9780691026435

Mathematics of Wave Propagation

by
  • ISBN13:

    9780691026435

  • ISBN10:

    0691026432

  • Format: Hardcover
  • Copyright: 2000-04-17
  • Publisher: Princeton Univ Pr

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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

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