Chapter
3.7 Rays and Waves in a Slowly Varying Environment
3.7.2 Wavepackets and the Group Speed Revisited
Chapter 4 Ray Optics: The Classical Rainbow
4.1 Physical Features and Historical Details: A Summary
4.2 Ray Theory of the Rainbow: Elementary Mathematical Considerations
4.2.1 Some Numerical Values
4.2.2 Polarization of the Rainbow
4.2.3 The Divergence Problem
4.3 Related Topics in Meteorological Optics
4.3.2 Coronas (Simplified)
4.3.3 Rayleigh Scattering—a Dimensional Analysis Argument
Chapter 5 An Improvement over Ray Optics: Airy’s Rainbow
5.1 The Airy Approximation
5.1.1 Some Ray Prerequisites
5.1.3 How Are Colors Distributed in the Airy Rainbow?
5.1.4 The Airy Wavefront: A Derivation for Arbitrary p
Chapter 6 Diffraction Catastrophes
6.1 Basic Geometry of the Fold and Cusp Catastrophes
6.2 A Better Approximation
6.2.1 The Fresnel Integrals
6.3 The Fold Diffraction Catastrophe
6.3.1 The Rainbow as a Fold Catastrophe
6.4 Caustics: The Airy Integral in the Complex Plane
6.4.1 The Nature of Ai(X)
Chapter 7 Introduction to the WKB(J) Approximation: All Things Airy
7.1.1 Elimination of the First Derivative Term
7.1.2 The Liouville Transformation
7.1.3 The One-Dimensional Schrödinger Equation
7.1.4 Physical Interpretation of the WKB(J) Approximation
7.1.5 The WKB(J) Connection Formulas
7.1.6 Application to a Potential Well
7.3 Matching Across a Turning Point
7.4 A Little More about Airy Functions
7.4.1 Relation to Bessel Functions
7.4.2 The Airy Integral and Related Topics
8.1 Straight and Parallel Depth Contours
8.1.1 Plane Wave Incident on a Ridge
8.1.2 Wave Trapping on a Ridge
8.2 Circular Depth Contours
9.1 Seismic Ray Equations
9.2 Ray Propagation in a Spherical Earth
9.2.1 A Horizontally Stratified Earth
9.2.2 The Wiechert-Herglotz Inversion
9.2.3 Further Properties of X in the Horizontally Stratified Case
10.2 Plane Wave Solutions
Chapter 11 Surface Gravity Waves
11.1 The Basic Fluid Equations
11.2 The Dispersion Relation
11.2.2 Shallow Water Waves
11.3.1 How Does Dispersion Affect the Wave Pattern Produced by a Moving Object?
11.3.2 Whitham’s Ship Wave Analysis
11.3.3 A Geometric Approach to Ship Waves and Wakes
11.3.4 Ship Waves in Shallow Water
11.5 Further Analysis for Surface Gravity Waves
Chapter 12 Ocean Acoustics
12.1 Ocean Acoustic Waveguides
12.1.1 The Governing Equation
12.1.2 Low Velocity Central Layer
12.2 One-Dimensional Waves in an Inhomogeneous Medium
12.2.1 An Eigenfunction Expansion
12.3 Model for a Stratified Fluid: Cylindrical Geometry
12.4 The Sech-Squared Potential Well
12.4.1 Positive Energy States
13.1 Mathematical Model of Tsunami Propagation (Transient Waves)
13.2 The Boundary-Value Problem
13.3 Special Case I: Tsunami Generation by a Displacement of the Free Surface
13.3.1 A Digression: Surface Waves on Deep Water (Again)
13.3.2 How Fast Does the Wave Energy Propagate?
13.4 Leading Waves Due to a Transient Disturbance
13.5 Special Case 2: Tsunami Generation by a Displacement of the Seafloor
Chapter 14 Atmospheric Waves
14.1 Governing Linearized Equations
14.2 A Mathematical Model of Lee/Mountain Waves over an Isolated Mountain Ridge
14.2.1 Basic Equations and Solutions
14.3 Billow Clouds, Wind Shear, and Howard’s Semicircle Theorem
14.4 The Taylor-Goldstein Equation
PART III CLASSICAL SCATTERING
Chapter 15 The Classical Connection
15.1 Lagrangians, Action, and Hamiltonians
15.2 The Classical Wave Equation
15.3 Classical Scattering: Scattering Angles and Cross Sections
15.3.2 The Classical Inverse Scattering Problem
Chapter 16 Gravitational Scattering
16.1 Planetary Orbits: Scattering by a Gravitational Field
16.1.1 Repulsive Case: k > 0
16.1.2 Attractive Case: k < 0
16.2 The Hamilton-Jacobi Equation for a Central Potential
16.2.1 The Kepler Problem Revisited
16.2.3 Hard Sphere Scattering
16.2.4 Rutherford Scattering
Chapter 17 Scattering of Surface Gravity Waves by Islands, Reefs, and Barriers
17.2 The Scattering Matrix S(α)
17.3 Trapped Modes: Imaginary Poles of S(α)
17.4 Properties of S(α) for α ∈ R
17.5 Submerged Circular Islands
17.6 Edge Waves on a Sloping Beach
17.6.1 One-Dimensional Edge Waves on a Constant Slope
17.6.2 Wave Amplication by a Sloping Beach
Chapter 18 Acoustic Scattering
18.1 Scattering by a Cylinder
18.2 Time-Averaged Energy Flux: A Little Bit of Physics
18.3 The Impenetrable Sphere
18.3.1 Introduction: Spherically Symmetric Geometry
18.3.2 The Scattering Amplitude Revisited
18.3.3 The Optical Theorem
18.3.4 The Sommerfeld Radiation Condition
18.4 Rigid Sphere: Small ka Approximation
18.5 Acoustic Radiation from a Rigid Pulsating Sphere
18.6 The Sound of Mountain Streams
18.6.2 Playing with Mathematical Bubbles
Chapter 19 Electromagnetic Scattering: The Mie Solution
19.1 Maxwell’s Equations of Electromagnetic Theory
19.2 The Vector Helmholtz Equation for Electromagnetic Waves
19.3 The Lorentz-Mie solution
19.3.1 Construction of the Solution
19.3.2 The Rayleigh Scattering Limit: A Condensed Derivation
19.3.3 The Radiation Field Generated by a Hertzian Dipole
Chapter 20 Diffraction of Plane Electromagnetic Waves by a Cylinder
20.1 Electric Polarization
20.2 More about Classical Diffraction
20.2.1 Huygen’s Principle
20.2.2 The Kirchhoff-Huygens Diffraction Integral
20.2.3 Derivation of the Generalized Airy Diffraction Pattern
PART IV SEMICLASSICAL SCATTERING
Chapter 21 The Classical-to-Semiclassical Connection
21.1 Introduction: Classical and Semiclassical Domains
21.2 Introduction: The Semiclassical Formulation
21.2.1 The Total Scattering Cross Section
21.2.2 Classical Wave Connections
21.3 The Scalar Wave Equation
21.3.1 Separation of Variables
21.3.2 Bauer’s Expansion Again
21.4 The Radial Equation: Further Details
21.5.1 Scattering by a One-Dimensional Potential Barrier
21.5.2 The Radially Symmetric Problem: Phase Shifts and the Potential Well
Chapter 22 The WKB(J) Approximation Revisited
22.1 The Connection Formulas revisited: An Alternative Approach
22.2 Tunneling: A Physical Discussion
22.3 A Triangular Barrier
22.4.2 Some Comments on Convergence
22.4.3 The Transition to Classical Scattering
22.5 Coulomb Scattering: The Asymptotic Solution
22.5.1 Parabolic Cylindrical Coordinates (ξ, η, φ)
22.5.2 Asymptotic Form of 1F1(−iμ, 1; ikξ)
22.5.3 The Spherical Coordinate System Revisited
22.6 Coulomb Scattering: The WKB(J) Approximation
22.6.2 Formal WKB(J) Solutions for the TIRSE
22.6.3 The Langer Transformation: Further Justification
Chapter 23 A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation
23.2.1 Bound-State Theorems
23.2.2 Complex Eigenvalues: Identities for Im(λn) and Re(λn)
23.3 Weyl’s Theorem: Limit Point and Limit Circle
PART V SPECIAL TOPICS IN SCATTERING THEORY
Chapter 24 The S-Matrix and Its Analysis
24.1 A Square Well Potential
24.1.2 Square Well Resonance: A Heuristic Derivation of the Breit-Wigner Formula
24.1.3 The Watson Transform and Regge Poles
24.2 More Details for the TIRSE
Chapter 25 The Jost Solutions: Technical Details
25.2 The Regular Solution Again
25.3 Poles of the S-Matrix
25.3.1 Wavepacket Approach
Chapter 26 One-Dimensional Jost Solutions: The S-Matrix Revisited
26.1 Transmission and Reflection Coefficients
26.1.1 Poles of the Transmission Coefficient: Zeros of c12(k)
26.2 The Jost Formulation on [0,∞): The Radial Equation Revisited
26.2.1 Jost Boundary Conditions at r = 0
26.2.2 Jost Boundary Conditions as r →∞
26.2.3 The Jost Function and the S-Matrix
26.2.4 Scattering from a Constant Spherical Inhomogeneity
Chapter 27 Morphology-Dependent Resonances: The Effective Potential
27.1 Some Familiar Territory
27.1.1 A Toy Model for l ≠ 0 Resonances: A Particle Analogy
Chapter 28 Back Where We Started
28.1 A Bridge over Colored Water
28.2 Ray Optics Revisited: Luneberg Inversion and Gravitational Lensing
28.2.1 Abel’s Integral Equation and the Luneberg Lens
28.2.2 Connection with Classical Scattering and Gravitational Lensing
Appendix A Order Notation: The “Big O,” “Little o,” and “~” Symbols
Appendix B Ray Theory: Exact Solutions
Appendix C Radially Inhomogeneous Spherically Symmetric Scattering: The Governing Equations
C.1 The Tranverse Magnetic Mode
C.2 The Tranverse Electric Mode
Appendix D Electromagnetic Scattering from a Radially Inhomogeneous Sphere
D.1 A classical/Quantum connection for Transverse Electric and Magnetic Modes
D.2 A Liouville Transformation
Appendix E Helmholtz’s Theorem
E.1 Proof of Helmholtz’s Theorem
Appendix F Semiclassical Scattering: A Précis (and a Few More Details)
Rays, Waves, and Scattering
：Topics in Classical Mathematical Physics
( Princeton Series in Applied Mathematics )
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