Title Mathematics of wave propagation / Julian L. Davis

Published Princeton, NJ : Princeton University Press, 2000

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Description 1 online resource (xv, 395 pages) : illustrations Contents Chapter 1 Physics of Propagating Waves 3 -- Discrete Wave-Propagating Systems 3 -- Approximation of Stress Wave Propagation in a Bar by a Finite System of Mass-Spring Models 4 -- Limiting Form of a Continuous Bar 5 -- Wave Equation for a Bar 5 -- Transverse Oscillations of a String 9 -- Speed of a Transverse Wave in a Siting 10 -- Traveling Waves in General 11 -- Sound Wave Propagation in a Tube 16 -- Superposition Principle 19 -- Sinusoidal Waves 19 -- Interference Phenomena 21 -- Reflection of Light Waves 25 -- Reflection of Waves in a String 27 -- Sound Waves 29 -- Doppler Effect 33 -- Dispersion and Group Velocity 36 -- Chapter 2 Partial Differential Equations of Wave Propagation 41 -- Types of Partial Differential Equations 41 -- Geometric Nature of the PDEs of Wave Phenomena 42 -- Directional Derivatives 42 -- Cauchy Initial Value Problem 44 -- Parametric Representation 49 -- Wave Equation Equivalent to Two First-Order PDEs 51 -- Characteristic Equations for First-Order PDEs 55 -- General Treatment of Linear PDEs by Characteristic Theory 57 -- Another Method of Characteristics for Second-Order PDEs 61 -- Geometric Interpretation of Quasilinear PDEs 63 -- Integral Surfaces 65 -- Nonlinear Case 67 -- Canonical Form of a Second-Order PDE 70 -- Riemann's Method of Integration 73 -- Chapter 3 Wave Equation 85 -- Part I One-Dimensional Wave Equation 85 -- Factorization of the Wave Equation and Characteristic Curves 85 -- Vibrating String as a Combined IV and B V Problem 90 -- D'Alembert's Solution to the IV Problem 97 -- Domain of Dependence and Range of Influence 101 -- Cauchy IV Problem Revisited 102 -- Solution of Wave Propagation Problems by Laplace Transforms 105 -- Laplace Transforms 108 -- Applications to the Wave Equation 111 -- Nonhomogeneous Wave Equation 116 -- Wave Propagation through Media with Different Velocities 120 -- Electrical Transmission Line 122 -- Part II Wave Equation in two and Three Dimensions 125 -- Two-Dimensional Wave Equation 125 -- Reduced Wave Equation in Two Dimensions 126 -- Eigenvalues Must Be Negative 127 -- Rectangular Membrane 127 -- Circular Membrane 131 -- Three-Dimensional Wave Equation 135 -- Chapter 4 Wave Propagation in Fluids 145 -- Part I Inviscid Fluids 145 -- Lagrangian Representation of One-Dimensional Compressible Gas Flow 146 -- Eulerian Representation of a One-Dimensional Gas 149 -- Solution by the Method of Characteristics: One-Dimensional Compressible Gas 151 -- Two-Dimensional Steady Flow 157 -- Bernoulli's Law 159 -- Method of Characteristics Applied to Two-Dimensional Steady Flow 161 -- Supersonic Velocity Potential 163 -- Hodograph Transformation 163 -- Shock Wave Phenomena 169 -- Part II Viscous Fluids 183 -- Elementary Discussion of Viscosity 183 -- Conservation Laws 185 -- Boundary Conditions and Boundary Layer 190 -- Energ Dissipation in a Viscous Fluid 191 -- Wave Propagation in a Viscous Fluid 193 -- Oscillating Body of Arbitrary Shape 196 -- Similarity Considerations and Dimensionless Parameters; Reynolds'Law 197 -- Poiseuille Flow 199 -- Stokes'Flow 201 -- Oseen Approximation 208 -- Chapter 5 Stress Waves in Elastic Solids 213 -- Fundamentals of Elasticity 214 -- Equations of Motion for the Stress 223 -- Navier Equations of Motion for the Displacement 224 -- Propagation of Plane Elastic Waves 227 -- General Decomposition of Elastic Waves 228 -- Characteristic Surfaces for Planar Waves 229 -- Time-Harmonic Solutions and Reduced Wave Equations 230 -- Spherically Symmetric Waves 232 -- Longitudinal Waves in a Bar 234 -- Curvilinear Orthogonal Coordinates 237 -- Navier Equations in Cylindrical Coordinates 239 -- Radially Symmetric Waves 240 -- Waves Propagated Over the Surface of an Elastic Body 243 -- Chapter 6 Stress Waves in Viscoelastic Solids 250 -- Internal Ftiction 251 -- Discrete Viscoelastic Models 252 -- Continuous Marwell Model 260 -- Continuous Voigt Model 263 -- Three-Dimensional VE Constitutive Equations 264 -- Equations of Motion for a VE Material 265 -- One-Dimensional Wave Propagation in VE Media 266 -- Radially Symmetric Waves for a VE Bar 270 -- ElectromechanicalAnalogy 271 -- Chapter 7 Wave Propagation in Thermoelastic Media 282 -- Duhamel-Neumann Law 282 -- Equations of Motion 285 -- Plane Harmonic Waves 287 -- Three-Dimensional Thermal Waves; Generalized Navier Equation 293 -- Chapter 8 Water Waves 297 -- Irrotational, Incompressible, Inviscid Flow; Velocity Potential and Equipotential Surfaces 297 -- Euler's Equations 299 -- Two-Dimensional Fluid Flow 300 -- Complec Variable Treatment 302 -- Vortex Motion 309 -- Small-Amplitude Gravity Waves 311 -- Water Waves in a Straight Canal 311 -- Kinematics of the Free Surface 316 -- Vertical Acceleration 317 -- Standing Waves 319 -- Two-Dimensional Waves of Finite Depth 321 -- Boundary Conditions 322 -- Formulation of a Typical Surface Wave Problem 324 -- Example of Instability 325 -- Approximation Aeories 327 -- Tidal Waves 337 -- Chapter 9 Variational Methods in Wave Propagation 344 -- Introduction; Fermat's PKnciple 344 -- Calculus of Variations; Euler's Equation 345 -- Configuration Space 349 -- Cnetic and Potential Eneigies 350 -- Hamilton's Variational Principle 350 -- PKnciple of Virtual Work 352 -- Transformation to Generalized Coordinates 354 -- Rayleigh's Dissipation Function 357 -- Hamilton's Equations of Motion 359 -- Cyclic Coordinates 362 -- Hamilton-Jacobi Theory 364 -- Extension of W to 2 n Degrees of Freedom 370 -- H-J Aeory and Wave P[similar]vpagation 372 -- Quantum Mechanics 376 -- An Analog between Geometric Optics and Classical Mechanics 377 -- Asymptotic Theory of Wave Propagation 380 -- Appendix Principle of Least Action 384 Summary Earthquakes, a plucked string, ocean waves crashing on the beach, the sound waves that allow us to recognize known voices. Waves are everywhere, and the propagation and classical properties of these apparently disparate phenomena can be described by the same mathematical methods: variational calculus, characteristics theory, and caustics. Taking a medium-by-medium approach, Julian Davis explains the mathematics needed to understand wave propagation in inviscid and viscous fluids, elastic solids, viscoelastic solids, and thermoelastic media, including hyperbolic partial differential equations and characteristics theory, which makes possible geometric solutions to nonlinear wave problems. The result is a clear and unified treatment of wave propagation that makes a diverse body of mathematics accessible to engineers, physicists, and applied mathematicians engaged in research on elasticity, aerodynamics, and fluid mechanics. This book will particularly appeal to those working across specializations and those who seek the truly interdisciplinary understanding necessary to fully grasp waves and their behavior. By proceeding from concrete phenomena (e.g., the Doppler effect, the motion of sinusoidal waves, energy dissipation in viscous fluids, thermal stress) rather than abstract mathematical principles, Davis also creates a one-stop reference that will be prized by students of continuum mechanics and by mathematicians needing information on the physics of waves Analysis Acoustic impedance Adiabatic condition Adjoint operators Bessel's equation Boundary layer Brachistochrome problem Bulk modulus Canonical transformations Caustic or focal curves Characteristic coordinates Characteristic line element Characteristics, method of Critical sound speed Cyclic coordinates Deformation Direction field Doppler effect Eigenfunctions Eigenvalues Epicycloid Fermat's principle Finite bar Friction, internal Generalized force Generalized velocity Hamilton-Jacoby theory Hamiltonian Ideal or perfect gas Integral surfaces Isothermal condition Jacobian Kinetic energy Lagrangian function Lame constants Laminar flow Memory function Minimizing curve Monge axis Monge cone Navier equations Oseen approximation Physics of propagating waves Plane elastic waves Poiseuille flow Progressing wave Quantum mechanics Radially symmetric waves Regressing wave Reynold's law Self-adjoint operator Sinusoidal waves Bibliography Includes bibliographical references (pages 389-390) and index Notes Print version record SUBJECT Welle, ... gnd Subject Wave-motion, Theory of. MATHEMATICS -- Applied. Wave-motion, Theory of Mathematische Physik Wellenausbreitung Welle WAVES. WAVE PROPAGATION. DIFFERENTIAL EQUATIONS. WAVE EQUATIONS. VISCOUS FLUIDS. Form Electronic book LC no. 2021694870 ISBN 9780691223377 0691223378

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