commit 04e79c6fff6cc50f761e2053b466eef3d9de71de
parent a9870ed993ec131d39230759273f0778b9029c3e
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Mon, 29 May 2017 18:17:38 +0300
Edit pressure section.
Diffstat:
| arma.org | | | 127 | ++++++++++++++++++++++++++++++++++++++++--------------------------------------- |
1 file changed, 65 insertions(+), 62 deletions(-)
diff --git a/arma.org b/arma.org
@@ -596,7 +596,7 @@ steps.
- Move maximum value of the resulting function to the origin by using
trigonometric identities to shift the phase.
-** Evaluation
+** Evaluation and discussion
In\nbsp{}cite:degtyarev2011modelling,degtyarev2013synoptic,boukhanovsky1997thesis
for AR model the following items were verified experimentally:
- probability distributions of different wave characteristics (wave heights,
@@ -606,7 +606,6 @@ for AR model the following items were verified experimentally:
In this work we repeat probability distribution tests for three-dimensional AR
and MA model.
-*** Verification of wavy surface integral characteristics
In\nbsp{}cite:rozhkov1990 the authors show that several ocean wave
characteristics (listed in table\nbsp{}[[tab-weibull-shape]]) have Weibull
distribution, and wavy surface elevation has Gaussian distribution. In order to
@@ -711,7 +710,6 @@ correspondence degree for wave elevation is obtained, because this is the
characteristic of the wavy surface (and corresponding AR or MA process) and is
not affected by the type of waves.
-** Discussion
ARMA model, owing to its non-physical nature, does not have the notion of ocean
wave; it simulates wavy surface as a whole instead. Motions of individual waves
and their shape are often rough, and the total number of waves can not be
@@ -731,7 +729,7 @@ work.
Analytic solutions to boundary problems in classical equations are often used to
study different properties of the solution, and for that purpose general
solution formula is too difficult to study, as it contains integrals of unknown
-functions. Fourier method is one of the methods to find analytic solutions to
+functions. Fourier method is one of the methods to find analytic solutions to a
PDE. It is based on application of Fourier transform to each part of PDE, which
reduces the equation to algebraic, and the solution is written as inverse
Fourier transform of some function (which may contain Fourier transforms of
@@ -741,16 +739,17 @@ is studied in different domains instead. At the same time, computing discrete
Fourier transforms on the computer is possible for any discretely defined
function and efficient when using FFT algorithms. These algorithms use symmetry
of complex exponentials to decrease asymptotic complexity from
-\(\mathcal{O}(n^2)\) to \(\mathcal{O}(n\log_{2}n)\). So, even if general solution
-contains Fourier transforms of unknown functions, they still can be computed
-numerically, and FFT family of algorithms makes this approach efficient.
+\(\mathcal{O}(n^2)\) to \(\mathcal{O}(n\log_{2}n)\). So, even if general
+solution contains Fourier transforms of unknown functions, they still can be
+computed numerically, and FFT family of algorithms makes this approach
+efficient.
-Alternative approach to solve PDE is to reduce it to difference equations, which
+Alternative approach to solve a PDE is to reduce it to difference equations, which
are solved by constructing various numerical schemes. This approach leads to
approximate solution, and asymptotic complexity of corresponding algorithms is
comparable to that of FFT. For example, stationary elliptic PDE transforms to
implicit numerical scheme which is solved by iterative method on each step of
-which a tridiagonal of five-diagonal system of algebraic equations is solved by
+which a tridiagonal or five-diagonal system of algebraic equations is solved via
Thomas algorithm. Asymptotic complexity of this approach is
\(\mathcal{O}({n}{m})\), where \(n\)\nbsp{}--- number of wavy surface grid
points, \(m\)\nbsp{}--- number of iterations. Despite their wide spread,
@@ -775,19 +774,20 @@ for it in general case is written as\nbsp{}cite:kochin1966theoretical
& \phi_t+\frac{1}{2} |\vec{\upsilon}|^2 + g\zeta=-\frac{p}{\rho}, & \text{на }z=\zeta(x,y,t),\label{eq-problem}\\
& D\zeta = \nabla \phi \cdot \vec{n}, & \text{на }z=\zeta(x,y,t),\nonumber
\end{align}
-where \(\phi\)\nbsp{}--- velocity potential, \(\zeta\)\nbsp{}--- elevation (\(z\) coordinate)
-of wavy surface, \(p\)\nbsp{}--- wave pressure, \(\rho\)\nbsp{}--- fluid density,
-\(\vec{\upsilon}=(\phi_x,\phi_y,\phi_z)\)\nbsp{}--- velocity vector, \(g\)\nbsp{}---
-acceleration of gravity, and \(D\)\nbsp{}--- substantial (Lagrange) derivative. The
-first equation is called continuity (Laplace) equation, the second one is the
-conservation of momentum law (the so called dynamic boundary condition); the
-third one is kinematic boundary condition for free wavy surface, which states
-that rate of change of wavy surface elevation (\(D\zeta\)) equals to the change of
-velocity potential derivative along the wavy surface normal
-(\(\nabla\phi\cdot\vec{n}\)).
+where \(\phi\)\nbsp{}--- velocity potential, \(\zeta\)\nbsp{}--- elevation
+(\(z\) coordinate) of wavy surface, \(p\)\nbsp{}--- wave pressure,
+\(\rho\)\nbsp{}--- fluid density,
+\(\vec{\upsilon}=(\phi_x,\phi_y,\phi_z)\)\nbsp{}--- velocity vector,
+\(g\)\nbsp{}--- acceleration of gravity, and \(D\)\nbsp{}--- substantial
+(Lagrange) derivative. The first equation is called continuity (Laplace)
+equation, the second one is the conservation of momentum law (the so called
+dynamic boundary condition); the third one is kinematic boundary condition for
+free wavy surface, which states that rate of change of wavy surface elevation
+(\(D\zeta\)) equals to the change of velocity potential derivative along the
+wavy surface normal (\(\nabla\phi\cdot\vec{n}\)).
Inverse problem of hydrodynamics consists in solving this system of equations
-for \(\phi\). In this formulation dynamic boundary condition becomes explicit
+for \(\phi\). In this formulation dynamic boundary condition becomes an explicit
formula to determine pressure field using velocity potential derivatives
obtained from the remaining equations. So, from mathematical point of view
inverse problem of hydrodynamics reduces to Laplace equation with mixed boundary
@@ -828,7 +828,7 @@ infinity.] \(v={i}{u}\) into the formula yields
\phi(x,z) = \InverseFourierY{e^{2\pi u z}E(u)}{x}.
\end{equation}
In order to make substitution \(z=\zeta(x,t)\) not interfere with Fourier
-transforms, we rewrite eqref:eq-guessed-sol-2d as a convolution:
+transforms, we rewrite\nbsp{}eqref:eq-guessed-sol-2d as a convolution:
\begin{equation*}
\phi(x,z)
=
@@ -836,10 +836,10 @@ transforms, we rewrite eqref:eq-guessed-sol-2d as a convolution:
\ast
\InverseFourierY{E(u)}{x},
\end{equation*}
-where \(\Fun{z}\)\nbsp{}--- a function, form of which is defined in section
-[[#sec:compute-delta]] and which satisfies equation
-\(\FourierY{\Fun{z}}{u}=e^{2\pi{u}{z}}\). Plugging formula \(\phi\) into the boundary
-condition yields
+where \(\Fun{z}\)\nbsp{}--- a function, form of which is defined in
+sec.\nbsp{}[[#sec:compute-delta]] and which satisfies equation
+\(\FourierY{\Fun{z}}{u}=e^{2\pi{u}{z}}\). Plugging formula \(\phi\) into the
+boundary condition yields
\begin{equation*}
\zeta_t
=
@@ -861,8 +861,8 @@ to both sides of this equation yields formula for coefficients \(E\):
\FourierY{\Fun{z}}{u}
}
\end{equation*}
-Finally, substituting \(z\) for \(\zeta(x,t)\) and plugging resulting equation into
-eqref:eq-guessed-sol-2d yields formula for \(\phi(x,z)\):
+Finally, substituting \(z\) for \(\zeta(x,t)\) and plugging resulting equation
+into\nbsp{}eqref:eq-guessed-sol-2d yields formula for \(\phi(x,z)\):
\begin{equation}
\label{eq-solution-2d}
\boxed{
@@ -879,12 +879,12 @@ eqref:eq-guessed-sol-2d yields formula for \(\phi(x,z)\):
}
\end{equation}
-Multiplier \(e^{2\pi{u}{z}}/(2\pi{u})\) makes graph of a function to which Fourier
-transform of which is applied asymmetric with respect to \(OY\) axis. This makes
+Multiplier \(e^{2\pi{u}{z}}/(2\pi{u})\) makes a graph of a function to which
+Fourier transform is applied asymmetric with respect to \(OY\) axis. This makes
it difficult to apply FFT which expects periodic function with nought on both
ends of the interval. Using numerical integration instead of FFT is not faster
than solving the initial system of equations with numerical schemes. This
-problem is alleviated by using formula eqref:eq-solution-2d-full for finite
+problem is alleviated by using formula\nbsp{}eqref:eq-solution-2d-full for finite
depth fluid with wittingly large depth \(h\). This formula is derived in the
following section.
@@ -909,7 +909,8 @@ Plugging \(\phi\) into the boundary condition on the sea bottom yields
hence \(C_1=\frac{1}{2}C{e}^{2\pi{u}{h}}\) and
\(C_2=-\frac{1}{2}C{e}^{-2\pi{u}{h}}\). Constant \(C\) may take arbitrary value
here, because after plugging it becomes part of unknown coefficients \(E(u)\).
-Plugging formulae for \(C_1\) and \(C_2\) into eqref:eq-guessed-sol-2d-full yields
+Plugging formulae for \(C_1\) and \(C_2\)
+into\nbsp{}eqref:eq-guessed-sol-2d-full yields
\begin{equation*}
\phi(x,z) = \InverseFourierY{ \Sinh{2\pi u (z+h)} E(u) }{x}.
\end{equation*}
@@ -937,17 +938,17 @@ previous section transformations yields final formula for \(\phi(x,z)\):
}
\label{eq-solution-2d-full}
\end{equation}
-where \(\FunSecond{z}\)\nbsp{}--- a function, form of which is defined in section
-[[#sec:compute-delta]] and which satisfies equation
+where \(\FunSecond{z}\)\nbsp{}--- a function, form of which is defined in
+sec.\nbsp{}[[#sec:compute-delta]] and which satisfies equation
\(\FourierY{\FunSecond{z}}{u}=\Sinh{2\pi{u}{z}}\).
**** Reducing to the formulae from linear wave theory.
Check the validity of derived formulae by substituting \(\zeta(x,t)\) with known
analytic formula for plain waves. Symbolic computation of Fourier transforms in
-this section were performed in Mathematica\nbsp{}cite:mathematica10. In the framework
-of linear wave theory assume that waves have small amplitude compared to their
-lengths, which allows us to simplify initial system of equations
-eqref:eq-problem-2d to
+this section were performed in Mathematica\nbsp{}cite:mathematica10. In the
+framework of linear wave theory assume that waves have small amplitude compared
+to their lengths, which allows us to simplify initial system of
+equations\nbsp{}eqref:eq-problem-2d to
\begin{align*}
& \phi_{xx}+\phi_{zz}=0,\\
& \zeta_t = -\phi_z & \text{на }z=\zeta(x,t),
@@ -962,17 +963,18 @@ solution to which is written as
}{x}
.
\end{equation*}
-Propagating wave profile is defined as \(\zeta(x,t)=A\cos(2\pi(kx-t))\). Plugging
-this formula into eqref:eq-solution-2d yields
-\(\phi(x,z,t)=-\frac{A}{k}\sin(2\pi(kx-t))\Sinh{2\pi{k}{z}}\). In order to reduce
-it to the formula from linear wave theory, rewrite hyperbolic sine in
-exponential form, discard the term containing \(e^{-2\pi{k}{z}}\) as contradicting
-condition \(\phi\underset{z\rightarrow-\infty}{\longrightarrow}0\). Taking real
-part of the resulting formula yields
+Propagating wave profile is defined as \(\zeta(x,t)=A\cos(2\pi(kx-t))\).
+Plugging this formula into\nbsp{}eqref:eq-solution-2d yields
+\(\phi(x,z,t)=-\frac{A}{k}\sin(2\pi(kx-t))\Sinh{2\pi{k}{z}}\). In order to
+reduce it to the formula from linear wave theory, rewrite hyperbolic sine in
+exponential form, discard the term containing \(e^{-2\pi{k}{z}}\) as
+contradicting condition
+\(\phi\underset{z\rightarrow-\infty}{\longrightarrow}0\). Taking real part of
+the resulting formula yields
\(\phi(x,z,t)=\frac{A}{k}e^{2\pi{k}{z}}\sin(2\pi(kx-t))\), which corresponds to
the known formula from linear wave theory. Similarly, under small-amplitude
-waves assumption the formula for finite depth fluid eqref:eq-solution-2d-full is
-reduced to
+waves assumption the formula for finite depth
+fluid\nbsp{}eqref:eq-solution-2d-full is reduced to
\begin{equation*}
\phi(x,z,t)
=
@@ -1001,16 +1003,16 @@ depth difference near free surface is negligible). So, for sufficiently large
depth any function (\(\cosh\) or \(\sinh\)) may be used for velocity potential
computation near free surface.
-Reducing eqref:eq-solution-2d и eqref:eq-solution-2d-full to the known formulae
-from linear wave theory shows, that formula for infinite depth
-eqref:eq-solution-2d is not suitable to compute velocity potentials with Fourier
-method, because it does not have symmetry, which is required for Fourier
-transform. However, formula for finite depth can be used instead by setting \(h\)
-to some characteristic water depth. For standing wave reducing to linear wave
-theory formulae is made under the same assumptions.
+Reducing\nbsp{}eqref:eq-solution-2d и\nbsp{}eqref:eq-solution-2d-full to the
+known formulae from linear wave theory shows, that formula for infinite
+depth\nbsp{}eqref:eq-solution-2d is not suitable to compute velocity potentials
+with Fourier method, because it does not have symmetry, which is required for
+Fourier transform. However, formula for finite depth can be used instead by
+setting \(h\) to some characteristic water depth. For standing wave reducing to
+linear wave theory formulae is made under the same assumptions.
** Three-dimensional case
-Three-dimensional version of eqref:eq-problem is written as
+Three-dimensional version of\nbsp{}eqref:eq-problem is written as
\begin{align}
\label{eq-problem-3d}
& \phi_{xx} + \phi_{yy} + \phi_{zz} = 0,\\
@@ -1058,7 +1060,7 @@ Plugging \(\phi\) into the boundary condition on the free surface yields
where \(f_1(x,y)={\zeta_x}/{\SqrtZeta{1+\zeta_x^2+\zeta_y^2}}-\zeta_x\) and
\(f_2(x,y)={\zeta_y}/{\SqrtZeta{1+\zeta_x^2+\zeta_y^2}}-\zeta_y\).
-Like in Section\nbsp{}[[#sec:pressure-2d]] we assume that
+Like in sec.\nbsp{}[[#sec:pressure-2d]] we assume that
\(\Sinh{2\pi{u}(z+h)}\approx\SinhX{2\pi{u}(z+h)}\) near free surface, but in
three-dimensional case this is not enough to solve the problem. In order to get
analytic formula for coefficients \(E\) we need to assume, that all Fourier
@@ -1072,7 +1074,7 @@ magnitude of the solution. Despite these two points, a use of more
mathematically rigorous approach would be preferable.
Making the replacement, applying Fourier transform to both sides of the equation
-and plugging the result into eqref:eq-guessed-sol-3d yields formula for
+and plugging the result into\nbsp{}eqref:eq-guessed-sol-3d yields formula for
\(\phi\):
\begin{equation*}
\phi(x,y,z,t) = \InverseFourierY{
@@ -1088,12 +1090,13 @@ where \(\FourierY{\mathcal{D}_3\left(x,y,z\right)}{u,v}=\Sinh{\smash{2\pi\Kvecle
:CUSTOM_ID: sec:compare-formulae
:END:
-Comparing obtained generic formulae eqref:eq-solution-2d and
-eqref:eq-solution-2d-full to the known formulae from linear wave theory allows
-to see the difference between velocity fields for both large and small amplitude
-waves. In general analytic formula for velocity potential in not known, even for
-plain waves, so comparison is done numerically. Taking into account conclusions
-of [[#sec:pressure-2d]], only finite depth formulae are compared.
+Comparing obtained generic formulae\nbsp}{}eqref:eq-solution-2d
+and\nbsp{}eqref:eq-solution-2d-full to the known formulae from linear wave
+theory allows to see the difference between velocity fields for both large and
+small amplitude waves. In general analytic formula for velocity potential in not
+known, even for plain waves, so comparison is done numerically. Taking into
+account conclusions of\nbsp{}[[#sec:pressure-2d]], only finite depth formulae are
+compared.
**** The difference with linear wave theory formulae.
In order to obtain velocity potential fields, ocean wavy surface was generated
