git clone https://git.igankevich.com/waves-16-arma.git
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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.

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