- Title Pages
- Epigraph
- Preface
- Acknowledgments
- Chapter One Introduction
- Chapter Two Introduction to the “Physics” of Rays
- Chapter Three Introduction to the Mathematics of Rays
- Chapter Four Ray Optics: The Classical Rainbow
- Chapter Five An Improvement over Ray Optics: Airy’s Rainbow
- Chapter Six Diffraction Catastrophes
- Chapter Seven Introduction to the WKB(J) Approximation: All Things Airy
- Chapter Eight Island Rays
- Chapter Nine Seismic Rays
- Chapter Ten Elastic Waves
- Chapter Eleven Surface Gravity Waves
- Chapter Twelve Ocean Acoustics
- Chapter Thirteen Tsunamis
- Chapter Fourteen Atmospheric Waves
- Chapter Fifteen The Classical Connection
- Chapter Sixteen Gravitational Scattering
- Chapter Seventeen Scattering of Surface Gravity Waves by Islands, Reefs, and Barriers
- Chapter Eighteen Acoustic Scattering
- Chapter Nineteen Electromagnetic Scattering: The Mie Solution
- Chapter Twenty Diffraction of Plane Electromagnetic Waves by a Cylinder
- Chapter Twenty-One The Classical-to-Semiclassical Connection
- Chapter Twenty-Two The WKB(J) Approximation Revisited
- Chapter Twenty-Three A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation
- Chapter Twenty-Four The <i>S</i>-Matrix and Its Analysis
- Chapter Twenty-Five The Jost Solutions: Technical Details
- Chapter Twenty-Six One-Dimensional Jost Solutions: The <i>S</i>-Matrix Revisited
- Chapter Twenty-Seven Morphology-Dependent Resonances: The Effective Potential
- Chapter Twenty-Eight Back Where We Started
- Appendix A Order Notation: The “Big <i>O</i>,” “Little <i>o</i>,” and “∼” Symbols
- Appendix B Ray Theory: Exact Solutions
- Appendix C Radially Inhomogeneous Spherically Symmetric Scattering: The Governing Equations
- Appendix D Electromagnetic Scattering from a Radially Inhomogeneous Sphere
- Appendix E Helmholtz’s Theorem
- Appendix F Semiclassical Scattering: A Précis (and a Few More Details)
- Bibliography
- Index
- Princeton Series in Applied Mathematics

# Introduction to the WKB(J) Approximation: All Things Airy

# Introduction to the WKB(J) Approximation: All Things Airy

- Chapter:
- (p.137) Chapter Seven Introduction to the WKB(J) Approximation: All Things Airy
- Source:
- Rays, Waves, and Scattering
- Author(s):
### John A. Adam

- Publisher:
- Princeton University Press

This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order in principle) with spatially varying coefficients. The WKB(J) approximation is closely related to the semiclassical approach in quantum mechanics in which the wavefunction is characterized by a slowly varying amplitude and/or phase. The chapter first introduces an inhomogeneous differential equation, from which the first derivative term is eliminated, before discussing the Liouville transformation and the one-dimensional Schrödinger equation. It then presents a physical interpretation of the WKB(J) approximation and its application to a potential well. It also considers the “patching region” in which the Airy function solution (the local turning point) is valid, the relation between Airy functions and Bessel functions, Airy integral and related topics, and related integrals.

*Keywords:*
differential equations, WKB(J) approximation, Liouville transformation, Schrödinger equation, potential well, Airy functions, Bessel functions, Airy integral, integrals

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- Title Pages
- Epigraph
- Preface
- Acknowledgments
- Chapter One Introduction
- Chapter Two Introduction to the “Physics” of Rays
- Chapter Three Introduction to the Mathematics of Rays
- Chapter Four Ray Optics: The Classical Rainbow
- Chapter Five An Improvement over Ray Optics: Airy’s Rainbow
- Chapter Six Diffraction Catastrophes
- Chapter Seven Introduction to the WKB(J) Approximation: All Things Airy
- Chapter Eight Island Rays
- Chapter Nine Seismic Rays
- Chapter Ten Elastic Waves
- Chapter Eleven Surface Gravity Waves
- Chapter Twelve Ocean Acoustics
- Chapter Thirteen Tsunamis
- Chapter Fourteen Atmospheric Waves
- Chapter Fifteen The Classical Connection
- Chapter Sixteen Gravitational Scattering
- Chapter Seventeen Scattering of Surface Gravity Waves by Islands, Reefs, and Barriers
- Chapter Eighteen Acoustic Scattering
- Chapter Nineteen Electromagnetic Scattering: The Mie Solution
- Chapter Twenty Diffraction of Plane Electromagnetic Waves by a Cylinder
- Chapter Twenty-One The Classical-to-Semiclassical Connection
- Chapter Twenty-Two The WKB(J) Approximation Revisited
- Chapter Twenty-Three A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation
- Chapter Twenty-Four The <i>S</i>-Matrix and Its Analysis
- Chapter Twenty-Five The Jost Solutions: Technical Details
- Chapter Twenty-Six One-Dimensional Jost Solutions: The <i>S</i>-Matrix Revisited
- Chapter Twenty-Seven Morphology-Dependent Resonances: The Effective Potential
- Chapter Twenty-Eight Back Where We Started
- Appendix A Order Notation: The “Big <i>O</i>,” “Little <i>o</i>,” and “∼” Symbols
- Appendix B Ray Theory: Exact Solutions
- Appendix C Radially Inhomogeneous Spherically Symmetric Scattering: The Governing Equations
- Appendix D Electromagnetic Scattering from a Radially Inhomogeneous Sphere
- Appendix E Helmholtz’s Theorem
- Appendix F Semiclassical Scattering: A Précis (and a Few More Details)
- Bibliography
- Index
- Princeton Series in Applied Mathematics