commit fee05346b32c1de6f8e92f90f777f03a895dc690
parent 14b36ed5e179578e46258c6523102cf039384eee
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Fri, 26 May 2017 11:00:29 +0300
Add introduction verbatim from Degtyarev&Reed paper.
Diffstat:
| arma.org | | | 127 | ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++- |
| preamble.tex | | | 3 | +++ |
2 files changed, 129 insertions(+), 1 deletion(-)
diff --git a/arma.org b/arma.org
@@ -2,7 +2,8 @@
#+AUTHOR: Ivan Gankevich, Alexander Degtyarev
#+LANGUAGE: en
#+LATEX_CLASS: scrartcl
-#+LATEX_CLASS_OPTIONS:
+#+LATEX_CLASS_OPTIONS:
+#+LATEX_HEADER_EXTRA: \input{preamble}
#+OPTIONS: H:2 num:0 todo:nil toc:nil
#+begin_abstract
@@ -22,6 +23,130 @@ pressures, an instance of which is included in the paper.
#+end_abstract
* Introduction
+Any research related to the study of the behaviour of a ship at sea, requires a
+description of wind waves, which are the major disturbance that displaces the
+vessel from equilibrium. Currently, the most popular models for describing wind
+waves, are models based on the linear expansion of a stochastic moving surface
+as a system of independent random variables. These include models by St. Denis &
+Pearson (1953), Rosenblatt (1957), Sveshnikov (1959), and Longuet-Higgins
+(1962). The most popular model is that of Longuet---Higgins, which is based on a
+stochastic approximation of the moving wave front as a superposition of
+elementary harmonic waves with random phases \(\epsilon_n\) and random
+amplitudes \(c_n\):
+\begin{equation}
+ \label{eq-lhmodel}
+ \zeta(x,y,t) = \sum\limits_n c_n \cos(u_n x + v_n y - \omega_n t + \epsilon_n),
+\end{equation}
+where the wave number \((u_n,v_n)\) is continuously
+distributed on the \((u,v)\) plane, i.e. the unit area
+\(du \times dv\) contains an infinite number of wave numbers.
+The frequency \(\omega_n\) associated with wave
+numbers \((u_n,v_n)\) is given by a dispersion relation
+\begin{equation*}
+ \omega_n = \omega(u_n,v_n).
+\end{equation*}
+The phase \(\epsilon_n\) are jointly independent random variables uniformly
+distributed in the interval \([0,2\pi]\).
+
+Longuet---Higgins showed that under the above conditions, the function
+\(\zeta(x,y,t)\) is a three-dimensional steady-state homogeneous ergodic
+Gaussian field, defined by
+\begin{equation*}
+ 2 E_{\zeta}(u,v) du dv = \sum\limits_{n} c_n^2,
+\end{equation*}
+where \(E_{\zeta}(u,v)\) is two dimensional spectral density of wave energy.
+
+Longuet---Higgins model is simple and easily computed. It incorporates the
+physical fundamentals of the process of wind waves and is fully consistent with
+the task of modelling ocean waves.
+
+Indeed, consider one of the simplest general conservation laws, the law of
+continuity:
+\begin{equation}
+ \label{eq-continuity}
+ \frac{\partial{\rho}}{\partial{t}} = \vec{\nabla} \cdot \left(\rho\vec{V}\right) = 0,
+\end{equation}
+where \(\rho\) is the density of the liquid, and \(V\) the
+fluid velocity.
+
+In relation to ocean waves, we can make the assumptions of incompressibility and
+isotropy within the marine environment. In this case eq.\nbsp{}eqref:eq-continuity
+reduces to
+\begin{equation}
+ \label{eq-continuity-2}
+ \mathoperator{div}\vec{V} =
+ \frac{\partial{V_x}}{\partial{x}} +
+ \frac{\partial{V_y}}{\partial{y}} +
+ \frac{\partial{V_z}}{\partial{z}} = 0
+\end{equation}
+The physics of wind waves is defined primarily by the action of gravitational
+forces, which simplifies nature of the phenomenon under investigation. This
+approach allows us to consider the irrotational motion of the fluid and
+introduce the velocity potential \(\phi\). Then eq.\nbsp{}eqref:eq-continuity &
+eqref:eq-continuity-2 reduce to Laplace's equation, which is the most general field
+equation for the problem of wave motions of a liquid:
+\begin{equation*}
+ \Delta\phi =
+ \frac{\partial{\phi_x}^2}{\partial{x^2}} +
+ \frac{\partial{\phi_y}^2}{\partial{y^2}} +
+ \frac{\partial{\phi_z}^2}{\partial{z^2}} = 0.
+\end{equation*}
+
+The difference between one formulation or another of the wave problem will be in
+the nature of the boundary conditions satisfied at the surface of the air-water
+interface. In our case, it is important to remember that the linear formulation
+of the boundary conditions is (a) applied on the surface of the undisturbed
+fluid \((z=0)\) and (b) all nonlinear terms in the boundary conditions are
+ignored. The Laplace equation is linear and its solution can be found using
+Fourier transforms. Thus, for plane waves a well-known solution is given in the
+form of a definite integral (Kochin, et al., 1964):
+\begin{equation*}
+ \phi(x,z,t) =
+ \int\limits_{0}^{\infty}
+ e^{kz} \left[ A(k,t) \cos{kx} + B(k,t) \sin{kx} \right] dk.
+\end{equation*}
+A similar, but slightly more complicated solution is obtained for the
+three-dimensional case. The constants A and B are determined from the boundary
+conditions on the surface. In the linear formulation the equation of the wave
+profile is
+\begin{align}
+ \zeta(x,t) &= -\frac{1}{g} \frac{\partial{\phi(x,0,t)}}{\partial{t}} \label{eq-integ-zeta}\\
+ &= \int\limits_{0}^{\infty}
+ \left[ \frac{\partial{A(k,t)}}{\partial{t}} \cos{kx} + \frac{\partial{B(k,t)}}{\partial{t}} \sin{kx} \right] dk \nonumber \\
+ &= \int\limits_{0}^{\infty} C_t(k,t) \cos\left(kx + \epsilon(k,t)\right).\nonumber
+\end{align}
+If we set \(c_n = C_t(k_n, t) dk\), then wave model eqref:eq-lhmodel may be
+associated with an approximation of integral eqref:eq-integ-zeta.
+
+Thus, Longuet---Higgins' model is distinguished by its considerable clarity and
+the simplicity of the computational algorithm. However, it is not without some
+serious shortcomings inherent in models of this class:
+- The Longuet---Higgins model is only designed to represent a stationary
+ Gaussian field. Normal distribution of the simulated process eqref:eq-lhmodel is
+ a consequence of the central limit theorem. Its application to the analysis of
+ more general problems (e.g., the evolution of ocean waves in a storm, or the
+ study of ocean waves distorted in shallow water) represents a significant
+ challenge.
+- Models of this class are periodic and need a very large set of frequencies to
+ re-creating long-term simulation.
+- In the numerical implementation of the Longuet-Higgins model, it appears that
+ the rate of convergence of eqref:eq-lhmodel is very slow. This is seen as a
+ distortion of the energy spectrum of the simulated process (i.e. not provided
+ by the convergence of \((u, v)\)), and the laws for the distribution of
+ elementary waves, especially in terms of extreme events (not ensured by the
+ convergence of \(\epsilon\)). This problem becomes especially significant when
+ simulating complex waves that have a broad spectrum with many-peaks.
+
+The latter point becomes particularly critical in numerical simulation. In a
+time domain computation of the responses of a vessel in a random seaway, the
+repeated evaluation of the apparently simple equation, eqref:eq-lhmodel at hundreds
+of points on the hull for thousands of time steps can becomes a major factor
+determining the execution speed of the code (Beck & Reed, 2001). This becomes an
+even more significant issue in a nonlinear computation where the wave model is
+even more complex. Thus identifying a significantly less time intensive method
+for modelling the ambient ocean-wave environment has the potential for
+significantly speeding the total simulation process.
+
* Related work
* Three-dimensional ARMA process as a sea wave simulation model
** Autoregressive (AR) process
diff --git a/preamble.tex b/preamble.tex
@@ -0,0 +1,2 @@
+\usepackage{amsmath}
+\usepackage{cite}+
\ No newline at end of file
