Mathematics of Wave Propagation

by Davis, Julian L.

• ISBN13: 9780691026435
• ISBN10: 0691026432
• Format: Hardcover
• Copyright: 2000-04-17
• Publisher: Princeton Univ Pr

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.

Table of Contents

Preface	xiii
Physics of Propagating Waves	3	(38)
Introduction	3	(1)
Discrete Wave-Propagating Systems	3	(1)
Approximation of Stress Wave Propagation in a Bar by a Finite System of Mass-Spring Models	4	(1)
Limiting Form of a Continuous Bar	5	(1)
Wave Equation for a Bar	5	(4)
Transverse Oscillations of a String	9	(1)
Speed of a Transverse Wave in a String	10	(1)
Traveling Waves in General	11	(5)
Sound Wave Propagation in a Tube	16	(3)
Superposition Principle	19	(1)
Sinusoidal Waves	19	(2)
Interference Phenomena	21	(4)
Reflection of Light Waves	25	(2)
Reflection of Waves in a String	27	(2)
Sound Waves	29	(4)
Doppler Effect	33	(3)
Dispersion and Group Velocity	36	(1)
Problems	37	(4)
Partial Differential Equations of Waves Propagation	41	(44)
Introduction	41	(1)
Types of Partial Differential Equations	41	(1)
Geometric Nature of the PDEs of Wave Phenomena	42	(1)
Directional Derivatives	42	(2)
Cauchy Initial Value Problem	44	(5)
Parametric Representation	49	(2)
Wave Equation Equivalent to Two First-Order PDEs	51	(4)
Characteristic Equations for First-Order PDEs	55	(2)
General Treatment of Linear PDEs by Characteristic Theory	57	(4)
Another Method of Characteristics for Second-Order PDEs	61	(2)
Geometric Interpretation of Quasilinear PDEs	63	(2)
Integral Surfaces	65	(2)
Nonlinear Case	67	(3)
Canonical Form of a Second-Order PDE	70	(3)
Riemann's Method of Integration	73	(9)
Problems	82	(3)
The Wave Equation	85	(60)
One-Dimensional Wave Equation	85	(1)
Factorization of the Wave Equation and Characteristic curves	85	(5)
Vibrating String as a Combined IV and BV Problem	90	(7)
D'Alembert's Solution to the IV Problem	97	(4)
Domain of Dependence and Range of Influence	101	(1)
Cauchy IV Problem Revisited	102	(3)
Solution of Wave Propagation Problems by Laplace Transforms	105	(3)
Laplace Transforms	108	(3)
Applications to the Wave Equation	111	(5)
Nonhomogeneous Wave Equation	116	(4)
Wave Propagation through Media with Different Velocities	120	(2)
Electrical Transmission Line	122	(3)
The Wave Equation in Two and Three Dimensions	125	(1)
Two-Dimensional Wave Equation	125	(1)
Reduced Wave Equation in Two Dimensions	126	(1)
The Eigenvalues Must Be Negative	127	(1)
Rectangular Membrane	127	(4)
Circular Membrane	131	(4)
Three-Dimensional Wave Equation	135	(5)
Problems	140	(5)
Wave Propagation in Fluids	145	(68)
Inviscid Fluids	145	(1)
Lagrangian Representation of One-Dimensional Compressible Gas Flow	146	(3)
Eulerian Representation of a One-Dimensional Gas	149	(2)
Solution by the Method of Characteristics: One-Dimensional Compressible Gas	151	(6)
Two-Dimensional Steady Flow	157	(2)
Bernoulli's Law	159	(2)
Method of Characteristics Applied to Two-Dimensional Steady Flow	161	(2)
Supersonic Velocity Potential	163	(1)
Hodograph Transformation	163	(6)
Shock Wave Phenomena	169	(14)
Viscous Fluids	183	(1)
Elementary Discussion of Viscosity	183	(2)
Conservation Laws	185	(5)
Boundary Conditions and Boundary Layer	190	(1)
Energy Dissipation in a Viscous Fluid	191	(2)
Wave Propagation in a Viscous Fluid	193	(3)
Oscillating Body of Arbitrary Shape	196	(1)
Similarity Considerations and Dimensionless Parameters; Reynolds' Law	197	(2)
Poiseuille Flow	199	(2)
Stokes' Flow	201	(7)
Ossen Approximation	208	(2)
Problems	210	(3)
Stress Waves in Elastic Solids	213	(37)
Introduction	213	(1)
Fundamentals of Elasticity	214	(9)
Equations of Motion for the Stress	223	(1)
Navier Equations of Motion for the Displacement	224	(3)
Propagation of Plane Elastic Waves	227	(1)
General Decomposition of Elastic Waves	228	(1)
Characteristic Surfaces for Planar Waves	229	(1)
Time-Harmonic Solutions and Reduced Wave Equations	230	(2)
Spherically Symmetric Waves	232	(2)
Longitudinal Waves in a Bar	234	(3)
Curvilinear Orthogonal Coordinates	237	(2)
The Navier Equations in Cylindrical Coordinates	239	(1)
Radially Symmetric Waves	240	(3)
Waves Propagated Over the Surface of an Elastic Body	243	(4)
Problems	247	(3)
Stress Waves in Viscoelastic Solids	250	(32)
Introduction	250	(1)
Internal Friction	251	(1)
Discrete Viscoelastic Models	252	(8)
Continuous Maxwell Model	260	(3)
Continuous Voigt Model	263	(1)
Three-Dimensional VE Constitutive Equations	264	(1)
Equations of Motion for a VE Material	265	(1)
One-Dimensional Wave Propagation in VE Media	266	(4)
Radially Symmetric Waves for a VE Bar	270	(1)
Electromechanical Analogy	271	(9)
Problems	280	(2)
Wave Propagation in Thermoelastic Media	282	(15)
Introduction	282	(1)
Duhamel-Neumann Law	282	(3)
Equations of Motion	285	(2)
Plane Harmonic Waves	287	(6)
Three-Dimensional Thermal Waves; Generalized Navier Equation	293	(4)
Water Waves	297	(47)
Introduction	297	(1)
Irrotational, Incompressible, Inviscid Flow; Velocity Potential and Equipotential Surfaces	297	(2)
Euler's Equations	299	(1)
Two-Dimensional Fluid Flow	300	(2)
Complex Variable Treatment	302	(7)
Vortex Motion	309	(2)
Small-Amplitude Gravity Waves	311	(1)
Water Waves in a Straight Canal	311	(5)
Kinematics of the Free Surface	316	(1)
Vertical Acceleration	317	(2)
Standing Waves	319	(2)
Two-Dimensional Waves of Finite Depth	321	(1)
Boundary Conditions	322	(2)
Formulation of a Typical Surface Wave Problem	324	(1)
Example of Instability	325	(2)
Approximation Theories	327	(10)
Tidal Waves	337	(5)
Problems	342	(2)
Variational Methods in Wave Propagation	344	(45)
Introduction; Fermat's Principle	344	(1)
Calculus of Variations; Euler's Equation	345	(4)
Configuration Space	349	(1)
Kinetic and Potential Energies	350	(1)
Hamilton's Variational Principle	350	(2)
Principle of Virtual Work	352	(2)
Transformation to Generalized Coordinates	354	(3)
Rayleigh's Dissipation Function	357	(2)
Hamilton's Equations of Motion	359	(3)
Cyclic Coordinates	362	(2)
Hamilton-Jacobi Theory	364	(6)
Extension of W to 2n Degrees of Freedom	370	(2)
H-J Theory and Wave Propagation	372	(4)
Quantum Mechanics	376	(1)
An Analogy between Geometric Optics and Classical Mechanics	377	(3)
Asymptotic Theory of Wave Propagation	380	(4)
Appendix: The Principle of Least Action	384	(3)
Problems	387	(2)
Bibliography	389	(2)
Index	391