbrace{\nabla \left(\mathbf{u}\left(\mathbf{u}(
+ \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}\right)
+\right)}_{\text{energy flow of the velocity field}} = 0.\label{eq:energy}
+\end{align}
+This is called the \textbf{energy equation} and is a general result for a
+inviscid and incompressible fluids, which we can apply to study water waves.
+We start off with replacing $\nabla = \nabla_\perp + \frac{\partial }{\partial
+z} $ and $\Omega = g z$ and multiplying by $\rho$, then our energy equation
+in \ref{eq:energy} becomes
+\begin{align}
+ \frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
+ g z\right) + \nabla_\perp\left( \mathbf{u}_\perp\left(
+ \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right) \right)
+ \frac{\partial}{\partial z} \left( w\left(
+ \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho g z \right) \right) = 0.
+\end{align}
+Integrating from bottom to top, i.e. from bed to free surface gets us to
+\begin{align}
+ &\int_b^h\frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
+ g z\right)\ dz \label{eq:e-int1}\\
+ &+ \int_b^h \nabla_\perp\left( \mathbf{u}_\perp\left(
+ \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right) \right)\
+ dz\label{eq:e-int2}\\
+ &+ \left(\frac{\partial}{\partial z} \left( w\left(
+ \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho g z \right)
+\right)\right)\Bigg|_b^h \label{eq:e-int3}
+ = 0.
+\end{align}
+For equation \ref{eq:e-int1} we use Leibniz Rule of Integration, leaving us
+with
+\begin{align}
+ \int_b^h\frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
+ g z\right)\ dz
+ &= \frac{\partial }{\partial t} \int_b^h
+ \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho gz \ dz\\
+ &+ \left( \frac{1}{2}\rho \mathbf{u}_s \mathbf{u}_s + \rho g h \right)
+ h_t\\
+ &- \left( \frac{1}{2}\rho \mathbf{u}_b \mathbf{u}_b + \rho g b \right)
+ b_t
+\end{align}
+For equation \ref{eq:e-int2} we again take note of the Leibniz Rule of
+Integration, getting
+\begin{align}
+ \int_b^h \nabla_\perp\left( \mathbf{u}_\perp\left(
+ \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right) \right)\
+ dz
+ &= \nabla_\perp \int_b^h \mathbf{u}_\perp\left(
+ \frac{1}{2}\rho\mathbf{u}\mathbf{u} + P + \rho g z \right) \ dz\\
+ &- \left( \frac{1}{2}\rho \mathbf{u}_s\mathbf{u}_s + P + \rho g h \right)
+ \left( \mathbf{u}_{\perp s} \nabla_\perp \right) h\\
+ &+\left( \frac{1}{2}\rho \mathbf{u}_b\mathbf{u}_b + P + \rho g b \right)
+ \left( \mathbf{u}_{\perp b} \nabla_\perp \right) b
+\end{align}
+Thereby transforming our equation into
+\begin{align}
+ \frac{\partial }{\partial t} \underbrace{\int_b^h \frac{1}{2}\rho
+ \mathbf{u}\mathbf{u}+\rho g z\ dz}_{=:\mathcal{E}}
+ + \nabla_\perp&\underbrace{\int_b^h
+ \mathbf{u}_\perp\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho g z
+\right)\ dz}_{:=\mathcal{F}}
++ \underbrace{P_s h_t - P_b b_t}_{:=\mathcal{P}} = 0\\
+\nonumber\\
+ &\frac{\partial \mathcal{E}}{\partial t}
+ + \nabla_\perp \mathcal{F} + \mathcal{P} = 0,
+\end{align}
+where $\mathcal{E}$ represents the energy in the flow per unit horizontal
+area, since we are integrating from bed to free surface. Where $\mathcal{F}$
+is the horizontal energy flux vector and lastly $\mathcal{P} = P_s h_t -
+P_b b_t$ is the net energy input due to the pressure forces doing work on the
+upper and lower boundaries, i.e. bottom and free surface of the fluid.
+Assuming stationary rigid bottom condition and constant surface pressure, we
+can set $P_s=0$, such that $\mathcal{P} =0$ leaving us with the equation
+\begin{align}
+ \frac{\partial \mathcal{E}}{\partial t}
+ + \nabla_\perp \mathcal{F} = 0.
+\end{align}
+We note that the assumption $P_s=0$ is only possible if the coefficient of
+surface tension is set to 0, which usually is not the case.
diff --git a/app_pde/chap2.tex b/app_pde/chap2.tex
@@ -0,0 +1,215 @@
+\section{Dimensional Analysis}
+Our derived model of fluid dynamics yields formal connections between
+physical quantities. These quantities bear units, e.g. the velocity of fluid
+particles $\mathbf{u}$ has the ``SI'' unites of $\frac{m}{s}$, meters per
+second. The idea is the make use of these scales and formulate a model, where
+the quantities are nondimensionalized, i.e. to get rid of physical units by
+scaling each quantity appropriately. The appropriate length scales are that
+of the typical water depth $h_0$ and the typical wavelength $\lambda$ of a
+surface wave.
+
+\subsection{Nondimensionalisation}
+In summary we use these adaptations
+\begin{itemize}
+ \item $h_0$ for the typical water depth
+ \item $\lambda$ for the typical wavelength
+ \item $\frac{\lambda}{\sqrt{g h_0}}$ time scale of wave propagation
+ \item $\sqrt{g h_0}$ velocity scale of waves in $(x, y)$
+ \item $\frac{h_0 \sqrt{g h_0} }{\lambda}$ velocity scale in the $z$
+ direction.
+\end{itemize}
+$(x, z, t)$, then
+\begin{align}
+ u = \psi _z, \qquad w = - \psi_x;
+\end{align}
+and the scale of $\psi$ must be $h_0\sqrt{g h_0}$. Additionally we write the
+boundary condition on the free surface as follows
+\begin{align}
+ h = h_0 + a \eta (\mathbf{x}_\perp, t) = z,
+\end{align}
+where $a$ is the typical amplitude and $\eta$ nondimensional function. All in
+all we have the following scaling for the physical quantities of our context
+\begin{align}
+ &x \rightarrow\ \lambda x, \quad u \rightarrow \sqrt{gh_0} u, \\
+ &y \rightarrow\ \lambda y, \quad v \rightarrow \sqrt{gh_0} v, \qquad
+ t\rightarrow \frac{\lambda}{\sqrt{gh_0}}t,\\
+ &z \rightarrow\ h_0 z, \quad w \rightarrow
+ \frac{h_0\sqrt{gh_0}}{\lambda} w.
+\end{align}
+with
+\begin{align}
+ h = h_0 + a \eta, \qquad b \rightarrow h_0 b.
+\end{align}
+The pressure is also rewritten into
+\begin{align}
+ P = P_a + \rho g(h_0 -z) + \rho g h_0 p,
+\end{align}
+where $P_a$ is the atmospheric pressure, the term $h_0-z$ represent the
+hydrostatic pressure distribution, i.e. pressure at depth and the term with the pressure
+variable $p$ measures the deviation from the hydrostatic pressure
+distribution. Indeed $p\neq 0 $ for wave propagation. Now we can perform a
+rescaling of the Euler's Equation of Motion, we introduce the notation
+\begin{align}
+ &t = \frac{\lambda}{\sqrt{gh_0}}\tau,\quad x = \lambda \xi,\quad u =
+ \sqrt{gh_0} \tilde{u}\\
+ &y = \lambda \chi,\quad v = \sqrt{gh_0} \tilde{v}\\
+ &z = h_0 \zeta, \quad w = \frac{h_0\sqrt{gh_0} }{\lambda}\tilde{w}.
+\end{align}
+We start off with the $x$ coordinate, substitute and apply the chain rule
+leading us to
+\begin{align}
+ \frac{Du}{Dt}
+ &= \frac{\partial u}{\partial t} +u \frac{\partial
+ u}{\partial x} \\
+ &= \sqrt{gh_{0}}\frac{\partial \tilde{u}}{\partial \tau} \frac{\partial
+ \tau}{\partial t} +gh_0 \tilde{u} \frac{\partial \tilde{u}}{\partial \xi}
+ \frac{\partial \xi}{\partial x} \\
+ &= \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau}
+ \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right),
+\end{align}
+on the other hand
+\begin{align}
+ \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau}
+ \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right)
+ &=-\frac{1}{\rho}\frac{1}{\lambda}\frac{\partial P}{\partial x} \\
+ &=-\frac{ g h_0 }{\lambda}\rho \frac{\partial p}{\partial \xi}.
+\end{align}
+Thereby the rescaling evolves to
+\begin{align}
+ \frac{D \tilde{u}}{D\tau} = -\frac{\partial p}{\partial \xi}.
+\end{align}
+Because of the same scaling in $y$ we get the same result as in $x$, that is
+\begin{align}
+ \frac{D \tilde{v}}{D\tau} = -\frac{\partial p}{\partial \chi}.
+\end{align}
+In the $z$ coordinate we have
+\begin{align}
+ \frac{Dw}{Dt}
+ &= \frac{\partial w}{\partial t} +w \frac{\partial
+ w}{\partial \zeta} \\
+ &= \frac{h_0\sqrt{gh_0}}{\lambda} \frac{\sqrt{gh_0}}{\lambda}
+ \frac{\partial \tilde{w}}{\partial \tau} + \frac{1}{h_0}
+ \frac{h_0\sqrt{gh_0} }{\lambda} \frac{h_0\sqrt{gh_0}}{\lambda}
+ \tilde{w}\frac{\partial \tilde{v}}{\partial \zeta}\\
+ &= \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
+ + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right) .
+\end{align}
+On the other side we have
+\begin{align}
+ \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
+ + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right)
+ &=
+ -\frac{1}{h_0\rho} \frac{\partial P}{\partial z} +g \\
+ &=-\frac{1}{h_0\rho}(-\rho gh_0 \frac{\partial \zeta}{\partial \zeta}
+ \rho gh_0
+ \frac{\partial p}{\partial \zeta} ) + g \\
+ &= -g \frac{\partial p}{\partial z}.
+\end{align}
+In total for the $z$ direction we get
+\begin{align}
+ \underbrace{\left( \frac{h_0}{\lambda} \right)^2}_{=: \delta^2}
+ \frac{Dw}{Dt} = -\frac{\partial p}{\partial z},
+\end{align}
+where $\delta$ is the \textbf{long wavelength} or \textbf{shallowness}
+parameter, a very important constant for developing model hierarchies. For
+clarity we resubstitute for $x, y, z, t, u, v$ and $w$, and for completeness
+the we display the equations again, which are
+\begin{align}\label{eq:nondim-motion}
+ \frac{Du}{Dt} = - \frac{\partial p}{\partial x}&, \quad
+ \frac{Dv}{Dt} = - \frac{\partial p}{\partial y}, \quad
+ \delta^2\frac{Dw}{Dt} = - \frac{\partial p}{\partial z}, \\
+ &\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}
+ +\frac{\partial w}{\partial z} = 0.
+\end{align}
+We can now turn our attention to the boundary conditions, on both free
+surface $z=h$ and the bottom $z=b$ we have $z \Rightarrow h_0 z$ and thereby
+\begin{align}
+ z = 1+
+ \underbrace{\frac{a}{h_0}}_{:=\varepsilon}\eta(\mathbf{x}_\perp,t) \quad
+ \text{and}\quad z= b,
+\end{align}
+where we arrive at our second very important parameter $\varepsilon$ called
+the \textbf{amplitude} parameter. As for the kinematic condition, we
+substitute the free surface $z=h = 1+\varepsilon \eta$ and get
+\begin{align}
+ \frac{Dz}{Dt} = \varepsilon\left(\eta_t + (\mathbf{u}_\perp
+ \nabla_\perp)\eta\right) \qquad \text{on}\;\; z= 1+\varepsilon \eta.
+\end{align}
+Respectively the bottom condition is not changed
+\begin{align}
+ w = b_t + (\mathbf{u}_\perp \nabla_\perp) b \quad \text{on}\;\; z= b.
+\end{align}
+The general dynamic condition for $h = h(x, y, t)$ yields a rescaling of the
+curvature in terms of
+\begin{align}
+ \frac{1}{R}
+ &= \frac{(1+h_y^2)h_{x x} + (1+h_x^2)h_yy - 2h_xh_yh_{xy}
+ }{\left(h_x^2+h_y^2 +1 \right)^{\frac{3}{2}} } \\
+ &= -\frac{\varepsilon h_0}{\lambda^2} \frac{(
+ 1+\varepsilon^2\delta^2\eta_y^2 )\eta_{x x}+
+ (1+\varepsilon^2\delta^2\eta_x^2)\eta_{yy} -
+ 2\varepsilon^2\delta^2\eta_x\eta_y\eta_{xy}}{\left(
+ 1+\varepsilon^2\delta^2\eta_x^2+\varepsilon^2\delta^2\eta_y^2
+ \right)^{\frac{3}{2}} },
+\end{align}
+together with the pressure difference
+\begin{align}
+ \Delta P = \rho g h_0(p - \varepsilon \eta) = \frac{\Gamma}{R},
+\end{align}
+leaving us ultimately with the dynamic condition
+\begin{align}
+ p-\varepsilon\eta= \varepsilon\left( \frac{\Gamma}{\rho g\lambda^2}
+ \right) \left(\frac{\lambda^2}{\varepsilon h_0}\frac{1}{R}\right),
+\end{align}
+where $W_e = \frac{\Gamma}{\rho g h_0^2}$ is the \textbf{Weber number}. This
+dimensionless parameter can be considered as a measure of the fluid's inertia
+compered to its surface tension, which satisfies the relation
+\begin{align}
+ \delta^2 W_e = \frac{\Gamma}{\rho g \lambda^2}.
+\end{align}
+\subsection{Scaling of Variables}
+Admits a simple observation of the governing equations in the last chapter we
+notice that $w$ and $p$ on the free surface $z = 1 + \varepsilon\eta$ are
+directly proportional to $\varepsilon$. Hence we want to ''scale this way``
+by introducing the following transformation
+\begin{align}
+ p \rightarrow \varepsilon p, \quad w \rightarrow \varepsilon w, \quad
+ \mathbf{u}_\perp \rightarrow \varepsilon \mathbf{u}_\perp.
+\end{align}
+Because of this scaling our material derivative changes slightly to
+\begin{align}\label{eq:mod-material}
+ \frac{D}{Dt} = \frac{\partial }{\partial t} + \varepsilon\left(u
+ \frac{\partial }{\partial x} + v \frac{\partial }{\partial y} + w
+ \frac{\partial }{\partial z} \right)
+\end{align}
+A simple recalculation yields the rescaled, nondimensionalized Euler's
+Equation of motion are the same as in equations \ref{eq:nondim-motion} with
+the modified material derivative from \ref{eq:mod-material}, and the boundary
+conditions are
+\begin{align}
+ p &= \eta - \frac{\delta^2\varepsilon h_0}{\lambda^2} \frac{W_e}{R}\\
+ w &= \frac{1}{\varepsilon}\eta_t + (\mathbf{u}_\perp \nabla_\perp)\eta
+ \quad \text{on}\;\; z = 1+\varepsilon\eta\\
+ w &=\frac{1}{\varepsilon}b_t + (\mathbf{u}_\perp \nabla_\perp)b \quad
+ \text{on}\;\; z=b
+\end{align}
+\subsection{Model Hierarchies}
+As we have derived a model of fluid dynamics, with small parameters
+$\varepsilon$ and $\delta$, we can conduct a series of classifications and
+perform asymptotic analysis on them. The main hierarchies important in this
+review are derived from the following problem classifications
+\begin{itemize}
+ \item $\varepsilon\rightarrow 0$: linearized problem, small amplitude
+ \item $\delta\rightarrow 0$: shallow Water, long-wave
+ \item$\delta \rightarrow 0;\; \varepsilon~1$: shallow Water, large
+ amplitude
+ \item $\delta\ll 1;\; \varepsilon~\delta$: shallow water, medium
+ amplitude
+ \item $\delta\ll 1;\; \varepsilon~\delta^2$: shallow water, small
+ amplitude
+ \item $\delta \gg 1;\; \varepsilon\delta\ll 1$: deep water, small
+ steepness.
+\end{itemize}
+
+
+
diff --git a/app_pde/chap3.tex b/app_pde/chap3.tex
@@ -0,0 +1,508 @@
+\section{The Solitary Wave and The KdV Equation}
+The solitary wave is a wave of translation, it is stable and can travel long
+distances additionally the speed depends on the size of the wave. An
+interesting feature is that two solitary waves do not merge together to form
+one solitary wave, rather the small wave is overtaken by a larger one. If a
+solitary wave is too big for the depth it splits into two, a big and a small
+one. Solitary waves arise in the region $\varepsilon=O(\delta^2)$.
+
+
+\subsection{Solitary Wave}
+To describe
+a solitary wave we begin with Euler's Equation of Motion, where we assume
+there is no surface tension we set $W_e = 0$ and additionally assume
+irrotational flow $\mathbf{\omega}=\nabla \times \mathbf{u} = 0$. This means
+that there exists a velocity potential $\phi(\mathbf{x},t)$ given
+by$\mathbf{u} = \nabla \phi$ satisfying the Laplace equation. In regard of a
+solitary wave being a plane wave, we rotate our coordinate system such that
+the propagation is in the $x$-direction and a stationary \& fixed bottom
+$b=0$. Ultimately leaving us with the following model
+\begin{align}\label{eq:soliton}
+\begin{drcases}
+ & \phi_{zz} + \delta \phi_{x x } = 0,\\
+ &\text{with the boundary conditions}\\
+ &\begin{drcases}
+ &\phi_z = \delta^2 (\eta_t + \varepsilon \phi_x \eta_x) \\
+ &\phi_t + \eta + \frac{1}{2}\varepsilon\left( \frac{1}{\delta^2}\phi^2_z
+ + \phi_x^2\right) =0
+ \end{drcases}\quad \text{on}\;\; z = 1+\varepsilon\eta,\\
+ &\text{and}\\
+ & \phi_z =0 \quad \text{on}\;\; z = b = 0.
+\end{drcases}
+\end{align}
+Since the model arises $\varepsilon = O(\delta^2)$, for convince we set
+$\varepsilon=1$. The fact of the matter is we are seeking a traveling wave
+solution, thereby we can go into the coordinate system of the traveling wave,
+one in the variable $\xi = x - ct$ for a from left to right traveling wave,
+where $c$ is the nondimensional speed of the wave. Our goal is to find the
+solution for the velocity potential $\phi(\xi, z)$ and the wave profile
+$\eta(\xi)$. The chain rule gives us
+\begin{align}
+ \frac{\partial }{\partial x} &= \frac{\partial \xi}{\partial x}
+ \frac{\partial }{\partial \xi} = \frac{\partial }{\partial \xi}, \\
+ \frac{\partial }{\partial t} &= \frac{\partial \xi}{\partial t}
+ \frac{\partial }{\partial \xi} = -c\frac{\partial }{\partial \xi}.
+\end{align}
+Together with the equations in \ref{eq:soliton} we obtain
+\begin{align}\label{eq:soliton-xi}
+ \begin{drcases}
+ & \phi_{zz} + \delta \phi_{\xi\xi} = 0,\\
+ &\text{with the boundary conditions}\\
+ &\begin{drcases}
+ &\phi_z = \delta^2 (\phi_\xi -c)\eta_\xi \\
+ &-c\phi_\xi + \eta + \frac{1}{2}\varepsilon\left( \frac{1}{\delta^2}\phi^2_z
+ + \phi_\xi^2\right) =0
+ \end{drcases}\quad \text{on}\;\; z = 1+\eta,\\
+ &\text{and}\\
+ & \phi_z =0 \quad \text{on}\;\; z = b = 0.
+ \end{drcases}
+\end{align}
+\subsubsection{Exponential Decay}
+We would like to analyze if the equation in \ref{eq:soliton-xi} gives viable a
+solution that decays exponentially, we make the ansatz
+\begin{align}
+ \eta \simeq a e^{-\alpha |\psi|},\quad \phi \simeq \psi(z)e^{-\alpha
+ |\xi|}, \qquad \mid \xi \mid \rightarrow \infty,
+\end{align}
+where $\alpha>0$ is the exponent. The equations in \ref{eq:soliton-xi}
+transforms to
+\begin{align}
+ \psi'' + \alpha^2 \delta^2\psi = 0.
+\end{align}
+The above equation is a standard well known ordinary differential equation
+reading
+\begin{align}
+ \psi = A \cos(\alpha\delta z),
+\end{align}
+where $A$ is the integration constant. On the free surface $z\simeq 1$ gives
+\begin{align}
+ &-cA\alpha\sin(\alpha\delta) = ca\alpha,\label{eq:sol1}\\
+ &cA\alpha \cos(\alpha\delta) = -a \label{eq:sol2}.
+\end{align}
+Dividing equation \ref{eq:sol1} with equation \ref{eq:sol2} gives
+\begin{align} \label{eq:soliton-dispersion}
+ c^2 = \frac{\tan\left(\alpha\delta \right) }{\alpha\delta}.
+\end{align}
+We conclude that the solution for such a wave exists provided that the
+dispersion relation on the wave propagation speed holds, thereby all solitary
+waves exhibit exponential decay in their tail and satisfy the dispersion
+relation in equation \ref{eq:soliton-dispersion}.
+\subsubsection{Asymptotic Analysis}
+The underlining equations in \ref{eq:soliton} extend from $-\infty$ to
+$\infty$, so the length scale is much greater than any finite depth of
+water. Therefore the classification $\delta \rightarrow 0$ is appropriate for
+a solitary wave, this however goes with the assumption
+$\varepsilon\rightarrow 0$ otherwise we cannot make an appropriate expansion.
+Let us look at the main equation
+\begin{align}\label{eq:sol-laplace}
+ \phi_{zz} + \delta \phi_{x x} = 0.
+\end{align}
+For small $\delta$ we conduct the $\delta^2 = O(\varepsilon)$ standard ansatz
+in asymptotic analysis
+\begin{align}
+ \phi_{\delta}(x, t, z) \simeq \sum_{n=0}^{\infty} \delta^{2n}\phi_n(x, t,
+ z).
+\end{align}
+Substituting $\phi_\delta$ into equation \ref{eq:sol-laplace} we get
+\begin{align}
+ \delta^{2\cdot 0}\left( \phi_{0zz} \right) + \delta^{2\cdot 1}\left(
+ \phi_{1zz}+\phi_{0 x x} \right) + \delta^{2\cdot 2}\left( \phi_{2zz}+
+ \phi_{1 x x} \right) + O(\delta^{2\cdot 3}) = 0.
+\end{align}
+We start off with $O(\delta^{2\cdot0}) $, which gives us an arbitrary function
+$\phi_{0} = \theta(x, t)$. Next we may generalize the results for all
+$O(\delta^{2\cdot n})$ in the means of
+\begin{align}
+ \phi_{n+1zz} = -\phi_{nx x}\qquad \forall n\in \mathbb{N} .
+\end{align}
+Therefore leaving us for $\phi_1$ and $\phi_2$ with
+\begin{align}
+ &\phi_1 = -\frac{1}{2} z^2 \theta_0(x,t) + \theta(x, t),\\
+ \Rightarrow& \phi_2 =
+ \frac{1}{24}z^4\theta_0(x,t)-\frac{1}{2}z^2\theta_1(x,t) + \theta_2(x,t).
+\end{align}
+The boundary condition on the bottom comes around to be
+\begin{align}
+ \phi_{nz} =0 \quad \text{on}\;\; z=0.
+\end{align}
+The free surface boundary condition $z= 1+\varepsilon\eta$ n evolves more calculation, we consider
+only terms up the order of $\delta^2$, initializing with
+\begin{align}
+ &\phi_z = \delta^2(\eta_t + \varepsilon\phi_x \eta_x)\\
+ \Leftrightarrow &\frac{1}{\delta}\phi_z = \eta_t + \varepsilon\phi_x
+ \eta_x,
+\end{align}
+substituting $\phi_\delta$ into the above proceeds to be
+\begin{align}
+ \frac{1}{\delta^2}\underbrace{\phi_{0z}}_{=0} + \phi_{1z}+ \delta^2\phi_{zz}
+ O(\delta^{2\cdot 2})
+ &= -z\theta_{x x} + \delta^2\left( \frac{1}{6}z^3\theta_{0 x x x x} - z
+ \theta_{0x x} \right) + O(\delta^{2\cdot 2})\\
+ &=-(1+\varepsilon\eta)\theta_{0 x x} + \delta^2\left(
+ \frac{1}{6}(1+\varepsilon\eta)^3\theta_{0 x x} -
+(1+\varepsilon\eta)\theta_{0 x x} \right) \label{eq:soliton-scale-boundary1}\\
+ &= \eta_t + \varepsilon\eta_x \left(
+ \theta_{0x}
+ \delta^2(\theta_{1x}-\frac{1}{2}( 1+ \varepsilon\eta)^2 \theta_{0x x
+ x}\label{eq:soliton-scale-boundary2}
+\right).
+\end{align}
+The second condition is
+\begin{align}
+ \phi_t + \eta + \frac{1}{2}\varepsilon \left( \frac{1}{\delta}\phi^2_z
+ \phi_x^2\right) = 0,
+\end{align}
+becomes after substitution
+\begin{align}
+ &\theta_{0t}+ \delta^2\left( -\frac{1}{2}(1+\varepsilon\eta)^2\theta_{0 x xt}
+ + \theta_{1t}\right) + \eta + O(\delta^{2\cdot 2})
+\label{eq:soliton-scale-boundary3}
+ \\&=-\frac{1}{2}\delta^2\varepsilon(-(1+\varepsilon\eta)\theta_{0 x x
+ })^2\label{eq:soliton-scale-boundary4}
+ -\frac{1}{2}\left( \theta_{0 x} + \delta^2\left( \theta_{1x} -
+ \frac{1}{2}(1+\varepsilon\eta)^2\theta_{0 x x x x} \right) \right) ^2
+\end{align}
+In the order of $O(\delta^{0})$ as $\varepsilon \rightarrow 0$ gives us the conditions
+\begin{align}
+ -\theta_{0 x x} &= \eta_t \simeq \text{and}\quad
+ \theta_{0t}\simeq-\eta\label{eq:solitonO0}\\
+ &\Rightarrow \theta_{0 x x} - \theta_{0 t t} \simeq 0.
+\end{align}
+This gives us the wave equation, a simple solution in the frame of the right
+moving wave dependent on $\xi = x -t$ the chain rule gives us
+\begin{align}
+ \frac{\partial \theta_0(\xi(x, t))}{\partial t}
+ &= \frac{\partial
+ \theta_0}{\partial \xi} \underbrace{\frac{\partial \xi}{\partial t}}_{=-1}
+ + \frac{\partial
+ \theta_0}{\partial t} \underbrace{\frac{\partial t}{\partial t}}_{=1}
+ + \frac{\partial \theta_0}{\partial x} \underbrace{\frac{\partial
+ x}{\partial t}}_{=0}\\
+ &=-\theta_{0\xi}+\theta_{0t}.
+\end{align}
+substituting into \label{eq:solitionO0} we get
+\begin{align}
+ &2\theta_{0t\xi}\simeq\theta_{0t t},\\
+ \Rightarrow\;\;&\eta= \theta_{0\xi}+O(\varepsilon).
+\end{align}
+As for the surface boundary condition we see that the derivatives in $t$ are
+''small``. So we can proceed by the scaling $\tau = \varepsilon t$ as
+$\varepsilon\rightarrow 0$, we proceed with equation given in
+\ref{eq:soliton-scale-boundary1} and \ref{eq:soliton-scale-boundary2} in the
+$O(\varepsilon), O(\delta^2)$
+\begin{align}\label{eq:soliton-scale-boundary5}
+ -(1+\varepsilon\eta)\theta_{0\xi\xi}+
+ \delta^2\left(\frac{1}{6}\theta_{0\xi\xi\xi\xi} - \theta_{1\xi\xi}\right)\simeq
+ \varepsilon\eta_\tau -\eta_\xi +\varepsilon\eta\theta_{0\xi}
+\end{align}
+and boundary equations in \ref{eq:soliton-scale-boundary3},
+\ref{eq:soliton-scale-boundary4} produce
+\begin{align}\label{eq:soliton-scale-boundary6}
+ \varepsilon\theta_{0\tau}-\theta_{0\xi}+\delta^2\left(
+ \frac{1}{2}\theta_{0\xi\xi\xi} - \theta_{1\xi} \right) +\eta \simeq
+ -\frac{1}{2}\varepsilon \theta^2_{0\xi}.
+\end{align}
+Doing the following operation to the above equations
+\ref{eq:soliton-scale-boundary5} $-$ $\frac{\partial }{\partial
+\xi}$\ref{eq:soliton-scale-boundary6} turns out to be
+\begin{align}
+ &-\theta_{0\xi\xi}-
+ \varepsilon\eta\theta_{0\xi\xi}+
+ \delta\left(\frac{1}{6}\theta_{0\xi\xi\xi\xi}-\theta_{1\xi\xi}\right)
+ - \varepsilon\theta_{0\xi\tau}+\theta_{0\xi\xi}-\delta^2\left(
+ \frac{1}{2}\theta_{0\xi\xi\xi\xi} -
+ \theta_{1\xi\xi}\right)+\eta_{\xi}\\
+ &\simeq \varepsilon\eta_t - \eta_\xi+
+ \varepsilon\eta\theta_{0\xi}+\varepsilon\theta_{0\xi\xi}\theta_{0\xi}.
+\end{align}
+In the above equation we can simplify, i.e. short some terms out and
+substitute $\eta = \theta_{0\xi} + O(\varepsilon)$ and because of $\delta^2 =
+O(\varepsilon)$ we set $\delta^2 = K\varepsilon$ for constant $K$, leaving us
+with the equation for the surface profile, called the \textbf{Korteweg-de
+Vries}, KdV equation (1895)
+\begin{align}
+ 2\eta_\tau + 3\eta\eta_\xi + \frac{K}{3}\eta_{\xi\xi\xi} = 0.
+\end{align}
+The KdV equation describes the balance between linearity and dispersion in
+the change of time of the wave profile. By rewriting $\eta = f(\xi-ct)$ we
+get
+\begin{align}
+ -2cf' + 3ff' + \frac{K}{3}f''' = 0\\
+ \text{with} \quad f, f', f''' \rightarrow 0\quad \text{as}\;\; |\xi-ct|
+ \Rightarrow \infty.
+\end{align}
+The solution is a $\text{sech}$ function
+\begin{align}
+ f = 2c\ \text{sech}^2\left( \sqrt{\frac{3}{2K}}(\xi-ct)\right)
+\end{align}
+\subsection{KdV Equation}
+In this section we will go over the more general prerequisites and therefore
+a more convincing expedition for the Korteweg-de Vries equation. We still
+want to derive the wave profile of a wave in shallow water, small amplitude
+regime $\delta^2 = O(\varepsilon)$, where the bottom is horizontal \&
+stationary. The propagating wave can be seen as a plane wave, therefore the
+coordinate system is rotated in such a way that the propagating direction is
+the $x$ direction. For irrotational, inviscid flow without surface tension
+$W_e=0$ that is for gravity waves, nondimensional and rescaled Euler's
+Equations of Motion for such a flow are
+\begin{align}
+ \begin{drcases}
+ \frac{Du}{Dt}=-p_x,\quad \quad \delta^2
+ \frac{Dw}{Dt} = -p_z,\\
+ \text{where}\\
+ \frac{D}{Dt} = \frac{\partial }{\partial t} + \varepsilon
+ \left(
+ u\frac{\partial u}{\partial x}
+ +w\frac{\partial w}{\partial z}\right)
+\\
+ \text{with}\\
+ \frac{\partial u}{\partial x} +\frac{\partial w}{\partial z} = 0
+ \end{drcases}
+\end{align}
+with free surface boundary conditions
+\begin{align}
+ \begin{drcases}
+ p=\eta\\
+ w=\eta_t+\varepsilon u \eta_x
+ \end{drcases}
+ \text{on}\;\; z= 1+\varepsilon\eta,
+\end{align}
+and bottom boundary condition
+\begin{align}
+ w = 0 \quad \text{on}\;\; z=b =0.
+\end{align}
+We note here thatthe soluition for such a wave is a solitary wave as in
+described in the previous section. In principel we expect to wind such waves
+reather rarely in nature, since $\delta^2 = O(\varepsilon)$ is a very special
+case. Never the less this is not the case. We demonstrate that $\forall\
+\delta$ as $\varepsilon$ goes to $0$ there exists a region in the position
+space $(x, t)$ where the KdV balance in terms of linearity and dispersion
+is observed. Indeed we can ''generate`` KdV solitary waves, provided a small
+enough amplitude in the sence of $\varepsilon$ goes to $0$. First of all we
+introduce a rescaling of the variables adjusted to our problem definition
+\begin{align}
+ x \rightarrow \frac{\delta}{\sqrt{\varepsilon} }\tilde{x}, \quad t
+ \rightarrow \frac{\delta}{\sqrt{\varepsilon} }\tilde{t}\\
+ w \rightarrow \frac{\sqrt{\varepsilon} }{\delta}\tilde{w}.
+\end{align}
+Then the material derivative is transformed to be
+\begin{align}
+ \frac{D}{Dt} = \frac{\sqrt{\varepsilon}}{\delta}(\frac{\partial
+ }{\partial \tilde{t}} +\varepsilon \tilde{\mathbf{u}} \nabla).
+\end{align}
+The initial equations become
+\begin{align}
+ \frac{Du}{Dt} = \frac{\sqrt{\varepsilon}}{\delta} =-
+ \frac{\sqrt{\varepsilon} }{\delta} p_{\tilde{x}}\;\; &\Rightarrow\;\;
+ u_{\tilde{t}} + \varepsilon(u u_{\tilde{x}} + wu_z)= -p_{\tilde{x}}.\\
+ \frac{Dw}{Dt} = \frac{\varepsilon}{\delta^2}
+ \frac{D\tilde{w}}{D\tilde{t}}=-p_z \;\;&\Rightarrow\;\;
+ \varepsilon\left(\tilde{w}_{\tilde{t}} + \varepsilon\left(
+ u\tilde{w}_{\tilde{x}}+ \tilde{w}\tilde{w}_z \right) \right) = -p_z,
+\end{align}
+with
+\begin{align}
+ &w
+ = \frac{\varepsilon}{\delta}\tilde{w}
+ = \frac{\sqrt{\varepsilon} }{\delta}
+ \eta_{\tilde{t}}+\varepsilon u\frac{\sqrt{\varepsilon}}{\delta}
+ \eta_{\tilde{x}}\\
+ &\Rightarrow\;\;
+ \begin{drcases}
+ \tilde{w} = \eta_{\tilde{t}}+ \varepsilon u
+ \eta_{\tilde{x}}\\
+ p=\eta
+ \end{drcases}
+ \text{on}\;\; z = 1+\varepsilon\eta
+\end{align}
+and
+\begin{align}
+ w = 0 \quad \text{on}\;\; z= b = 0.
+\end{align}
+Now we replace the region $\delta^2$ with $\varepsilon = \delta^2$, while we
+let $\varepsilon$ go to $0$. We conclude to the following equations
+\begin{align}\label{eq:kdv3}
+ \begin{drcases}
+ u_t = -p_x, \quad p_z = 0\\
+ u_x + w_z = 0,\\
+ \text{with}\\
+ w=\eta_t \quad p=\eta \quad \text{on}\;\; z= 1\\
+ w = 0 \quad \text{on}\;\; z= 0.
+ \end{drcases}
+\end{align}
+Modification to these equations on the boundary condition, i.e. on $z=1$
+leaves us with
+\begin{align}
+ u = -p_x = -\eta_x \quad \Rightarrow \quad u_t + \eta_x = 0
+ \label{eq:kdv1}\\
+ w = -zu_x\Big|_{z=1} = -u_x = \eta_t \quad \Rightarrow \quad u_x + \eta_t
+ =0.\label{eq:kdv2}
+\end{align}
+By doing differentiation \ref{eq:kdv1} with respect to $x$ and subtracting
+the equation \ref{eq:kdv2} differentiated with respect tot $t$ we get the
+standard wave equation for the profile of the wave
+\begin{align}
+ \eta_{x x} - \eta_{t t} = 0 .
+\end{align}
+We choose a solution for a right going wave and go into the frame of the
+moving wave by a coordinate transformation as in the last section to $\xi =
+x- t$. Additionally we want to introduce a long term variable, since we have
+a uniformity as $t$ (or $x$) goes to infinity. This is usually done by
+rescaling $t = \varepsilon \tau$. In summary we have that $\xi = O(1)$ as
+well as $\tau = O(1)$. This is for \textbf{far field region} of the wave, and
+therefore the region, where we expect KdV type balance, between dispersion
+and linearity. The fact of this matter can be rigorously proven, it needs to
+be show that any sufficiently fast decaying smooth solution will eventually
+split into a finite superposition of two solitary waves traveling to the
+right and a decaying dispersive part traveling to the left. However will not
+go into this here. To transform the equations in \ref{eq:kdv3}, we look at
+the chain rule w.r.t $\xi ,\tau$ evolving to
+\begin{align}
+ \frac{\partial }{\partial t} &= -\frac{\partial }{\partial \xi}
+ +\varepsilon \frac{\partial }{\partial \tau} \\
+ \frac{\partial }{\partial x} &= \frac{\partial }{\partial \xi}.
+\end{align}
+Then we get
+\begin{align}\label{eq:kdv5}
+ \begin{drcases}
+ -u_\xi + \varepsilon\left(u_\tau + uu_\xi + w u_z \right) =
+ -p_\xi\\
+ \varepsilon\left( -w_\xi + \varepsilon\left( w_\tau + u w_\xi + w w_z
+ \right) \right) = - p_z\\
+ u_\xi + w_z = 0\\
+ \text{with}\\
+ \begin{drcases}
+ w = -\eta_\xi+\varepsilon(\eta_\tau+u \eta_\xi)\\
+ p=\eta
+ \end{drcases}
+ \text{on} \;\; z=1+\varepsilon\eta\\
+ \text{and}\\
+ w = 0 \quad \;\; z = b =0.
+ \end{drcases}
+\end{align}
+The crucial part now is to consider an asymptotic expansion of in
+$\varepsilon$ for velocity of the fluid particles $u, w$ and also the wave
+profile $\eta$ and for the pressure variable $p$. The general asymptotic
+ansatz is of the form
+\begin{align}
+ q\left( \xi, \tau, z; \varepsilon \right) = \sum_{n=0}^{\infty}
+ \varepsilon^n q_n\left( \xi, \tau, z \right).
+\end{align}
+The first equation in \ref{eq:kdv5} up to the order of $O(\varepsilon^2)$ is
+of the form
+\begin{align}
+ \varepsilon^0\left( p_{0\xi} - u_{0\xi}\right) + \varepsilon^1\left(
+ p_{1\xi} - u_{1\xi} + u_{0\tau} + u_0 u_{0\xi} + w_0u_{0z} \right)
+ +O(\varepsilon^2) = 0,
+\end{align}
+with the main condition $p_{0\xi} = u_{0\xi}$. For the second equation in
+\ref{eq:kdv5} becomes
+\begin{align}
+ \varepsilon^0\left( p_{0z} \right)
+ +\varepsilon^1\left( p_{1z}-w_{0\xi} + w_{0\tau} + u_0w_{0\xi}+w_0w_{0z} \right)
+ + O(\varepsilon^2) = 0,
+\end{align}
+the main condition $p_{0z} =0 $. The third equation in \ref{eq:kdv5} is the
+following
+\begin{align}
+ \varepsilon^0(u_{0\xi}+w_{0z}) + \varepsilon^1\left( u_{1\xi}+w_{1z}
+ \right)
+ O(\varepsilon^2) =0,
+\end{align}
+where the main condition satisfies $u_{n\xi} = -w_{n\xi}$ for all $n \in
+\mathbb{N}$. Further the surface condition is expanded into
+\begin{align}
+ p_n = \eta_n \qquad \forall\ n \in \mathbb{N},
+\end{align}
+and
+\begin{align}
+ \varepsilon^0\left(w_0 + \eta_{0\xi}\right)+
+ \varepsilon^1\left( w_1 + \eta_{1\xi} + \eta_{0\tau} + \eta_0 \eta_{0\xi}\right)
+ + O(\varepsilon^2) = 0,
+\end{align}
+Do note that the condition for for $\varepsilon^0$ is $z=1$ and for
+$\varepsilon^1$ is $z=\varepsilon\eta$. The main conclusion from the above
+equation is however $w_0 = -\eta_{0\xi}$. And lastly the bottom condition
+remains unchanged for all $n$ as
+\begin{align}
+ w_n = 0 \quad \text{on}\;\; z= b=0
+\end{align}
+In essence $O(\varepsilon^0)$ give us the equations
+\begin{align}
+ u_{0\xi}=p_{0\xi},\quad p_{0z} =0,\quad u_{0\xi} + w_{0z} = 0,
+\end{align}
+with
+\begin{align}
+ p_0 = \eta_0, \quad w_0 = -\eta_{0\xi} \quad \text{on}\;\; z=1\\
+ w_0 = 0 \quad \text{on}\;\; z=b=0.
+\end{align}
+They lead us tot he following solution which satisfies the boundary
+\begin{align}
+ p_0 = \eta_0, \quad u_0 = \eta_0, \quad w_0 = -z\eta_{0\xi} \quad
+ \forall\ z\in[0, 1].
+\end{align}
+Do notice who $w_0 = -z\eta_{0\xi}$ automatically satisfies the boundary
+conditions for both $z=0$ and $z=1$. Before we go on to consider
+$O(\varepsilon)$, we expand $u, w$ and $p$ around $z=1$ via Taylor expansion.
+This makes only since in the case $\varepsilon\rightarrow 0$
+\begin{align}
+ \begin{drcases}
+ p_0 + \varepsilon\eta_0 p_{0z} + \varepsilon p_1 = \eta_0
+ \varepsilon\eta_1 + O(\varepsilon^2)\\
+ w_0 +\varepsilon\eta_0w_{0z} + \varepsilon w_1 = -\eta_{0\xi} -
+ \varepsilon\eta_{1\xi} + \varepsilon\left( \eta_0 + u_0 \eta_{0\xi}
+ \right) +O\left(\varepsilon^2 \right).
+ \end{drcases} \text{on}\;\; z=1
+\end{align}
+Right off the equations of order $O(\varepsilon^1)$ become
+\begin{align}
+ -u_{1\xi} + u_{0\tau} + u_0u_{0\xi} + w_{0}u_{0z} = -p_{1\xi},\\
+ p_{1z} = w_{0\xi} \quad \text{and} \quad u_{1\xi} + w_{1z} = 0,
+\end{align}
+with the boundary conditions
+\begin{align}
+ \begin{drcases}
+ p_1 + \eta_0 p_{0z} = \eta_1\\
+ w_1 + \eta_0 w_{0z} = -\eta_{1\xi} + \eta_{0t} + u
+ \end{drcases}
+ \text{on}\;\; z=1\\
+ w_1 = 0 \quad \text{on}\;\; z =b = 0.
+\end{align}
+Thus
+\begin{align}
+ &p_{1z} = w_{0\xi} = -z\eta_{0\xi}\\
+ \Rightarrow p_1 = -\frac{1}{2}z^2 \eta_{0\xi\xi} +c,
+\end{align}
+where $c$ is the integration constant, together with the boundary condition
+on $z=1$ we get that
+\begin{align}
+ c = \eta_1 + \frac{1}{2} \eta_{0\xi\xi},
+\end{align}
+for $p_1$ leaving is with
+\begin{align}
+ p_1 = \frac{1}{2} \left( 1-z^2 \right) \eta_{0\xi\xi} +\eta_1
+\end{align}
+As for the condition $w_{1z} = -u_{1\xi}$ we get
+\begin{align}
+ w_{1z} &= -u_{1\xi} = -p_{1\xi} - u_{0\tau} - u_0u_{0\xi} - u_0u_{0z} \\
+ &=\frac{1}{2} (1-z^2)\eta_{0\xi\xi\xi} - \eta_{1\xi} -\eta_{0\tau}
+ -\eta_{0\xi}.
+\end{align}
+By integration and evaluation on $z=1$ of the above equation we get
+\begin{align}\label{eq:kdv6}
+ w_1\Big|_{z=1} = -\frac{1}{3} \eta_{0\xi\xi\xi} - \eta_{1\xi} -
+ \eta_{0\tau} -\eta_0\eta_{0\xi},
+\end{align}
+on the other hand we have the original boundary condition
+\begin{align}\label{eq:kdv7}
+ w_1\Big|_{z=1} = -\eta_{1\xi} + \eta_{0\tau} +2\eta_{0}\eta_{0\xi} .
+\end{align}
+Subtracting equation \ref{eq:kdv6} from \ref{eq:kdv7} we get the KdV equation
+\begin{align}
+ \frac{1}{3} \eta_{0\xi\xi\xi} - 2\eta_{0\tau} - 3\eta_0\eta_{0\xi} = 0.
+\end{align}
+
+
+
+
+
diff --git a/app_pde/main.tex b/app_pde/main.tex
@@ -0,0 +1,28 @@
+\include{./preamble.tex}
+
+\usepackage{amsmath}
+\numberwithin{equation}{section}
+
+\begin{document}
+
+\maketitle
+\tableofcontents
+
+\include{./chap1.tex}
+\include{./chap2.tex}
+\include{./chap3.tex}
+
+
+\newpage
+\include{appendix.tex}
+
+
+
+\nocite{johnson_1997}
+\nocite{vallis_2017}
+\nocite{constantin_tsunami}
+\nocite{rupert_2009}
+\nocite{mathe-physik}
+\printbibliography
+
+\end{document}
