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31 
Numerics of Elastic and Acoustic Wave MotionVirta, Kristoffer January 2016 (has links)
The elastic wave equation describes the propagation of elastic disturbances produced by seismic events in the Earth or vibrations in plates and beams. The acoustic wave equation governs the propagation of sound. The description of the wave fields resulting from an initial configuration or time dependent forces is a valuable tool when gaining insight into the effects of the layering of the Earth, the propagation of earthquakes or the behavior of underwater sound. In the most general case exact solutions to both the elastic wave equation and the acoustic wave equation are impossible to construct. Numerical methods that produce approximative solutions to the underlaying equations now become valuable tools. In this thesis we construct numerical solvers for the elastic and acoustic wave equations with focus on stability, high order of accuracy, boundary conditions and geometric flexibility. The numerical solvers are used to study wave boundary interactions and effects of curved geometries. We also compare the methods that we have constructed to other methods for the simulation of elastic and acoustic wave motion.

32 
Existence of a Solution for a Wave Equation and an Elliptic Dirichlet ProblemSumalee Unsurangsie 05 1900 (has links)
In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a nonpositone Dirichlet problem in a bounded subset of R^n.

33 
An analysis of the symmetries and conservation laws of some classes of nonlinear wave equations in curved spacetime geometryJamal, S 08 August 2013 (has links)
A thesis submitted to the Faculty of Science, University of the
Witwatersrand, in requirement for the degree Doctor of Philosophy,
Johannesburg, 2013. / The (1+3) dimensional wave and KleinGordon equations are constructed using
the covariant d'Alembertian operator on several spacetimes of interest.
Equations on curved geometry inherit the nonlinearities of the geometry. These
equations display interesting properties in a number of ways. In particular, the
number of symmetries and therefore, the conservation laws reduce depending
on how curved the manifold is. We study the symmetry properties and
conservation laws of wave equations on FreidmannRobertsonWalker, Milne,
Bianchi, and de Sitter universes. Symmetry structures are used to reduce the
number of unknown functions, and hence contribute to nding exact solutions
of the equations. As expected, properties of reduction procedures using symmetries,
variational structures and conservation laws are more involved than
on the well known
at (Minkowski) manifold.

34 
Atratores para equações de ondas em domínios de fronteira móvel / Attractors for a weakly damped semilinear wave equation on timevarying domainsChuño, Christian Manuel Surco 09 June 2014 (has links)
Este trabalho contém um estudo sobre equações de ondas fracamente dissipativas definidas em domínios de fronteira móvel ∂2u/∂t2/ + η∂u/∂t  Δu + g(u) = f(x,t), (x,t) ∈ ^Dτ, onde ^Dτ = &CUP;t∈(τ,+ ∞) Ot X . Dizemos que domínio Dτ possui fronteira móvel se admitirmos que a fronteira Γt de de Ot varia em relação a t. Nossa contribuição é dividida em três etapas. 1  Provamos que o problema munido da condição de fronteira de Dirichlet é bem posto no sentido de Hadamard (existência global, unicidade e dependência contínua dos dados) para soluções fortes e fracas. Nessa etapa utilizamos um método clássico que transforma o domínio dependente de t em um domínio fixo. Como consequência observamos que o sistema é essencialmente não autônomo. 2  Buscamos uma teoria de sistemas dinâmicos não autônomos para estudar o operador solução do problema como um processo U(t; τ) : Xτ → Xτ, t≥ τ, definido em espaços de fase Xt = H01(Ot) × L2(Ot) que são dependentes do tempo t. 3  No contexto da dinâmica de longo prazo encontramos hipóteses para garantir que o sistema dinâmico associado ao problema de ondas em domínios de fronteira móvel possui um atrator pullback. Basicamente admitimos que o domínio é crescente e \"timelike\". Salientamos que o nosso trabalho é o primeiro que estuda tais equações de ondas sob o ponto de vista de sistemas dinâmicos nãoautônomos. Para equações parabólicas, resultados no mesmo contexto foram obtidos anteriormente por Kloeden, MarínRubio e Real [JDE 244 (2008) 20622090] e Kloeden, Real e Sun [JDE 246 (2009) 47024730]. Entretanto o nosso problema á hiperbólico e nã possui a regularidade das equações parabólicas. / In this work we study a weakly dissipative wave equation defined in domains with moving boundary ∂2u/∂t2/ + η∂u/∂t  Δu + g(u) = f(x,t), (x,t) ∈ Dτ, where D&tau> = &CUP;t∈(τ,+ ∞) Ot X . We says that a domain D&tau has moving boundary if the boundary &Gama;t of Ot varies with respect to t. Our contribution is threefold. 1  We prove that the wave equation equipped with Dirichlet boundary condition is wellposed in the sense of Hadamard (global existence, uniqueness and continuous dependence with respect to data) for weak and strong solutions. This is done by using a classical argument that transforms the time dependent domain in a fixed domain. As a consequence we see that the problem is essentially nonautonomous. 2 We find a theory of nonautonomous dynamical systems in order to study the solution operator as a process U(t; τ) : Xτ → Xsub>t, t≥τ, defined in time dependent phase spaces Xt = H01 (Ot) × L2.(Ot. 3  In the context of longtime behavior of solutions we find suitable conditions to guarantee the existence of a pullback attractor. Roughly speaking, we assume the domain Q is expanding and timelike. We emphasize that our work is the first one that consider wave equations in noncylindrical domains as nonautonomous dynamical systems. With respect to parabolic equations, similar results were early obtained by Kloeden, MarínRubio and Real [JDE 244 (2008) 20622090] and Kloeden, Real and Sun [JDE 246 (2009) 47024730]. However our problem is hyperbolic and does not enjoy regularity properties as the parabolic ones.

35 
The Nonisospectral and variable coefficient Kortewegde Vries equation.January 1992 (has links)
by Li Kam Shun. / Thesis (M.Phil.)Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaf 65). / Chapter CHAPTER 1  Soliton Solutions of the Nonisospectral and Variable Coefficient Kortewegde Vries Equation / Chapter §1.1  Introduction  p.4 / Chapter §1.2  Inverse Scattering  p.6 / Chapter §1.3  NSoliton Solution  p.11 / Chapter §1.4  OneSoliton Solutions  p.15 / Chapter §1.5  TwoSoliton Solutions  p.18 / Chapter §1.6  Oscillating and Asymptotically Standing Solitons  p.23 / Chapter CHAPTER 2  Asymptotic Behaviour of Nonsoliton Solutions of the Nonisospectral and Variable Coefficient Kortewegde Vries Equation / Chapter §2.1  Introduction  p.31 / Chapter §2.2  Main Results  p.36 / Chapter §2.3  Lemmas  p.39 / Chapter §2.4  Proof of the Main Results  p.59 / References  p.65

36 
Properties of quasinormal modes in open systems.January 1995 (has links)
by Tong Shiu Sing Dominic. / Parallel title in Chinese characters. / Thesis (Ph.D.)Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 236241). / Acknowledgements  p.iv / Abstract  p.v / Chapter 1  Open Systems and Quasinormal Modes  p.1 / Chapter 1.1  Introduction  p.1 / Chapter 1.1.1  NonHermitian Systems  p.1 / Chapter 1.1.2  Optical Cavities as Open Systems  p.3 / Chapter 1.1.3  Outline of this Thesis  p.6 / Chapter 1.2  Simple Models of Open Systems  p.10 / Chapter 1.3  Contributions of the Author  p.14 / Chapter 2  Completeness and Orthogonality  p.16 / Chapter 2.1  Introduction  p.16 / Chapter 2.2  Green's Function of the Open System  p.19 / Chapter 2.3  High Frequency Behaviour of the Green's Function  p.24 / Chapter 2.4  Completeness of Quasinormal Modes  p.29 / Chapter 2. 5  Method of Projection  p.31 / Chapter 2.5.1  Problems with the Usual Method of Projection  p.31 / Chapter 2.5.2  Modified Method of Projection  p.33 / Chapter 2.6  Uniqueness of Representation  p.38 / Chapter 2.7  Definition of Inner Product and QuasiStationary States  p.39 / Chapter 2.7.1  Orthogonal Relation of Quasinormal Modes  p.39 / Chapter 2.7.2  Definition of Hilbert Space and State Vectors  p.41 / Chapter 2.8  Hermitian Limits  p.43 / Chapter 2.9  Numerical Examples  p.45 / Chapter 3  TimeIndependent Perturbation  p.58 / Chapter 3.1  Introduction  p.58 / Chapter 3.2  Formalism  p.60 / Chapter 3.2.1  Expansion of the Perturbed QuasiStationary States  p.60 / Chapter 3.2.2  Formal Solution  p.62 / Chapter 3.2.3  Perturbative Series  p.66 / Chapter 3.3  Diagrammatic Perturbation  p.70 / Chapter 3.3.1  Series Representation of the Green's Function  p.70 / Chapter 3.3.2  Eigenfrequencies  p.73 / Chapter 3.3.3  Eigenfunctions  p.75 / Chapter 3.4  Numerical Examples  p.77 / Chapter 4  Method of Diagonization  p.81 / Chapter 4.1  Introduction  p.81 / Chapter 4.2  Formalism  p.82 / Chapter 4.2.1  Matrix Equation with Nonunique Solution  p.82 / Chapter 4.2.2  Matrix Equation with a Unique Solution  p.88 / Chapter 4.3  Numerical Examples  p.91 / Chapter 5  Evolution of the Open System  p.97 / Chapter 5.1  Introduction  p.97 / Chapter 5.2  Evolution with Arbitrary Initial Conditions  p.99 / Chapter 5.3  Evolution with the Outgoing Plane Wave Condition  p.106 / Chapter 5.3.1  Evolution Inside the Cavity  p.106 / Chapter 5.3.2  Evolution Outside the Cavity  p.110 / Chapter 5.4  Physical Implications  p.112 / Chapter 6  TimeDependent Perturbation  p.114 / Chapter 6.1  Introduction  p.114 / Chapter 6.2  Inhomogeneous Wave Equation  p.117 / Chapter 6.3  Perturbative Scheme  p.120 / Chapter 6.4  Energy Changes due to the Perturbation  p.128 / Chapter 6.5  Numerical Examples  p.131 / Chapter 7  Adiabatic Approximation  p.150 / Chapter 7.1  Introduction  p.150 / Chapter 7.2  The Effect of a Varying Refractive Index  p.153 / Chapter 7.3  Adiabatic Expansion  p.156 / Chapter 7.4  Numerical Examples  p.167 / Chapter 8  Generalization of the Formalism  p.176 / Chapter 8. 1  Introduction  p.176 / Chapter 8.2  Generalization of the Orthogonal Relation  p.180 / Chapter 8.3  Evolution with the Outgong Wave Condition  p.183 / Chapter 8.4  Uniform Convergence of the Series Representation  p.193 / Chapter 8.5  Uniqueness of Representation  p.200 / Chapter 8.6  Generalization of Standard Calculations  p.202 / Chapter 8.6.1  TimeIndependent Perturbation  p.203 / Chapter 8.6.2  Method of Diagonization  p.206 / Chapter 8.6.3  Remarks on Dynamical Calculations  p.208 / Appendix A  p.209 / Appendix B  p.213 / Appendix C  p.225 / Appendix D  p.231 / Appendix E  p.234 / References  p.236

37 
Waves in a cavity with an oscillating boundary =: 振動空腔中的波動. / 振動空腔中的波動 / Waves in a cavity with an oscillating boundary =: Zhen dong kong qiang zhong de bo dong. / Zhen dong kong qiang zhong de bo dongJanuary 1999 (has links)
by Ho Yum Bun. / Thesis (M.Phil.)Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 9394). / Text in English; abstracts in English and Chinese. / by Ho Yum Bun. / List of Figures  p.3 / Abstract  p.9 / Chinese Abstract  p.10 / Acknowledgement  p.11 / Chapter 1  Introduction  p.12 / Chapter 1.1  Motivation  p.12 / Chapter 1.2  What is Sonoluminescence?  p.13 / Chapter 1.3  The Main Task of this Project  p.13 / Chapter 1.4  Organization of this Thesis  p.13 / Chapter 2  Reviews on Onedimensional Dynamical Cavity Problem  p.15 / Chapter 2.1  Introduction  p.15 / Chapter 2.2  Formulation  p.15 / Chapter 2.3  Moore's R Function Method  p.18 / Chapter 2.4  Mode Expansion Method  p.19 / Chapter 2.5  Transformation method  p.20 / Chapter 26  Summary  p.21 / Chapter 3  Numerical Results For Onedimensional Dynamical Cavity Prob lem  p.22 / Chapter 3.1  Introduction  p.22 / Chapter 3.2  Evolution of a Cavity System  p.23 / Chapter 3.3  Motion of the Moving Mirror  p.23 / Chapter 3.4  R(z) Function  p.24 / Chapter 3.4.1  Construction of R(z) Function  p.24 / Chapter 3.4.2  Numerical R(z) Function  p.27 / Chapter 3.5  Results  p.27 / Chapter 3.5.1  Results with Moore's R(z) Function Method  p.27 / Chapter 3.5.2  Results with the Mode Expansion Method  p.29 / Chapter 3.5.3  Results with the Transformation Method  p.36 / Chapter 3.6  Summary  p.36 / Chapter 4  Spherical Dynamical Cavity Problem  p.37 / Chapter 4.1  Introduction  p.37 / Chapter 4.2  Formulation  p.37 / Chapter 4.3  Motion of a Moving Spherical Mirror  p.39 / Chapter 4.4  Summary  p.40 / Chapter 5  The G(z) Function Method  p.41 / Chapter 5.1  Introduction  p.41 / Chapter 5.2  G(z) Function  p.42 / Chapter 5.2.1  Ideas of Deriving the G(z) Function  p.42 / Chapter 5.2.2  Formalism  p.42 / Chapter 5.2.3  Initial G(z) Function  p.45 / Chapter 5.3  Construction of the G(z) Function  p.46 / Chapter 5.3.1  Case I : l=0  p.46 / Chapter 5.3.2  Case II : l > 0  p.49 / Chapter 5.4  Asymptotic Series Solution of G(z)  p.50 / Chapter 5.5  Application to Resonant Mirror Motion  p.52 / Chapter 5.6  Regularization of G(z)  p.58 / Chapter 5.7  Behaviors of the Fields  p.58 / Chapter 5.7.1  z vs tf Graph  p.61 / Chapter 5.7.2  Case 1: l= 0  p.61 / Chapter 5.7.3  "Case2: l= 1,2"  p.62 / Chapter 5.7.4  Case 3: l= 3  p.73 / Chapter 5.7.5  Section Summary  p.73 / Chapter 5.8  Summary  p.73 / Chapter 6  Threedimensional Mode Expansion Method and Transforma tion Method  p.75 / Chapter 6.1  Introduction  p.75 / Chapter 6.2  Mode Expansion Method  p.75 / Chapter 6.2.1  Formalism  p.75 / Chapter 6.2.2  Application of Floquet's Theory  p.78 / Chapter 6.2.3  Results  p.80 / Chapter 6.3  The Transformation Method  p.80 / Chapter 6.3.1  The Method  p.80 / Chapter 6.3.2  Numerical Schemes  p.86 / Chapter 6.3.3  Results  p.89 / Chapter 6.4  Summary  p.89 / Chapter 7  Conclusion  p.90 / Chapter 7.1  The Onedimensional Dynamical Cavity Problem  p.90 / Chapter 7.2  The Dynamical Spherical Cavity Problem  p.91 / Chapter 7.3  Numerical Methods  p.91 / Chapter 7.4  Further Investigation  p.92 / Bibliography  p.93

38 
A survey on linearized method for inverse wave equations.January 2012 (has links)
在本文中， 我們將主要討論一種在求解一類波動方程反問題中很有價值的數值方法：線性化方法。 / 在介紹上述的數值方法之前， 我們將首先討論波動方程的一些重要的特質，主要包括四類典型的波動方程模型，方程的基本解和一般解，以及波動方程解的性質。 / 接下來，在本文的第二部分中，我們會首先介紹所求解的模型以及其反問題。此反問題主要研究求解波動方程[附圖]中的系數c. 線性化方法的主要思想在於將速度c分解成兩部分：c₁ 和c₂ ，並且滿足關系式：[附圖]，其中c₁ 是一個小的擾動量。另一方面，上述波動方程的解u 可以被線性表示：u = u₀ + u₁ ，其中u₀ 和u₁ 分別是一維問題和二維問題的解。相應的，我們將運用有限差分方法和傅利葉變換方法求解上述一維問題和二維問題，從而分別求解c₁ 和c₂ ，最終求解得到係數c. 在本文的最後，我們將進行一些數值試驗，從而驗證此線性化方法的有效性和可靠性。 / In this thesis, we will discuss a numerical method of enormous value, a linearized method for solving a certain kind of inverse wave equations. / Before the introduction of the abovementioned method, we shall discuss some important features of the wave equations in the first part of the thesis, consisting of four typical mathematical models of wave equations, there fundamental solutions, general solutions and the properties of those general solutions. / Next, we shall present the model and its inverse problem of recovering the coefficient c representing the propagation velocity of wave from the wave equation [with mathematic formula] The linearized method aims at dividing the velocity c into two parts, c₀ and c₁, which satisfying the relation [with mathematic formula], where c₁ is a tiny perturbation. On the other hand, the solution u can be represented in the linear form, u = u₀ + u₁, where u₀ and u₁ are the solutions to onedimensional problem and two dimensional problem respectively. Accordingly, we can use the numerical methods, finite difference method and Fourier transform method to solve the onedimensional forward problem and twodimensional inverse problem respectively, thus we can get c₀ and c₁, a step before we recover the velocity c. In the numerical experiments, we shall test the proposed linearized numerical method for some special examples and demonstrate the effectiveness and robustness of the method. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Xu, Xinyi. / Thesis (M.Phil.)Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 6465). / Abstracts also in Chinese. / Chapter 1  Fundamental aspects of wave equations  p.6 / Chapter 1.1  Introduction  p.6 / Chapter 1.1.1  Four important wave equations  p.6 / Chapter 1.1.2  General form of wave equations  p.10 / Chapter 1.2  Fundamental solutions  p.11 / Chapter 1.2.1  Fourier transform  p.11 / Chapter 1.2.2  Fundamental solution in threedimensional space  p.14 / Chapter 1.2.3  Fundamental solution in twodimensional space  p.16 / Chapter 1.3  General solution  p.19 / Chapter 1.3.1  Onedimensional wave equations  p.19 / Chapter 1.3.2  Two and three dimensional wave equations  p.26 / Chapter 1.3.3  n dimensional case  p.28 / Chapter 1.4  Properties of solutions to wave equation  p.31 / Chapter 1.4.1  Properties of Kirchhoff’s solutions  p.31 / Chapter 1.4.2  Properties of Poisson’s solutions  p.33 / Chapter 1.4.3  Decay of the solutions to wave equation  p.34 / Chapter 2  Linearized method for wave equations  p.36 / Chapter 2.1  Introduction  p.36 / Chapter 2.1.1  Background  p.36 / Chapter 2.1.2  Forward and inverse problem  p.38 / Chapter 2.2  Basic ideas of Linearized Method  p.39 / Chapter 2.3  Theoretical analysis on linearized method  p.41 / Chapter 2.3.1  Onedimensional forward problem  p.42 / Chapter 2.3.2  Twodimensional forward problem  p.43 / Chapter 2.3.3  Existence and uniqueness of solutions to the inverse problem  p.45 / Chapter 2.4  Numerical analysis on linearized method  p.45 / Chapter 2.4.1  Discrete analog of the inverse problem  p.46 / Chapter 2.4.2  Fourier transform  p.48 / Chapter 2.4.3  Direct methods for inverse and forward problems  p.52 / Chapter 2.5  Numerical Simulation  p.54 / Chapter 2.5.1  Special Case  p.54 / Chapter 2.5.2  General Case  p.59 / Chapter 3  Conclusion  p.63 / Bibliography  p.64

39 
On a shallow water equation.January 2001 (has links)
Zhou Yong. / Thesis (M.Phil.)Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 5153). / Abstracts in English and Chinese. / Acknowledgments  p.i / Abstract  p.ii / Chapter 1  Introduction  p.2 / Chapter 2  Preliminaries  p.10 / Chapter 3  Periodic Case  p.22 / Chapter 4  Nonperiodic Case  p.35 / Bibliography  p.51

40 
Approximations hyperboliques des équations de NavierStokes / Hyperbolic approximations of the NavierStokes equationsHachicha, Imène 15 November 2013 (has links)
Dans cette thèse, nous nous intéressons à deux approximations hyperboliques des équations de NavierStokes incompressibles en dimensions 2 et 3 d'espace. Dans un premier temps, on considère une perturbation hyperbolique de l'équation de la chaleur, introduite par Cattaneo en 1949, pour remédier au paradoxe de la propagation instantanée de cette équation. En 2004, Brenier, Natalini et Puel remarquent que la même perturbation, qui consiste à rajouter ε∂tt à l'équation, intervient en relaxant les équations d'Euler. En dimension 2, les auteurs montrent que, pour des sonnées régulières et sous certaines hypothèses de petitesse, la solution globale de la perturbation converge vers l'unique solution globale de (NS). En 2007, Paicu et Raugel améliorent les résultats de [BNP] en étendant la théorie à la dimension 3 et en prenant des données beaucoup moins régulières. Nous avons obtenu des résultats de convergence, avec données de régularité quasicritique, qui complètent et prolongent ceux de [BNP] et [PR]. La seconde approximation que l'on considère est un nouveau modèle hyperbolique à vitesse de propagation finie. Ce modèle est obtenu en pénalisant la contrainte d'incompressibilité dans la perturbation de Cattaneo. Nous démontrons que les résultats d'existence globale et de convergence du précédent modèle sont encore vérifiés pour celuici. / In this work, we are interested in two hyperbolic approximations of the 2D and 3D NavierStokes equations. The first model we consider comes from Cattaneo's hyperbolic perturbation of the heat equation to obtain a finite speed of propagation equation. Brenier, Natalini and Puel studied the same perturbation as a relaxed version of the 2D Euler equations and proved that the solution to this relaxation converges towards the solution to (NS) with smooth data, provided some smallness assumptions. Later, Paicu and Raugel improved their results, extending the theory to the 3D setting and requiring significantly less regular data. Following [BNP] and [PR], we prove global existence and convergence results with quasicritical regularity assumptions on the initial data. In the second part, we introduce a new hyperbolic model with finite speed of propagation, obtained by penalizing the incompressibility constraint in Cattaneo's perturbation. We prove that the same global existence and convergence results hold for this model as well as for the first one.

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