commit a1e896da15085d307d9cf066aeb70fa02f25096b
parent 381b8ae3559e9dfa5e009da020fc2051043c0e67
Author: miksa234 <milutin@popovic.xyz>
Date: Wed, 8 Jun 2022 15:35:38 +0200
fix mistakes
Diffstat:
5 files changed, 88 insertions(+), 89 deletions(-)
diff --git a/app_pde/build/main.pdf b/app_pde/build/main.pdf
Binary files differ.
diff --git a/app_pde/chap1.tex b/app_pde/chap1.tex
@@ -1,5 +1,5 @@
\section{Governing Equations of Fluid Dynamics}
-We first start of with a fluid with a density
+We first start off with a fluid with a density given by
\begin{align}
\rho(\mathbf{x}, t),
\end{align}
@@ -16,7 +16,7 @@ Additionally it is also described by its pressure
\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
+direction (depending on how we set up our system $z$ pointing up or down
respectively).
The general assumption in fluid dynamics is the \textbf{Continuum
@@ -24,7 +24,7 @@ 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.
+fluid dynamics.
\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
@@ -32,7 +32,7 @@ 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
+Let us now think 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 $\rho(\mathbf{x}, t)$,
@@ -74,31 +74,30 @@ The figure bellow \ref{fig:volume}, expresses the above described picture.
\end{figure}
Since we want to figure out the fluid's dynamics, we can consider the rate
-of change in the completely arbitrary $V$
+of change in the completely arbitrary volume $V$, by
\begin{align}
\frac{d}{dt}\left( \int_V \rho(\mathbf{x}, t)\ dV \right) = \int_V
\frac{\partial \rho(\mathbf{x}, t)}{\partial t} \ dV
\end{align}
-On the other hand we have that density is mass over volume or
+On the other hand we have that density is mass over volume, meaning
\begin{align}
dm = \rho \cdot dV.
\end{align}
The infinitesimal volume has the base area $dS$ with hight $h$, which is the
distance in the direction perpendicular to to the base area, leaving us with
-$dV = h dS$. By definition $\mathbf{n}$ is perpendicular to $dS$, we have
+$dV = h\ dS$. By definition $\mathbf{n}$ is perpendicular to $dS$, we have
that $h = l \mathbf{n}$. Where $l$ is the unit of length, or velocity times
-time $l = \mathbf{u} dt$, since mass is flowing out of the surface we
+time $l = \mathbf{u}\ dt$, since mass is flowing out of the surface we
change the sign of the flow leading us to
\begin{align}
- dm = -\rho dV = -\rho \mathbf{u} \cdot \mathbf{n} dt dS,
+ dm = -\rho dV = -\rho \mathbf{u} \cdot \mathbf{n}\ dt\ dS,
\end{align}
All together we have
\begin{align}
- \frac{dm}{dt} = -\int_{\partial V} \rho(\mathbf{x},t) dS
- \mathbf{u}\cdot\mathbf{n}.
+ \frac{dm}{dt} = -\int_{\partial V} \rho(\mathbf{x},t)
+ \mathbf{u}\cdot\mathbf{n}\ dS.
\end{align}
-Putting both equations on one side leaves us with a variation of the mass
-conservation equation
+Putting both equations on one side leaves us with the equation
\begin{align}\label{eq:mass balance}
\int_V \frac{\partial \rho(\mathbf{x}, t)}{\partial t}\ dV
+\int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u}\cdot\mathbf{n}\ dS
@@ -116,7 +115,7 @@ integration law to acquire an integral over the volume
\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.
+ \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
@@ -126,14 +125,14 @@ called the \textbf{Continuity Equation}
\end{align}
In light of the results of the equation of mass conservation
-in \ref{eq:continuity}, an product rule gives
+in \ref{eq:continuity}, the 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.
+ \frac{D}{Dt} = \frac{\partial }{\partial t} + \mathbf{u}\nabla.
\end{align}
With the material derivative the equation of mass conservation reads
\begin{align}
@@ -156,9 +155,10 @@ 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
+In the inviscid case, the local force retains the pressure $P$, producing a
+normal force, with respect to the surface, acting onto the infinitesimal
+element in the fluid. Summing over all infinitesimal pieces of the fluid,
+gives us an integral formulation of the force
\begin{align}
\int_V \rho \mathbf{F}\ dV - \int_S P\mathbf{n}\ dV.
\end{align}
@@ -173,14 +173,14 @@ and thus the total force per unit volume on the other hand is
\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
+Newton's Second Law for a fluid in a 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) =
+ +(\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
@@ -205,19 +205,19 @@ For now we have separated two simplifications, that define an
\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
+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
+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
+of vorticity has only minor contributions. 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
+\ref{appendix:diff identity}, rewriting the motion derivative in terms of
\begin{align}
\frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t}
- \nabla(\frac{1}{2}\mathbf{u}\mathbf{u)}
+ +\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
@@ -229,7 +229,7 @@ Thus the equations of motion become
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}.
+At this point we may differentiate between \textbf{steady and unsteady flow}.
For \textbf{Steady Flow} we assume that $\mathbf{u}, P$ and $\Omega$ are time
independent, thus we get
\begin{align}
@@ -258,8 +258,7 @@ where $\phi$ needs to satisfy the Laplace equation
\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
+}{\partial t}$ and $\nabla$, giving us the expression
\begin{align}
\nabla\left( \frac{\partial \phi}{\partial t} +\frac{1}{2}
\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega \right) = 0
@@ -271,15 +270,14 @@ function $f(\mathbf{x}, t)$, writing
\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.
+$\phi \rightarrow \phi + \int f(\mathbf{x}, t)\ dt$.
\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
+fluid 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
@@ -293,8 +291,8 @@ 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
+The logical step now is to define boundary conditions on the bed of the
+fluid. In 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
@@ -312,7 +310,7 @@ 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.
+ \frac{D}{Dt}\left(z - h(\mathbf{x}_\perp, t ) \right) = 0.
\end{align}
Upon expanding the derivative we get
\begin{align}
@@ -336,15 +334,15 @@ 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.
+ + \frac{P_a}{\rho}+gh = f(t) \qquad \text{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
+equilibrium and is thereby stationary, meaning it has no motion and the
+pressure is $P=P_a$, the surface is a constant $h = h_0$ and therefore
\begin{align}
f(t) = \frac{P_a}{\rho}+gh_0.
\end{align}
-The simplest description for the \textbf{dynamic condition} may be written as
+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.
@@ -358,7 +356,7 @@ dynamic condition by introducing the pressure difference known as the
\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
+function is $z - h(\mathbf{x}_\perp, t)=0$ 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} -
@@ -413,10 +411,11 @@ 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
+ + 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
+To simplify the equation we resort to the Leibniz Rule of Integration
+\ref{appendix:leigniz},
\begin{align}
\int_b^h \nabla_\perp\mathbf{u}_\perp =
\nabla_\perp \int_b^h \mathbf{u}_\perp\ dz - (\mathbf{u}_{\perp \text{s}}
@@ -461,7 +460,7 @@ Then we may add
\end{align}
to above not changing anything. Thereby getting
\begin{align}
- \frac{\partial }{\partial t} (\frac{1}{2}\mathbf{u}\mathbf{u})
+ \frac{\partial }{\partial t} \left(\frac{1}{2}\mathbf{u}\mathbf{u}\right)
+(\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)
diff --git a/app_pde/chap2.tex b/app_pde/chap2.tex
@@ -2,14 +2,15 @@
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
+second. The idea is to 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\label{sec:nondim}}
-In summary we use these adaptations
+In summary we use these adaptations for the scales
+
\begin{itemize}
\item $h_0$ for the typical water depth
\item $\lambda$ for the typical wavelength
@@ -18,11 +19,7 @@ In summary we use these adaptations
\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
+Additionally we write the
boundary condition on the free surface as follows
\begin{align}
h = h_0 + a \eta (\mathbf{x}_\perp, t) = z,
@@ -47,7 +44,7 @@ The pressure is also rewritten into
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
+distribution. We have $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 =
@@ -56,7 +53,7 @@ rescaling of the Euler's Equation of Motion, we introduce the notation
&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
+leaving us with
\begin{align}
\frac{Du}{Dt}
&= \frac{\partial u}{\partial t} +u \frac{\partial
@@ -65,12 +62,12 @@ leading us to
\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),
+ + \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)
+ +\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}
@@ -96,24 +93,24 @@ In the $z$ coordinate we have
\end{align}
On the other side we have
\begin{align}
- \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
+ \frac{h_0^2g}{\lambda^2}\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}.
+ &=-\frac{1}{h_0\rho}\left(-\rho gh_0 \frac{\partial \zeta}{\partial
+ \zeta} + \rho gh_0
+ \frac{\partial p}{\partial \zeta} \right) + g \\
+ &= -g \frac{\partial p}{\partial \zeta}.
\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},
+ \frac{Dw}{Dt} = -\frac{\partial p}{\partial \zeta},
\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
+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
@@ -187,10 +184,12 @@ 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
+ \begin{drcases}
+ 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
+ \end{drcases} \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}
@@ -201,13 +200,13 @@ 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
+ \item$\delta \rightarrow 0;\; \varepsilon\approx1$: shallow Water, large
amplitude
- \item $\delta\ll 1;\; \varepsilon~\delta$: shallow water, medium
+ \item $\delta\ll 1;\; \varepsilon\approx\delta$: shallow water, medium
amplitude
- \item $\delta\ll 1;\; \varepsilon~\delta^2$: shallow water, small
+ \item $\delta\ll 1;\; \varepsilon\approx\delta^2$: shallow water, small
amplitude
- \item $\delta \gg 1;\; \varepsilon\delta\ll 1$: deep water, small
+ \item $\delta \gg 1;\; \varepsilon\cdot\delta\ll 1$: deep water, small
steepness.
\end{itemize}
diff --git a/app_pde/chap3.tex b/app_pde/chap3.tex
@@ -1,8 +1,8 @@
\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
+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
+one solitary wave, but 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 $\varepsilon=O(\delta^2)$.
@@ -13,7 +13,7 @@ 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
+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
@@ -30,10 +30,10 @@ $b=0$. Ultimately leaving us with the following model
& \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
+Since the model arises in $\varepsilon = O(\delta^2)$, 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,
+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
@@ -84,7 +84,7 @@ Dividing equation \ref{eq:sol1} with equation \ref{eq:sol2} gives
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
+dispersion relation on the wave propagation speed holds, thereby solitary
waves exhibit exponential decay in their tail and satisfy the dispersion
relation in equation \ref{eq:soliton-dispersion}.
\subsubsection{Asymptotic Analysis}
@@ -125,7 +125,7 @@ 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
+The free surface boundary condition $z= 1+\varepsilon\eta$ involves 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)\\
@@ -150,7 +150,7 @@ substituting $\phi_\delta$ into the above proceeds to be
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,
+ +\phi_x^2\right) = 0,
\end{align}
becomes after substitution
\begin{align}
@@ -272,7 +272,7 @@ and bottom boundary condition
w = 0 \quad \text{on}\;\; z=b =0.
\end{align}
We note here that the solution for such a wave is a solitary wave as in
-described in the previous section. In principle we expect to wind such waves
+described in the previous section. In principle we expect to find such waves
rather 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
@@ -320,7 +320,8 @@ and
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
+let $\varepsilon$ go to $0$. We conclude to the following equations, after
+replacing to $x, t$ and $w$
\begin{align}\label{eq:kdv3}
\begin{drcases}
u_t = -p_x, \quad p_z = 0\\
@@ -339,7 +340,7 @@ leaves us with
=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
+the equation \ref{eq:kdv2} differentiated with respect to $t$ we get the
standard wave equation for the profile of the wave
\begin{align}
\eta_{x x} - \eta_{t t} = 0 .
@@ -471,7 +472,7 @@ with the boundary conditions
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,
+ \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
diff --git a/app_pde/chap4.tex b/app_pde/chap4.tex
@@ -2,7 +2,7 @@
\subsection{Description}
On the 26. December 2004, time 7:58 a powerful earthquake generated a tsunami
killing more than 275000 people and leaving millions homeless. The
-hypocenter f the earthquake was 30 km under the floor of the Indian Ocean,
+hypocenter of the earthquake was 30 km under the floor of the Indian Ocean,
100 km away from Sumatra, an island in Indonesia. The earthquake displaced
an enormous amount of water, sending tsunami waves westwards across the
Indian Ocean to Sri Lanka and India and eastwards across the Andaman Basin.
@@ -26,12 +26,12 @@ amplitude of $10\ \text{m}$. Observations tell us that as the tsunami waves
reached the shore the shape of the initial disturbance was not altered, which
is supported by measurements by a radar altimeter two hours after the
earthquake showing first an wave of elevation and then a wave of depression
-westwards and respectively vice versa eastwards. A conclusion is made that t
-he shape of the tsunami remained approximately constant. Additionally it
+westwards and respectively vice versa eastwards. The conclusion is made that
+the shape of the tsunami remained approximately constant. Additionally it
should be mentioned that the tsunami waves reach very high amplitudes due to
the diminishing depth effect as they approach the shore, yet at open sea are
barely noticeable. A boat on open sea positioned at high depth in the region
-of the tsunami during the which the tsunami waves passed, captured the raise
+of the tsunami during which the tsunami waves passed, captured the raise
from $\pm 0.8\ \text{m}$ of the boat over a period of $10\ \text{min}$.
This means that the wavelength of the tsunami wave was about $100\
\text{km}$.
@@ -54,7 +54,7 @@ tallest in from front followed by an oscillatory tail. The KdV is the
proper equation, for our modeling purposes of tsunami waves. The main
question arises if KdV enters the regime of validity, in our case for the
$2004$ tsunami in the Andaman Basin. Or in other words are the involved
-geophysical scales leading to time and space scales that are compatible with
+geophysical scales leading to time and space scales, compatible with
the KdV weak nonlinearity balance.
\subsection{Governing equations}
The last two section show the derivation of the modeling equation for fluid
@@ -109,10 +109,10 @@ giving us
\end{align}
giving us a $\beta \simeq 6,4 = O(1)$ for $\varepsilon = \beta \delta^2$.
The main issue is if the KdV balance can occur within the geophysical scales
-.Thee conditions $x-t = O(1)$ and $\tau = O(1)$ give
+.The conditions $x-t = O(1)$ and $\tau = O(1)$ give
\begin{align}
- \frac{x - t \sqrt{gh_0} }{\lambda} = 0(1),\quad \frac{\varepsilon t
- \sqrt{gh_0} }{\lambda} = 0(1).
+ \frac{x - t \sqrt{gh_0} }{\lambda} = O(1),\quad \frac{\varepsilon t
+ \sqrt{gh_0} }{\lambda} = O(1).
\end{align}
Combining the above equations, we have
\begin{align}
