Selected Topics In Nonlinear Wave Mechanics

This book gives an overview ofthe current state of nonlinear wave mechanics with emphasis on strong discontinuities (shock waves) and localized self preserving shapes (solitons) in both elastic and fluid media. The exposition is intentionallyat a detailed mathematical and physical level, our expectation being that the reader will enjoy coming to grips in a concrete manner with advances in this fascinating subject. Historically, modern research in nonlinear wave mechanics began with the famous 1858 piston problem paper of Riemann on shock waves and con tinued into the early part of the last century with the work of Hadamard, Rankine, and Hugoniot. After WWII, research into nonlinear propagation of dispersive waves rapidly accelerated with the advent of computers. Works of particular importance in the immediate post-war years include those of von Neumann, Fermi, and Lax. Later, additional contributions were made by Lighthill, Glimm, Strauss, Wendroff, and Bishop. Dispersion alone leads to shock fronts of the propagating waves. That the nonlinearity can com pensate for the dispersion, leading to propagation with a stable wave having constant velocity and shape (solitons) came as a surprise. A solitary wave was first discussed by J. Scott Russell in 1845 in "Report of British Asso ciations for the Advancement of Science. " He had, while horseback riding, observed a solitary wave travelling along a water channel and followed its unbroken progress for over a mile.

This book gathers contributions on various aspects of the theory and applications of linear and nonlinear waves and associated phenomena, as well as approaches developed in a global partnership of researchers with the national Centre of Excellence in Nonlinear Studies (CENS) at the Department of Cybernetics of Tallinn University of Technology in Estonia. The papers chiefly focus on the role of mathematics in the analysis of wave phenomena. They highlight the complexity of related topics concerning wave generation, propagation, transformation and impact in solids, gases, fluids and human tissues, while also sharing insights into selected mathematical methods for the analytical and numerical treatment of complex phenomena. In addition, the contributions derive advanced mathematical models, share innovative ideas on computing, and present novel applications for a number of research fields where both linear and nonlinear wave problems play an important role. The papers are written in a tutorial style, intended for non-specialist researchers and students. The authors first describe the basics of a problem that is currently of interest in the scientific community, discuss the state of the art in related research, and then share their own experiences in tackling the problem. Each chapter highlights the importance of applied mathematics for central issues in the study of waves and associated complex phenomena in different media. The topics range from basic principles of wave mechanics up to the mathematics of Planet Earth in the broadest sense, including contemporary challenges in the mathematics of society. In turn, the areas of application range from classic ocean wave mathematics to material science, and to human nerves and tissues. All contributions describe the approaches in a straightforward manner, making them ideal material for educational purposes, e.g. for courses, master class lectures, or seminar presentations.

The aim of this book is to give a self-contained introduction to the mathe matical analysis and physical explanations of some basic nonlinear wave phe nomena. This volume grew out of lecture notes for graduate courf;!es which I gave at the University of Alberta, the University of Saskatchewan, ·and Texas A&M University. As an introduction it is not intended to be exhaustive iQ its choice of material, but rather to convey to interested readers a basic; yet practical, methodology as well as some of the more important results obtained since the 1950's. Although the primary purpose of this volume is to serve as a textbook, it should be useful to anyone who wishes to understand or conduct research into nonlinear waves. Here, for the first time, materials on X-ray crystallography and the forced Korteweg-de Vries equation are incorporated naturally into a textbook on non linear waves. Another characteristic feature of the book is the inclusion of four symbolic calculation programs written in MATHEMATICA. They emphasize outcomes rather than numerical methods and provide certain symbolic and nu merical results related to solitons. Requiring only one or two commands to run, these programs have user-friendly interfaces. For example, to get the explicit expression of the 2-soliton of the Korteweg-de Vries equation, one only needs to type in soliton[2] when using the program solipac.m.

This unique volume presents an original approach to physical acoustics with additional emphasis on the most useful surface acoustic waves on solids. The study is based on foundational work of Léon Brillouin, and application of the celebrated invariance theorem of Emmy Noether to an element of volume that is representative of the wave motion. This approach provides an easy interpretation of typical wave motions of physical acoustics in bulk, at surfaces, and across interfaces, in the form of the motion of associated quasi-particles. This type of motion, Newtonian or not, depends on the wave motion considered, and on the original modeling of the continuum that supports it. After a thoughtful review of Brillouin's fundamental ideas related to radiative stresses, wave momentum and action, and the necessary reminder on modern nonlinear continuum thermomechanics, invariance theory and techniques of asymptotics, a variety of situations and models illustrates the power and richness of the approach and its strong potential in applications. Elasticity, piezoelectricity and new models of continua with nonlinearity, viscosity and some generalized features (microstructure, weak or strong nonlocality) or unusual situations (bounding surface with energy, elastic thin film glued on a surface waveguide), are considered, exhibiting thus the versatility of the approach. This original book offers an innovative vision and treatment of the problems of wave propagation in deformable solids. It opens up new horizons in the theoretical and applied facets of physical acoustics. Contents:Pro;egomena: Wave Momentum and Radiative Stresses in 1D in the Line of BrillouinElements of Continuum ThermomechanicsPseudomomentum and Eshelby StressAction, Phonons and Wave MechanicsTransmission-Reflection ProblemsApplication to Dynamic MaterialsElastic Surface Waves in Terms of Quasi-ParticlesElectroelastic Surface Waves in Terms of Quasi-ParticlesWaves Generalized Elastic ContinuaExamples of Solitonic Systems Readership: Graduate students and researchers in applied physics and mathematics, as well as accousticians. Key Features:Originality of approach to physical acousticsInnovative vision of the problem of wave propagation in deformable solidsEnriching interaction between mathematical physics and wave theoryKeywords:Waves;Physical Acoustics;Surface Waves;Quasi-Particles;Elasticity;Invariance Theorems

Exploring recent developments in continuum mechanics, Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics presents the general framework for configurational forces. It also covers a range of applications in engineering and condensed matter physics. The author presents the fundamentals of accepted standard continuum mechanics, before introducing Eshelby material stress, field theory, variational formulations, Noether’s theorem, and the resulting conservation laws. In the chapter on complex continua, he compares the classical perspective of B.D. Coleman and W. Noll with the viewpoint linked to abstract field theory. He then describes the important notion of local structural rearrangement and its relationship to Eshelby stress. After looking at the relevance of Eshelby stress in the thermodynamic description of singular interfaces, the text focuses on fracture problems, microstructured media, systems with mass exchanges, and electromagnetic deformable media. The concluding chapters discuss the exploitation of the canonical conservation law of momentum in nonlinear wave propagation, the application of canonical-momentum conservation law and material force in numerical schemes, and similarities of fluid mechanics and aerodynamics. Written by a long-time researcher in mechanical engineering, this book provides a detailed treatment of the theory of configurational forces—one of the latest and most fruitful advances in macroscopic field theories. Through many applications, it shows the depth and efficiency of this theory.

This book consists of select proceedings of the National Conference on Wave Mechanics and Vibrations (WMVC 2018). It covers recent developments and cutting-edge methods in wave mechanics and vibrations applied to a wide range of engineering problems. The book presents analytical and computational studies in structural mechanics, seismology and earthquake engineering, mechanical engineering, aeronautics, robotics and nuclear engineering among others. This book can be useful for students, researchers, and professionals interested in the wide-ranging applications of wave mechanics and vibrations.

This textbook gives a detailed explanation of waves and oscillations in classical physics. These classical phenomena are dealt with at a more advanced level than is customary for second-year courses. All aspects of classical wave physics are presented, including the mathematical and physical basis needed for extended understanding. Finally several chapters are devoted to important topics in current wave physics. Special attention is given to nonlinear waves, solitons, chaotic behavior and associated phenomena. The new edition contains improvements such as full development of Greens functions, a broadening of the treatment of wave mechanics and a closer integration with classical mechanics, plus more examples and problems.