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E-raamat: Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

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Rays, Waves, and Scattering: Topics in Classical Mathematical PhysicsSuurem pilt
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This one-of-a-kind book presents many of the mathematical concepts, structures, and techniques used in the study of rays, waves, and scattering. Panoramic in scope, it includes discussions of how ocean waves are refracted around islands and underwater ridges, how seismic waves are refracted in the earth's interior, how atmospheric waves are scattered by mountains and ridges, how the scattering of light waves produces the blue sky, and meteorological phenomena such as rainbows and coronas.

Rays, Waves, and Scattering is a valuable resource for practitioners, graduate students, and advanced undergraduates in applied mathematics, theoretical physics, and engineering. Bridging the gap between advanced treatments of the subject written for specialists and less mathematical books aimed at beginners, this unique mathematical compendium features problems and exercises throughout that are geared to various levels of sophistication, covering everything from Ptolemy's theorem to Airy integrals (as well as more technical material), and several informative appendixes.

  • Provides a panoramic look at wave motion in many different contexts
  • Features problems and exercises throughout
  • Includes numerous appendixes, some on topics not often covered
  • An ideal reference book for practitioners
  • Can also serve as a supplemental text in classical applied mathematics, particularly wave theory and mathematical methods in physics and engineering
  • Accessible to anyone with a strong background in ordinary differential equations, partial differential equations, and functions of a complex variable

Arvustused

"A tour de force of the mathematical description of waves. . . . I sincerely wish I had encountered such a book early in my teaching career. The material presented in it would have provided a very useful enhancement to a number of courses I have taught to undergraduate physics majors over the years."---James A. Lock, American Journal of Physics

Preface xvii
Acknowledgments xxiii
Chapter 1 Introduction
1(14)
1.1 The Rainbow Directory
3(2)
1.1.1 The Multifaceted Rainbow
3(2)
1.2 A Mathematical Taste of Things to Come
5(12)
1.2.1 Rays
5(1)
1.2.2 Waves
6(1)
1.2.3 Scattering (Classical)
7(2)
1.2.4 Scattering (Semiclassical)
9(2)
1.2.5 Caustics and Diffraction Catastrophes
11(4)
Part I: Rays 15(172)
Chapter 2 Introduction to the "Physics" of Rays
17(16)
2.1 What Is a Ray?
17(8)
2.1.1 Some Mathematical Definitions
18(1)
2.1.2 Geometric Wavefronts
19(2)
2.1.3 Fermat's Principle
21(1)
2.1.4 The Intensity Law
21(2)
2.1.5 Heuristic Derivation of Snell's Laws
23(1)
2.1.6 Generalization
24(1)
2.2 Geometric and Other Proofs of Snell's Laws of Reflection and Refraction
25(8)
2.2.1 The Law of Reflection
25(1)
2.2.2 The Law of Refraction
26(2)
2.2.3 A Wave-Theoretic Proof
28(1)
2.2.4 An Algebraic Proof
29(4)
Chapter 3 Introduction to the Mathematics of Rays
33(43)
3.1 Background
33(1)
3.2 The Method of Characteristics
34(3)
3.3 Introduction to Hamilton-Jacobi Theory
37(9)
3.3.1 Hamilton's Principle
39(1)
3.3.2 Rays and Characteristics
39(4)
3.3.3 The Optical Path Length Revisited
43(3)
3.4 Ray Differential Geometry and the Eikonal Equation Again
46(8)
3.4.1 The Mirage Theorem for Horizontally Stratified Media
49(2)
3.4.2 A Return to Spherically Symmetric Media: n (r) Continuous
51(3)
3.5 Dispersion Relations: A Wave-Ray Connection
54(10)
3.5.1 Fourier Transforms and Dispersion Relations
55(1)
3.5.2 The Bottom Line
56(5)
3.5.3 Applications to Atmospheric Waves
61(3)
3.6 General Solution of the Linear Wave Equation: Some Asymptotics
64(6)
3.6.1 Stationary Phase
64(1)
3.6.2 Asymptotics for Oscillatory Sources: Wavenumber Surfaces
65(5)
3.7 Rays and Waves in a Slowly Varying Environment
70(6)
3.7.1 Some Consequences
71(4)
3.7.2 Wavepackets and the Group Speed Revisited
75(1)
Chapter 4 Ray Optics: The Classical Rainbow
76(19)
4.1 Physical Features and Historical Details: A Summary
76(2)
4.2 Ray Theory of the Rainbow: Elementary Mathematical Considerations
78(11)
4.2.1 Some Numerical Values
84(1)
4.2.2 Polarization of the Rainbow
85(2)
4.2.3 The Divergence Problem
87(2)
4.3 Related Topics in Meteorological Optics
89(6)
4.3.1 The Glory
89(3)
4.3.2 Coronas (Simplified)
92(1)
4.3.3 Rayleigh Scattering-a Dimensional Analysis Argument
93(2)
Chapter 5 An Improvement over Ray Optics: Airy's Rainbow
95(18)
5.1 The Airy Approximation
95(18)
5.1.1 Some Ray Prerequisites
95(5)
5.1.2 The Airy Wavefront
100(4)
5.1.3 How Are Colors Distributed in the Airy Rainbow?
104(1)
5.1.4 The Airy Wavefront: A Derivation for Arbitrary p
105(8)
Chapter 6 Diffraction Catastrophes
113(24)
6.1 Basic Geometry of the Fold and Cusp Catastrophes
114(8)
6.1.1 The Fold
114(1)
6.1.2 The Cusp
115(7)
6.2 A Better Approximation
122(4)
6.2.1 The Fresnel Integrals
124(2)
6.3 The Fold Diffraction Catastrophe
126(4)
6.3.1 The Rainbow as a Fold Catastrophe
128(2)
6.4 Caustics: The Airy Integral in the Complex Plane
130(7)
6.4.1 The Nature of Ai(X)
133(4)
Chapter 7 Introduction to the WKB(J) Approximation: All Things Airy
137(25)
7.1 Overview
137(12)
7.1.1 Elimination of the First Derivative Term
139(2)
7.1.2 The Liouville Transformation
141(2)
7.1.3 The One-Dimensional Schrodinger Equation
143(1)
7.1.4 Physical Interpretation of the WKB(J) Approximation
144(1)
7.1.5 The WKB(J) Connection Formulas
145(3)
7.1.6 Application to a Potential Well
148(1)
7.2 Technical Details
149(3)
7.3 Matching Across a Turning Point
152(1)
7.4 A Little More about Airy Functions
153(9)
7.4.1 Relation to Bessel Functions
154(2)
7.4.2 The Airy Integral and Related Topics
156(3)
7.4.3 Related Integrals
159(3)
Chapter 8 Island Rays
162(11)
8.1 Straight and Parallel Depth Contours
163(4)
8.1.1 Plane Wave Incident on a Ridge
164(2)
8.1.2 Wave Trapping on a Ridge
166(1)
8.2 Circular Depth Contours
167(2)
8.3 Constant Phase Lines
169(1)
8.3.1 Case 1
169(1)
8.3.2 Case 2
170(1)
8.3.3 Case 3
170(1)
8.4 Waves and Currents
170(3)
Chapter 9 Seismic Rays
173(14)
9.1 Seismic Ray Equations
173(16)
9.2 Ray Propagation in a Spherical Earth
175(3)
9.2.1 A Horizontally Stratified Earth
178(1)
9.2.2 The Wiechert-Herglotz Inversion
179(2)
9.2.3 Further Properties of X in the Horizontally Stratified Case
181(6)
Part II: Waves 187(112)
Chapter 10 Elastic Waves
189(11)
10.1 Basic Notation
190(3)
10.2 Plane Wave Solutions
193(2)
10.3 Surface waves
195(3)
10.4 Love Waves
198(2)
Chapter 11 Surface Gravity Waves
200(37)
11.1 The Basic Fluid Equations
201(2)
11.2 The Dispersion Relation
203(11)
11.2.1 Deep Water Waves
203(1)
11.2.2 Shallow Water Waves
204(1)
11.2.3 Instability
205(5)
11.2.4 Group Speed Again
210(2)
11.2.5 Wavepackets
212(2)
11.3 Ship Waves
214(15)
11.3.1 How Does Dispersion Affect the Wave Pattern Produced by a Moving Object?
214(4)
11.3.2 Whitham's Ship Wave Analysis
218(3)
11.3.3 A Geometric Approach to Ship Waves and Wakes
221(6)
11.3.4 Ship Waves in Shallow Water
227(2)
11.4 A Discrete Approach
229(2)
11.4.1 Long Waves
229(1)
11.4.2 Short Waves
230(1)
11.5 Further Analysis for Surface Gravity Waves
231(6)
Chapter 12 Ocean Acoustics
237(18)
12.1 Ocean Acoustic Waveguides
237(4)
12.1.1 The Governing Equation
237(2)
12.1.2 Low Velocity Central Layer
239(1)
12.1.3 Leaky Modes
240(1)
12.2 One-Dimensional Waves in an Inhomogeneous Medium
241(6)
12.2.1 An Eigenfunction Expansion
242(3)
12.2.2 Poles
245(2)
12.3 Model for a Stratified Fluid: Cylindrical Geometry
247(3)
12.4 The Sech-Squared Potential Well
250(5)
12.4.1 Positive Energy States
250(3)
12.4.2 Bound States
253(2)
Chapter 13 Tsunamis
255(18)
13.1 Mathematical Model of Tsunami Propagation (Transient Waves)
255(2)
13.2 The Boundary-Value Problem
257(1)
13.3 Special Case I: Tsunami Generation by a Displacement of the Free Surface
258(10)
13.3.1 A Digression: Surface Waves on Deep Water (Again)
263(2)
13.3.2 How Fast Does the Wave Energy Propagate?
265(2)
13.3.3 Kinematics Again
267(1)
13.4 Leading Waves Due to a Transient Disturbance
268(2)
13.5 Special Case 2: Tsunami Generation by a Displacement of the Seafloor
270(3)
Chapter 14 Atmospheric Waves
273(26)
14.1 Governing Linearized Equations
274(11)
14.2 A Mathematical Model of Lee/Mountain Waves over an Isolated Mountain Ridge
285(7)
14.2.1 Basic Equations and Solutions
286(2)
14.2.2 An Isolated Ridge
288(2)
14.2.3 Trapped Lee Waves
290(2)
14.3 Billow Clouds, Wind Shear, and Howard's Semicircle Theorem
292(4)
14.4 The Taylor-Goldstein Equation
296(3)
Part III: Classical Scattering 299(114)
Chapter 15 The Classical Connection
301(15)
15.1 Lagrangians, Action, and Hamiltonians
301(3)
15.2 The Classical Wave Equation
304(4)
15.3 Classical Scattering: Scattering Angles and Cross Sections
308(8)
15.3.1 Overview
308(5)
15.3.2 The Classical Inverse Scattering Problem
313(3)
Chapter 16 Gravitational Scattering
316(16)
16.1 Planetary Orbits: Scattering by a Gravitational Field
317(8)
16.1.1 Repulsive Case: k > 0
318(1)
16.1.2 Attractive Case: k < 0
319(1)
16.1.3 The Orbits
319(6)
16.2 The Hamilton-Jacobi Equation for a Central Potential
325(7)
16.2.1 The Kepler Problem Revisited
326(1)
16.2.2 Generalizations
327(1)
16.2.3 Hard Sphere Scattering
328(1)
16.2.4 Rutherford Scattering
329(3)
Chapter 17 Scattering of Surface Gravity Waves by Islands, Reefs, and Barriers
332(16)
17.1 Trapped Waves
333(1)
17.2 The Scattering Matrix S(a)
334(3)
17.3 Trapped Modes: Imaginary Poles of S(alpha)
337(1)
17.4 Properties of S(alpha) for element of a in Real numbers
338(2)
17.5 Submerged Circular Islands
340(2)
17.6 Edge Waves on a Sloping Beach
342(6)
17.6.1 One-Dimensional Edge Waves on a Constant Slope
345(1)
17.6.2 Wave Amplication by a Sloping Beach
345(3)
Chapter 18 Acoustic Scattering
348(23)
18.1 Scattering by a Cylinder
350(2)
18.2 Time-Averaged Energy Flux: A Little Bit of Physics
352(2)
18.3 The Impenetrable Sphere
354(5)
18.3.1 Introduction: Spherically Symmetric Geometry
354(2)
18.3.2 The Scattering Amplitude Revisited
356(2)
18.3.3 The Optical Theorem
358(1)
18.3.4 The Sommerfeld Radiation Condition
358(1)
18.4 Rigid Sphere: Small ka Approximation
359(2)
18.5 Acoustic Radiation from a Rigid Pulsating Sphere
361(3)
18.6 The Sound of Mountain Streams
364(7)
18.6.1 Bubble Collapse
367(2)
18.6.2 Playing with Mathematical Bubbles
369(2)
Chapter 19 Electromagnetic Scattering: The Mie Solution
371(26)
19.1 Maxwell's Equations of Electromagnetic Theory
378(1)
19.2 The Vector Helmholtz Equation for Electromagnetic Waves
379(4)
19.3 The Lorentz-Mie solution
383(14)
19.3.1 Construction of the Solution
386(6)
19.3.2 The Rayleigh Scattering Limit: A Condensed Derivation
392(2)
19.3.3 The Radiation Field Generated by a Hertzian Dipole
394(3)
Chapter 20 Diffraction of Plane Electromagnetic Waves by a Cylinder
397(16)
20.1 Electric Polarization
398(8)
20.2 More about Classical Diffraction
406(9)
20.2.1 Huygen's Principle
406(1)
20.2.2 The Kirchhoff-Huygens Diffraction Integral
406(3)
20.2.3 Derivation of the Generalized Airy Diffraction Pattern
409(4)
Part IV: Semiclassical Scattering 413(62)
Chapter 21 The Classical-to-Semiclassical Connection
415(19)
21.1 Introduction: Classical and Semiclassical Domains
415(1)
21.2 Introduction: The Semiclassical Formulation
416(4)
21.2.1 The Total Scattering Cross Section
418(1)
21.2.2 Classical Wave Connections
419(1)
21.3 The Scalar Wave Equation
420(3)
21.3.1 Separation of Variables
420(2)
21.3.2 Bauer's Expansion Again
422(1)
21.4 The Radial Equation: Further Details
423(3)
21.5 Some Examples
426(8)
21.5.1 Scattering by a One-Dimensional Potential Barrier
426(2)
21.5.2 The Radially Symmetric Problem: Phase Shifts and the Potential Well
428(6)
Chapter 22 The WKB(J) Approximation Revisited
434(25)
22.1 The Connection Formulas revisited: An Alternative Approach
435(2)
22.2 Tunneling: A Physical Discussion
437(1)
22.3 A Triangular Barrier
438(2)
22.4 More Nuts and Bolts
440(8)
22.4.1 The Phase Shift
445(1)
22.4.2 Some Comments on Convergence
445(1)
22.4.3 The Transition to Classical Scattering
446(2)
22.5 Coulomb Scattering: The Asymptotic Solution
448(5)
22.5.1 Parabolic Cylindrical Coordinates xi, eta, phi)
449(1)
22.5.2 Asymptotic Form of 1F1 (-imu, 1;ikxi)
450(1)
22.5.3 The Spherical Coordinate System Revisited
451(2)
22.6 Coulomb Scattering: The WKB(J) Approximation
453(6)
22.6.1 Coulomb Phases
453(1)
22.6.2 Formal WKB(J) Solutions for the TIRSE
454(2)
22.6.3 The Langer Transformation: Further Justification
456(3)
Chapter 23 A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrodinger Equation
459(16)
23.1 Various Theorems
460(3)
23.2 Bound States
463(8)
23.2.1 Bound-State Theorems
463(4)
23.2.2 Complex Eigenvalues: Identities for Im(lambdan) and Re(lambdan)
467(1)
23.2.3 Further Theorems
468(3)
23.3 Weyl's Theorem: Limit Point and Limit Circle
471(4)
Part V: Special Topics In Scattering Theory 475(62)
Chapter 24 The S-Matrix and Its Analysis
477(14)
24.1 A Square Well Potential
477(10)
24.1.1 The Bound States
480(1)
24.1.2 Square Well Resonance: A Heuristic Derivation of the Breit-Wigner Formula
480(1)
24.1.3 The Watson Transform and Regge Poles
481(6)
24.2 More Details for the TIRSE
487(2)
24.3 Levinson's Theorem
489(2)
Chapter 25 The Jost Solutions: Technical Details
491(13)
25.1 Once More the TIRSE
491(3)
25.2 The Regular Solution Again
494(4)
25.3 Poles of the S-Matrix
498(6)
25.3.1 Wavepacket Approach
501(3)
Chapter 26 One-Dimensional Jost Solutions: The S-Matrix Revisited
504(8)
26.1 Transmission and Reflection Coefficients
504(3)
26.1.1 Poles of the Transmission Coefficient: Zeros of c12(k)
506(1)
26.2 The Jost Formulation on [ 0,infinity): The Radial Equation Revisited
507(5)
26.2.1 Jost Boundary Conditions at r = 0
507(1)
26.2.2 Jost Boundary Conditions as r goes to infinity
508(1)
26.2.3 The Jost Function and the S-Matrix
508(1)
26.2.4 Scattering from a Constant Spherical Inhomogeneity
509(3)
Chapter 27 Morphology-Dependent Resonances: The Effective Potential
512(11)
27.1 Some Familiar Territory
512(11)
27.1.1 A Toy Model for l does not equal 0 Resonances: A Particle Analogy
516(5)
27.1.2 Resonances
521(2)
Chapter 28 Back Where We Started
523(16)
28.1 A Bridge over Colored Water
523(8)
28.2 Ray Optics Revisited: Luneberg Inversion and Gravitational Lensing
531(6)
28.2.1 Abel's Integral Equation and the Luneberg Lens
531(3)
28.2.2 Connection with Classical Scattering and Gravitational Lensing
534(3)
Appendix A: Order Notation: The "Big 0," "Little o," and "~" Symbols 537(2)
Appendix B: Ray Theory: Exact Solutions 539(11)
B.1 Profile 1
540(1)
B.2 Profile 2
541(1)
B.3 Profile 3
542(1)
B.4 Profile 4
543(1)
B.5 Profile 5
543(1)
B.6 Profile 6
544(1)
B.7 Profile 7
545(1)
B.8 Profile 8
546(1)
B.9 Profile 9
546(1)
B.10 Profile 10
547(3)
Appendix C: Radially Inhomogeneous Spherically Symmetric Scattering: The Governing Equations 550(3)
C.1 The Tranverse Magnetic Mode
550(1)
C.2 The Tranverse Electric Mode
551(2)
Appendix D: Electromagnetic Scattering from a Radially Inhomogeneous Sphere 553(6)
D.1 A classical/Quantum connection for Transverse Electric and Magnetic Modes
553(3)
D.2 A Liouville Transformation
556(3)
Appendix E: Helmholtz's Theorem 559(3)
E.1 Proof of Helmholtz's Theorem
559(1)
E.2 Lame's Theorem
560(2)
Appendix F:Semiclassical Scattering: A Precis (and a Few More Details) 562(5)
Bibliography 567(18)
Index 585
John A. Adam is professor of mathematics at Old Dominion University. His books include X and the City: Modeling Aspects of Urban Life, Mathematics in Nature: Modeling Patterns in the Natural World, and Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin (all Princeton).
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