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Cette thèse a pour objet l’étude des solutions à l’équation des vagues en régime dit toit rigide lorsque l’amplitude des vagues tend vers zéro. Plus précisément, l’équation des vagues modélise le mouvement d’un fluide à surface libre borné en dessous par un fond fixe. Les équations dépendent de plusieurs paramètres physiques, notamment du rapport epsilon entre l’amplitude des vagues et la profondeur. Le modèle asymptotique toit rigide consiste à changer l’échelle de temps d’un rapport epsilon, puis de faire tendre ce paramètre, et donc l’amplitude des vagues, vers zéro. L’étude mathématique de cette limite correspond à un problème de perturbation singulière d’une équation dispersive. Dans cette thèse, on commence par utiliser des outils de résolution d’équations aux dérivées partielles de type hyperbolique pour démontrer un résultat d’existence locale pour l’équation des vagues en temps long. Ceci est suivi par un résultat de dispersion sur l’équation des vagues, utilisant des techniques de type phase stationnaire et décomposition de Paley-Littlewood pour l’étude des intégrales oscillantes. Enfin, la dernière partie de la thèse utilise les résultats obtenus ci-dessus pour étudier un défaut de compacité dans la convergence faible (mais non forte) des solutions de l’équation des vagues lorsque l’amplitude tend vers 0. / In this thesis, we study the behavior of the solutions of the Water-Waves equations in the rigid lid regime as the amplitude of the waves goes to zero. More precisely, the Water-Waves equations investigate the dynamic of a free surface fluid, bounded from below by a fixed bottom. The equations depends on many physical parameters, as the ratio epsilon between the wave amplitude and the deepness of the water. The rigid lid model consists in scaling the time by an epsilon factor and taking the limit epsilon goes to zero, simulating a situation where the amplitude of the waves goes to zero. The mathematical study of this limit correspond to a singular perturbation problem of a dispersive equation. In this thesis, we first use classical tools of hyperbolics equations to prove a long time existence result for the Water-Waves equations. We then prove a dispersion result for these equations, using stationary phase methods and Paley-Littlewood decomposition. We then combine these results to highlight the lack of compactness in the weak (but non strong) convergence of the solutions of the Water-Waves equations as the amplitude goes to zero.

This dissertation revolves around various mathematical aspects of nonlinear wave motion in viscoelasticity and free surface flows. The introduction is devoted to the physical derivation of the stress-strain constitutive relations from the first principles of Newtonian mechanics and is accessible to a broad audience. This derivation is not necessary for the analysis carried out in the rest of the thesis, however, is very useful to connect the different-looking partial differential equations (PDEs) investigated in each subsequent chapter. In the second chapter we investigate a multi-dimensional scalar wave equation with memory for the motion of a viscoelastic material described by the most general linear constitutive law between the stress, strain and their rates of change. The model equation is rewritten as a system of first-order linear PDEs with relaxation and the well-posedness of the Cauchy problem is established. In the third chapter we consider the Euler equations describing the evolution of a perfect, incompressible, irrotational fluid with a free surface. We focus on the Hamiltonian description of surface waves and obtain a recursion relation which allows to expand the Hamiltonian in powers of wave steepness valid to arbitrary order and in any dimension. In the case of pure gravity waves in a two-dimensional flow there exists a symplectic coordinate transformation that eliminates all cubic terms and puts the Hamiltonian in a Birkhoff normal form up to order four due to the unexpected cancellation of the coefficients of all fourth order non-generic resonant terms. We explain how to obtain higher-order vanishing coefficients. Finally, using the properties of the expansion kernels we derive a set of nonlinear evolution equations for unidirectional gravity waves propagating on the surface of an ideal fluid of infinite depth and show that they admit an exact traveling wave solution expressed in terms of Lambert’s W-function. The only other known deep fluid surface waves are the Gerstner and Stokes waves, with the former being exact but rotational whereas the latter being approximate and irrotational. Our results yield a wave that is both exact and irrotational, however, unlike Gerstner and Stokes waves, it is complex-valued.

Tato diplomová práce se zabývá matematickým modelováním vodních vln v blízkosti pobřeží pomocí parciálních diferenciálních rovnic. Cílem této práce je formulace pohybových rovnic a jejich následné numerické řešení s grafickou interpretací dosažených výsledků.

The behaviour of free-surface gravity waves on small Froude number fluid flow past some obstacle cannot be determined using ordinary asymptotic power series methods, as the amplitude of the waves is exponentially small. An exponential asymptotic method is used by Chapman and Vanden-Broeck (2006) to consider the problem of two-dimensional, steady flow past a submerged obstacle in the small Froude number limit, finding that a steady downstream wavetrainis switched on rapidly across a curve known as a Stokes line. Here, equivalent wavetrains on three-dimensional and unsteady flow configurations are considered, and Stokes switching causedby the interaction between exponentially small free-surface components is shown to play an important role in both cases. The behaviour of free-surface gravity waves is introduced by considering the problem of steady free-surface flow due to a line source. A steady wavetrain is shown to exist in the far field, and the behaviour of these waves is compared to existing numerical results. The problem of unsteady flow over a step is subsequently investigated, with the flow behaviour formulated in terms of Lagrangian coordinates so that the position of the free surface is fixed. Initially, the problem is linearized in the step-height, and the steady wavetrain is shown to spread downstream over time. The position of the wavefront is determined by considering the full Stokes structure present in the problem. The equivalent fully-nonlinear problem is then considered, with the position of the Stokes lines, and hence the wavefront, being determined numerically. Finally, linearized three-dimensional free-surface flow past an obstacle is considered in both the steady and unsteady case. The surface is shown to contain downstream longitudinal and transverse waves. These waves are shown to propagate downstream in the unsteady case, with the position of the wavefront again determined by considering the full Stokes structure of the problem.

Cet ouvrage présente de nouveaux aspects de l’analogie entre temps et espace pour les ondes, à travers des concepts de contrôle temporel de la propagation des ondes, qui peuvent être interprétés comme la transposition au domaine temporel d’opérations standard du domaine spatial. Toute l’étendue de cette analogie est étudiée dans le cadre deux phénomènes ondulatoires bien connus (dans leurs versions spatiales), et dont nous montrons qu’ils sont étroitement liés : la réflexion des ondes et leur transformation par un cristal. En particulier, il est démontré expérimentalement que l’analogue temporel de la réflexion par un miroir génère une onde retournée temporellement, c’est-à-dire dont l’évolution temporelle est inversée. Une variante monofréquentielle de ce miroir temporel instantané, qui possède des liens étroits avec le concept de conjugaison de phase utilisé pour le retournement temporel d'ondes monochromatiques en optique, est également étudiée. Nous montrons que la modulation temporelle périodique du milieu mise en jeu dans ces expériences constitue l'équivalent temporel d'un cristal pour les ondes et étudions les propriétés générales des ondes dans ces milieux. Nous cherchons naturellement à sonder les limites de ces analogies spatiotemporelles, dont, de toute évidence, le principe de causalité est un élément majeur. Les phénomènes étudiés sont généraux et communs à toutes les ondes classiques, gouvernées en général par l’équation de d’Alembert ou par d’autres équations similaires. Les ondes à la surface d’un liquide sont utilisées comme système ondulatoire modèle dans nos expériences / This work presents new aspects of the analogy between time and space in wave phenomena, through new concepts of time control of wave propagation which can be interpreted as the transposition onto the time domain of standard spatial operations. The depth of this analogy is thoroughly studied in the framework of two well-known wave phenomena: reflection of waves on a mirror and their transformation by a crystal. More precisely, we experimentally demonstrate that the time analog of reflection by a mirror generates a time reverse wave that is whose time evolution is inverted. A monofrequency variant of this instantaneous time mirror, which has strong connections to the concept of wave phase conjugation used for time reversing monochromatic waves in optics, is also studied. We show that the periodic time modulation of the medium involved in the latter experiments constitutes the time equivalent of a crystal for waves and we study the general properties of waves in such media. We address of course the issue of the limits of theses space time analogies, of which, the principle of causality is evidently a major constituent. The phenomena studied here are general and apply to all classical waves (usually governed by d’Alembert’s equation or by similar ones). Waves at the surface of a liquid are used as a model wave system in our experiments / Die Universalitat der klassischen Wellenphanomenen lasst sich weitgehend durch die d’Alembertsche Struktur der Wellengleichungen beschreiben. In letzterer spielen die Zeit- und Raumvariabeln eine ahnliche Rolle. Wir betrachten in diesem Werk diese bekannte Analogie unter einem neuen Licht, indem wir neue Konzepte der Zeitkontrolle von der Wellenpropagation introduzieren, die als Transposition zum Bereich der Zeit von standarden Wellenphanomen des Raumes interpretiert werden konnen, wobei die raumliche Variation der Eigenschaften des Mediums, die sie bestimmen, durch eine zeitliche ersetzt wird. Wir bringen den experimentellen Beweis der Relevanz dieses Ansatzes, wobei wir die Wellen an der Oberflache einer Flussigkeit als Modelwellensystem verwenden und zeigen durch theoretische Erkenntnisse, dass er auf alle Wellensysteme generalisiert werden kann, die durch eine Wellengleichung beschrieben wird, deren 10 Struktur die der d’Alembertschen Gleichung ahnelt. Die ganze Reichweite dieser Analogie wird im Rahmen zweier langst bekannten Wellenphanomenen (in deren raumlichen Version), von denen wir zeigen, dass sie eng verbunden sind : die Spiegelung einer Welle und deren Umwandlung durch einen Kristal. Das Equivalent eines Spiegels fuhrt zur augenblicklichen Erscheinung aus dem gesammten Raum einer zeitumgekehrten Welle, das heist einer Wellenform, derer zeitliche Entwicklung im Vergleich zu der originellen Welle umgekehrt ist. Eine Einzelfrequenzvariante dieses augenblicklichen Zeitspiegels wird auch vorgestellt, die enge Verbindungen mit dem fur die Zeitumkehr monochromatischer Wellen angewendeten Konzept der optischen "Phase Conjugation" (Phasenkonjugierung) besitzt. Wir zeigen, dass die dazugehorige periodische Zeitmodulation des Mediums das zeitliche Equivalent eines Kristals fur die Welle bildet, und wir studieren die generellen Eigenschaften der Wellen in solchen Medien. Wir versuchen, die Grenzen dieser raumzeitlichen Analogien zu erkundigen, in derer Bestimmung selbstverstandlich das Kausalitatsprinzip eine masgebliche Rolle spielt, und die interessanten Unterschiede zwischen den vorgestellten Konzepten und deren raumlichen Pendants offenbaren / La universalidad de los fenomenos ondulatorios clasicos es ampliamente descrita por la estructura de la ecuacion de d’Alembert. En esta ecuacion, las variables espaciales y temporales desempenan un papel similar. En esa obra, revisitamos esta analogia bien conocida a traves de nuevos conceptos de control temporal de la propagacion de las ondas, quienes pueden ser interpretados como transposiciones de fenomenos espaciales estandartes al ambito temporal, en los cuales las variaciones espaciales de las propiedades del medio son reemplazadas por las correspondientes variaciones temporales. Hacemos la prueba, usando ondas a la superficie de un liquido, de la relevancia de este enfoque y mostramos teoreticamente su generalizacion a todo tipo de onda clasica, es decir gobernada por una ecuacion cuya estructura es similar a la de d’Alembert. Toda la extension de esta analogia queda estudiada en el cuadro de los dos fenomenos clasicos (en sus versiones espaciales) que son la reflexion de una onda sobre un espejo y su trasformacion en un cristal. Ademas, mostramos que los dos son intimamente relacionados. El equivalente temporal de un espejo produje instantaneamente desde el medio entero una onda retornada en el tiempo, es decir cuya evolucion temporal es invertida comparado a la onda inicial. Una variante monofrecuencial de este espejo queda estudiada tambien. Posee estrechos vinculos con el concepto de Phase Conjugation (conjugacion de fase), usado en Optica para hacer retornamiento temporal de ondas monocromaticas. Mostramos que la modulacion temporal implicada constituye el equivalente de un cristal para las ondas et estudiamos las caracteristicas generales de ondas en estos medios. Sondeamos los limites de esas analogias espaciotemporales de cuyos obviamente el principio de causalidad es un elemento mayor y que revelan diferencias interesantes entre los conceptos presentados y sus equivalentes espaciales / L’universalita dei fenomeni ondulatori classici e in larga misura descritta dalla struttura dell’equazione di d’Alembert. In quest’equazione, le variabili spaziali e temporali svolgono ruoli analoghi. Nell’opera seguente rivisitiamo questa ben nota analogia introducendo nuovi concetti sul controllo temporale della propagazione delle onde. Questi concetti possono essere interpretati come trasposizione di fenomeni ondulatori spaziali standard nell’ambito temporale, sostituendo le variazioni spaziali delle proprieta del mezzo con le variazioni temporali corrispondenti. Usando delle onde sulla superficie di un liquido come modello fisico, facciamo fede della rilevanza di quell’approccio e mostriamo teoricamente la generalizzazione a tutti i tipi di onde classiche, governate da equazioni simili a quella di d’Alembert. Questa analogia viene studiata nell’ambito di due fenomeni ondulatori ben noti (nella loro versione spaziale) : la riflessione delle onde generata da un specchio e la loro trasformazione generata da un cristallo. Mostriamo inoltre che i due concetti sono intimamente vincolati. L’equivalente temporale di un specchio porta alla generazione in tutto lo spazio di un’onda restituita nel tempo, vale a dire un’onda di cui l’evoluzione temporale e invertita in relazione all’onda originale. In questa tesi viene presentata anche una variante monofrequenziale di questo specchio temporale istantaneo che possiede legami stretti con il concetto di coniugazione di fase usato in ottica per invertire nel tempo onde monocromatiche. Mostriamo in questo lavoro che la modulazione temporale periodica del mezzo in questione costituisce, per le onde, l’equivalente temporale di un cristallo e ne studiamo le proprieta generali. In questa tesi, cerchiamo di sondare i limiti dell’analogia spaziotemporale, di cui il principio di causalita ne e l’essenziale e che rivelano differenze interessanti tra i concetti presentati e i loro equivalenti spaziali

This paper examines the regional longshore sediment transport pattern of the seaward coast of the United States and Gulf of Mexico from northern Maine to Tampa Bay, Florida. From previous studies it is known that along the coast there are variations in direction of sediment transport known as nodal zones as well as variations in sediment transport rate. Wave Information Study (WIS) hindcast data for the interval 1976 through 1995 (United States Army Corps of Engineers, 2003) provide a spatially continuous model of the regional longshore current directions in the study area. In chapter one, all available published field studies of longshore current direction and sediment transport directions and rates are compiled to create a description of the direction and, whenever possible, magnitude of longshore transport. A detailed compilation of regional and local published studies are provided in tables. An interpretation of sediment transport rates and directions is provided in eight regional maps of the study area. In chapter two the results of the literature compilations are compared with gross and net potential sediment transport directions and rates modeled using WIS hindcast data. The WIS deep-water wave characteristics are used to predict the directions and rate of longshore sediment transport at local outer coast positions using the method of Ashton et al. (2003a). The WIS-derived transport directions, including nodal zones, generally agree with the published field studies, although there are a few local inconsistencies particularly near inlets, shoals and irregular bathymetry. Trends in longshore transport rates, such as increases and decreases in gross transport rates are well represented by the WIS-derived potential transport rates. The discrepencies between the published field studies and WIS results are apparently primarily due to assumptions in the WIS model, such as assuming shore-parallel bathymetric contours.

A novel video observational method for the space-time stereoscopic reconstruction of dynamic surfaces representable as graphs, such as ocean waves, is developed. Variational optimization algorithms combining image processing, computer vision and partial differential equations are designed to address the problem of the recovery of the shape of an object's surface from sequences of synchronized multi-view images. Several theoretical and numerical paths are discussed to solve the problem. The variational stereo method developed in this thesis has several advantages over existing 3-D reconstruction algorithms. Our method follows a top-down approach or object-centered philosophy in which an explicit model of the target object in the scene is devised and then related to image measurements. The key advantages of our method are the coherence (smoothness) of the reconstructed surface caused by a coherent object-centered design, the robustness to noise due to a generative model of the observed images, the ability to handle surfaces with smooth textures where other methods typically fail to provide a solution, and the higher resolution achieved due to a suitable graph representation of the object's surface. The method provides competitive results with respect to existing variational reconstruction algorithms. However, our method is based upon a simplified but complete physical model of the scene that allows the reconstruction process to include physical properties of the object's surface that are otherwise difficult to take into account with existing reconstruction algorithms. Some initial steps are taken toward incorporating the physics of ocean waves in the stereo reconstruction process. The developed method is applied to empirical data of ocean waves collected at an off-shore oceanographic platform located off the coast of Crimea, Ukraine. An empirically-based physical model founded upon current ocean engineering standards is used to validate the results. Our findings suggest that this remote sensing observational method has a broad impact on off-shore engineering to enrich the understanding of sea states, enabling improved design of off-shore structures. The exploration of ways to incorporate dynamical properties, such as the wave equation, in the reconstruction process is discussed for future research.

The instabilities of shallow shear flows are analyzed to study exchanges processes across shear flows in inland and coastal waters, coastal and ocean currents, and winds across the thermal-and-moisture fronts. These shear flows observed in nature are driven by gravity and governed by the shallow water equations (SWE). A highly accurate, and robust, computational scheme has been developed to solve these SWE. Time integration of the SWE was carried out using the fourth-order Runge-Kutta scheme. A third-order upwind bias finite difference approximation known as QUICK (Quadratic Upstream Interpolation of Convective Kinematics) was employed for the spatial discretization. The numerical oscillations were controlled using flux limiters for Total Variation Diminishing (TVD). Direct numerical simulations (DNS) were conducted for the base flow with the TANH velocity profile, and the base flow in the form of a jet with the SECH velocity profile. The depth across the base flows was selected for the' balance of the driving forces. In the rotating flow simulation, the Coriolis force in the lateral direction was perfectly in balance with the pressure gradient across the shear flow during the simulation. The development of instabilities in the shear flows was considered for a range of convective Froude number, friction number, and Rossby number. The DNS of the SWE has produced linear results that are consistent with classical stability analyses based on the normal mode approach, and new results that had not been determined by the classical method. The formation of eddies, and the generation of shocklets subsequent to the linear instabilities were computed as part of the DNS. Without modelling the small scales, the simulation was able to produce the correct turbulent spreading rate in agreement with the experimental observations. The simulations have identified radiation damping, in addition to friction damping, as a primary factor of influence on the instability of the shear flows admissible to waves. A convective Froude number correlated the energy lost due to radiation damping. The friction number determined the energy lost due to friction. A significant fraction of available energy produced by the shear flow is lost due the radiation of waves at high convective Froude number. This radiation of gravity waves in shallow gravity-stratified shear flow, and its dependence on the convective Froude number, is shown to be analogous to the Mach-number effect in compressible flow. Furthermore, and most significantly, is the discovery from the simulation the crucial role of the radiation damping in the development of shear flows in the rotating earth. Rings and eddies were produced by the rotating-flow simulations in a range of Rossby numbers, as they were observed in the Gulf Stream of the Atlantic, Jet Stream in the atmosphere, and various fronts across currents in coastal waters.

Investigations into the characteristics and underlying mechanism of the circulation of water near the shore are reported. The two main types of circulation, one a cellular system resulting from Haves propagated nearly normally to the shore, and the other an essentially alongshore flow associated with oblique waves, are treated separately. The cellular circulation studies were made in the field at Virginia Beach and more extensively in wave tanks. From the field experiments data were collected about the dimensions of the cells, the way in which the Hater circulated, the rate of exchance of surf zone water and the extent of recycling. A method for measuring the changes in the mean sea level over intervals of time greater than those of the wave periods, was developed. The model experiments carried out in uniform wave tanks showed that the cellular circulations could be well simulated. Measurements were made of the cell dimensions, the velocity of the longshore and rip currents, and of the recycling regime. A finding from the wave tank studies Has the presence of standing waves formed by transverse edge waves. The interaction of these standing waves with the gene rated waves normal to the shore could be the initial cause of rip currents and the cellular circulation. Studies of the alongshore system were made in the field only. A method for measuring the volume of flow of longshore currents was developed, tested, and applied. Calculated volumes of flow using a theory based on continuity and the solitary wave theory (as proposed by Inman and Bagnold) compared tolerably well with the field observations. The calculations of volume of flow required a knowledge of the wave height spectra in the surf. This was established by making wave height recordings in the between-breaker zone . It was found that the characteristics of the spectra compared reasonably well with those pr e dict ed by the Longuet-Higgins theory, previously assumed to apply to deep water waves only. A mechanism for the transition from cellular to alongshore system is proposed. / Thesis (Ph.D.)-University of Natal, 1967.