In 1966 Mark Kac posed an interesting question regarding vibrating membranes and the sounds they make. His article entitled "Can One Hear the Shape of a Drum?", which appeared in The American Mathematical Monthly, generated much interest and scholarly debate. The evolution of Kac's intriguing question will be the subject of this project.

Thesis (MScEng)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The various consequences of climate change pose a significant threat to developments near the coast. These threats include saltwater intrusion, coastal erosion and flooding. In the coastal context, the climate change effect often raising the most concern is that of sea-level rise. Much work has therefore been done on the linear setback caused by a rise in sea-level. In order to get the full picture of possible changes caused by sea-level rise, the secondary effects of a rising sea-level also need to be considered. Sea-level rise could cause changes to the nearshore wave climate and could have impacts such as coastal erosion and changes to the coastline shape. The primary objective of this study was therefore to investigate the effects of sea-level rise on the nearshore wave climate and, consequently, the coastline stability. Other consequences of climate change considered in this study include increasing average wave heights and a rotation of offshore wave directions. The many headland-bay beaches on the South African coastline are generally in a state of dynamic equilibrium and find their planforms based on the local wave climate. Changes to the wave climate may therefore disrupt the equilibrium shapes of these bays. This study was therefore also aimed at investigating the effects of the changes to the wave climate on the stability of headland-bay beaches. The three consequences of climate change expected to affect the nearshore wave climate were identified as (1) sea-level rise; (2) an increased wave height; and (3) changing offshore wave angles. Although changes to storm frequency and intensity are also possible, the impacts of these changes were not studied. In order to assess the impacts of the three considered changes on a typical headland-bay beach, two numerical models were set up for Mossel Bay – a headland-bay beach on the southern coast of South Africa. The modelling approach included a wave transformation model to calculate nearshore wave climates from offshore data and a coastline model to assess the stability of the bay under the changed nearshore wave climates. The model results indicated that the rising sea-level alone would cause changes in the nearshore wave direction. These changes were shown to alter the longshore sediment transport regime such that rotations are expected in the south-western corner and eastern end of Mossel Bay. These rotations do not include the cross-shore effects of inundation and erosion, as suggested by models such as the Bruun Model. The results for an increased offshore wave height were inconclusive. The southerly rotation in offshore wave angles was shown to affect the nearshore wave angles. These changes affected the longshore transport regime such that the outward sediment transports were reduced. A minor accretion resulted in the centre of the bay for a 1° southerly rotation in offshore wave angles. For a 2° rotation, the extent of accretion increased and shifted towards the eastern end of the bay, primarily due to the dominance of south-westerly waves in the local wave climate. A valuable observation was made regarding the current stability of Mossel Bay. Inter-tidal reefs are present along three sections of the bay. These reefs protect the coastline such that the current bay shape contains sharp bends between the reefs. Under a rising sea-level, however, the effect of the reefs will become less pronounced. If a water level should be reached where these reefs become less significant, the planform of the bay is expected to smooth out through a significant redistribution of sediment. This smoothing effect was shown to cause erosion of the coastline in the order of 80m near the town of Klein Brak River. / AFRIKAANSE OPSOMMING: Die verskeie gevolge van klimaatsverandering bied ‘n merkwaardige bedreiging vir ontwikkelings naby die kus. Hierdie bedreigings sluit die versouting van varswaterbronne, kuserosie en oorstromings in. Vir kusgebiede is seevlakstyging gereeld die effek van klimaatsverandering wat die meeste kommer wek. Dus is heelwat navorsing rakende die direkte erosie as gevolg van seevlakstyging reeds gedoen. Om die volle beeld van die gevolge van ‘n stygende seevlak te verkry, is dit egter nodig om ook die sekondêre effekte hiervan in ag te neem. Seevlakstyging kan veranderinge in die golfklimaat naby die kus veroorsaak, en kan impakte soos kuserosie en veranderende baaivorms tot gevolg hê. Die primêre doel van hierdie studie is dus om die effek van seevlakstyging op die golfklimaat by die kus en gevolglik die stabiliteit van die kuslyn, te ondersoek. Benewens die styging van die seevlak word die effekte van groter gemiddelde golfhoogtes en die rotasie van diepsee golfrigtings ook in hierdie studie ondersoek. Die vele landpunt-baaie (headlandbay beaches) op die Suid-Afrikaanse kus is meestal in ‘n dinamiese ekwilibriumtoestand, waarvan die vorm deur die lokale golfklimaat bepaal word. Veranderinge aan dié golfklimaat mag dus die ekwilibrium vorms van sulke baaie versteur. Hierdie studie het dus ook die stabilititeit van landpuntbaaie onder ‘n veranderende golfklimaat ondersoek. Die drie gevolge van klimaatsverandering wat verwag word om die golfklimaat naby die kus te beïnvloed is geïdentifiseer as (1) seevlakstyging; (2) vergrote golfhoogtes; en (3) veranderende diepsee golfhoeke. Veranderinge aan die frekwensie en intensiteit van storms is ook moontlike gevolge van klimaatsverandering, maar die impakte hiervan is nie in die studie ondersoek nie. Twee numeriese modelle is toegepas om die impak van die drie bogenoemde gevolge op Mosselbaai – ‘n tipiese landpunt-baai aan die suidkus van Suid-Afrika – te ondersoek. ‘n Golfmodel is ingespan om die golfklimaat naby die kus te bepaal waarna ‘n kuslynmodel gebruik is om die stabiliteit van die baai onder die veranderde golfklimaat te ondersoek. Die resultate van die studie dui daarop dat die golfhoeke naby die kus beïnvloed word deur seevlakstyging. Daar is aangetoon dat dié veranderinge die langsstroomvervoer sodanig sal verander dat kuslynrotasies in die suid-westelike hoek asook die oostelike rand van Mosselbaai verwag word. Hierdie rotasies sluit nie die lineêre landwaartse verplasing van die kuslyn as gevolg van erosie en oorstroming in nie. Die effek van vergrote golfhoogtes kon nie met akkuraatheid ondersoek word nie. Daar is wel gevind dat die suidwaartse rotasie van diepsee golfhoeke rotasies in die golfklimaat naby die kus veroorsaak. Hierdie rotasies verander die langsstroom sedimentvervoer sodanig dat die uitwaartse sedimentvervoer verminder word en ‘n klein opbou van sediment in die middel van die baai vir ‘n 1° diepsee rotasie verwag word. Vir ‘n 2° suidwaartse rotasie is daar ‘n groter opbou van sediment wat verder ooswaarts veplaas is. Die ooswaartse veplasing is primêr ‘n gevolg van die oorheersing van suid-westelike golftoestande in die golfklimaat. ‘n Waardevolle gevolgtrekking rakende die huidige stabiliteit van Mosselbaai is ook gemaak. Langs drie gedeeltes van die Mosselbaaise kus word riwwe in die gebied tussen hoog- en laagwater aangetref. Hierdie riwwe beskerm die kus sodanig dat skerp kinkels in die vorm van die baai tussen die riwwe gesien kan word. Wanneer die seevlak styg, word die beskermende effek van die riwwe egter minder doeltreffend. Indien ‘n watervlak bereik word waar dié effek genoegsaam verminder is, word daar verwag dat die baai deur ‘n merkwaardige verplasing van sediment die kinkels sal uitstryk. Deur hierdie proses word erosie in die orde van 80m naby die dorp van Klein Brakrivier verwag.

Made available in DSpace on 2016-04-28T14:16:34Z (GMT). No. of bitstreams: 1 Douglas Guilherme Schmidt.pdf: 995968 bytes, checksum: 1a8192151fc433a54ec422f421249ac4 (MD5) Previous issue date: 2008-09-15 / Secretaria da Educação do Estado de São Paulo / This historical research analyses the initial period of construction of wave mechanics, as propounded by the Austrian physicist Erwin Schrödinger (1887- 1961), emphasizing his work from December 1925 to February 1926. During these months Schrödinger created the basis of his quantum theory and wrote his two earlier papers on this subject. They were published in the Annalen der Physik. The present dissertation analyses those two papers, and some of their precedents and immediate consequences. Special attention is given to the influence of the theory of matter waves of the French physicist Louis de Broglie (1892-1987) upon the development of Schrödinger s theory, as well as other works and relevant circumstances that contributed to the creation of wave mechanics. The dissertation also discusses the interpretation given by Schrödinger to his own theory, comparing it to the approach of other physicists of that time / O presente trabalho histórico investiga a fase inicial da construção da mecânica ondulatória formulada pelo físico austríaco Erwin Schrödinger (1887- 1961), dando especial atenção aos trabalhos por ele realizados de dezembro de 1925 até fevereiro de 1926. Nesse período, Schrödinger concebeu as bases de sua teoria quântica e redigiu os dois primeiros artigos sobre o assunto, publicados na revista Annalen der Physik. Esta dissertação analisa esses dois artigos, bem como alguns de seus precedentes e repercussões. É analisada em especial a influência da teoria de ondas de matéria do físico francês Louis de Broglie (1892- 1987) no desenvolvimento da teoria de Schrödinger, bem como outros trabalhos e circunstâncias importantes que contribuíram para a elaboração da mecânica ondulatória. Discute-se também a interpretação que o próprio Schrödinger deu à sua teoria, comparando-a com o enfoque adotado por outros físicos da época

Acompanha: Encarte do produto educacional: a física da música e a pluralidade didática / CAPES / Esta dissertação visa apresentar uma proposta de sequência didática para explorar significativamente conteúdos de ondulatória, acústica e conteúdos relacionados à sua compreensão, utilizando elementos da música como agente motivacional ao aprendizado de física, bem como uma gama de recursos didáticos com o objetivo de oferecer uma pluralidade didática adaptada a partir dos fundamentos da pluralidade metodológica de Paul Feyerabend. Estes recursos são desde quadro giz, a inventos, instrumentos musicais, experimentos, softwares, simuladores, imagens, animações, videoaulas, músicas, poesias e alguns aspectos epistemológicos da física da música. / This thesis aims to present a proposal for didactic sequence to significantly explore waves, acoustics and related content to their understanding, using elements of music as a motivational agent to the physics learning, as well as a range of instructional resources with the aim of offering a didactic plurality it was adapted from the fundamentals of methodological plurality of Paul Feyerabend. These features are since chalk framework, to inventions, musical instruments, experiments, softwares, simulators, images, animations, video lessons, songs, poetry and some epistemological aspects of the physics of music.

The analysis of wave propagation in hyperelastic waveguides has significant applications in various branches of engineering like Non-Destructive Testing and Evaluation, impact analysis, material characterization and damage detection. Linear elastic models are typically used for wave analysis since they are sufficient for many applications. However, certain solids exhibit inherent nonlinear material properties that cannot be adequately described with linear models. In the presence of large deformations, geometric nonlinearity also needs to be incorporated in the analysis. These two forms of nonlinearity can have significant consequences on the propagation of stress waves in solids. A detailed analysis of nonlinear wave propagation in solids is thus necessary for a proper understanding of these phenomena. The current research focuses on the development of novel algorithms for nonlinear finite element analysis of stress wave propagation in hyperelastic waveguides. A full three-dimensional(3D) finite element analysis of stress wave propagation in waveguides is both computationally difficult and expensive, especially in the presence of nonlinearities. By definition, waveguides are solids with special geometric features that channel the propagation of stress waves along certain preferred directions. This suggests the use of kinematic waveguide models that take advantage of the special geometric features of the waveguide. The primary advantage of using waveguide models is the reduction of the problem dimension and hence the associated computational cost. Elementary waveguide models like the Euler-Bernoulli beam model, Kirchoff plate model etc., which are developed primarily within the context of linear elasticity, need to be modified appropriately in the presence of material/geometric nonlinearities and/or loads with high frequency content. This modification, besides being non-trivial, may be inadequate for studying nonlinear wave propagation and higher order waveguide models need to be developed. However, higher order models are difficult to formulate and typically have complex governing equations for the kinematic modes. This reflects in the relatively scarce research on the development of higher order waveguide models for studying nonlinear wave propagation. The formulation is difficult primarily because of the complexity of both the governing equations and their linearization, which is required as part of a nonlinear finite element analysis. One of the primary contributions of this thesis is the development and implementation of a general, flexible and efficient framework for automating the finite element analysis of higher order kinematic models for nonlinear waveguides. A hierarchic set of higher order waveguide models that are compatible with this formulation are proposed for this purpose. This hierarchic series of waveguide models are similar in form to the kinematic assumptions associated with standard waveguide models, but are different in the sense that no conditions related to the stress distribution specific to a waveguide are imposed since that is automatically handled by the proposed algorithm. The automation of the finite element analysis is accomplished with a dexterous combination of a nodal degrees-of-freedom based assembly algorithm, automatic differentiation and a novel procedure for numerically computing the finite element matrices directly from a given waveguide model. The algorithm, however, is quite general and is also developed for studying nonlinear plane stress configurations and inhomogeneous structures that require a coupling of continuum and waveguide elements. Significant features of the algorithm are the automatic numerical derivation of the finite element matrices for both linear and nonlinear problems, especially in the context of nonlinear plane stress and higher order waveguide models, without requiring an explicit derivation of their algebraic forms, automatic assembly of finite element matrices and the automatic handling of natural boundary conditions. Full geometric nonlinearity and the hyperelastic form of material nonlinearity are considered in this thesis. The procedures developed here are however quite general and can be extended for other types of material nonlinearities. Throughout this thesis, It is assumed that the solids under investigation are homogeneous and isotropic. The subject matter of the research is developed in four stages: First, a comparison of different finite element discretization schemes is carried out using a simple rod model to choose the most efficient computational scheme to study nonlinear wave propagation. As part of this, the frequency domain Fourier spectral finite element method is extended for a special class of weakly nonlinear problems. Based on this comparative study, the Legendre spectral element method is identified as the most efficient computational tool. The efficiency of the Legendre spectral element is also illustrated in the context of a nonlinear Timoshenko beam model. Since the spectral element method is a special case of the standard nonlinear finite element Method, differing primarily in the choice of the element basis functions and quadrature rules, the automation of the standard nonlinear finite element method is undertaken next. The automatic finite element formulation and assembly algorithm that constitutes the most significant contribution of this thesis is developed as an efficient numerical alternative to study the physics of wave propagation in nonlinear higher order structural models. The development of this algorithm and its extension to a general automatic framework for studying a large class of problems in nonlinear solid mechanics forms the second part of this research. Of special importance are the automatic handling of nonlinear plane stress configurations, hierarchic higher order waveguide models and the automatic coupling of continuum and higher order structural elements using specially designed transition elements that enable an efficient means to study waveguides with local inhomogeneities. In the third stage, the automatic algorithm is used to study wave propagation in hyperelastic waveguides using a few higher order 1D kinematic models. Two variants of a particular hyperelastic constitutive law – the6-constantMurnaghanmodel(for rock like solids) and the 9-constant Murnaghan model(for metallic solids) –are chosen for modeling the material nonlinearity in the analysis. Finally, the algorithm is extended to study energy-momentum conserving time integrators that are derived within a Hamiltonian framework, thus illustrating the extensibility of the algorithm for more complex finite element formulations. In short, the current research deals primarily with the identification and automation of finite element schemes that are most suited for studying wave propagation in hyper-elastic waveguides. Of special mention is the development of a novel unified computational framework that automates the finite element analysis of a large class of problems involving nonlinear plane stress/plane strain, higher order waveguide models and coupling of both continuum and waveguide elements. The thesis, which comprises of 10 chapters, provides a detailed account of various aspects of hyperelastic wave propagation, primarily for 1D waveguides.