Classical asymptotic expansions, while producing a good approximation for the diffracted fields in general, appear hardly applicable in the case of extremely elongated bodies. Thus, there are problems that are on the one hand too difficult for numerical solvers due to large system size, and on the other hand make the description with classical asymptotic methods hard. The book explains why this happens and suggests the way out. By defining the characteristics of a strongly elongated body it introduces a special class of asymptotic approximations, which are in some sense uniform with respect to the rate of body elongation.
Chapter 1 briefly describes the results of V. A. Fock and further developments of his approach towards the problems of diffraction by elongated obstacles. It formulates the cases of moderately and strongly elongated bodies. The rest of the book describes the approach of special parabolic equations, which lead to new asymptotic approximations for the diffracted fields. Chapters 2, 3 and 4 discuss diffraction by bodies of elliptical shape: The elliptic cylinder with a strongly elongated cross section and prolate spheroid with a high aspect ratio. Chapter 5 generalizes the approach to some other shapes such as narrow cones and narrow hyperboloids. Mathematical formulas for the Whittaker functions widely used in the book are collected in the Appendix.
The concise derivations are supplied with numerous test examples that compare asymptotic approximations with numerically computed fields and clarify the specifics of high frequency diffraction by strongly elongated bodies. The reference solutions presented in the book enable one to validate the newly developed numerical solvers.
Author(s): Ivan Andronov
Series: Springer Series in Optical Sciences, 243
Publisher: Springer
Year: 2023
Language: English
Pages: 193
City: Singapore
Contents
List of Figures
1 High-Frequency Diffraction and Elongated Bodies
1.1 High-Frequency Methods
1.2 Diffraction by a Smooth Convex Body
1.2.1 Analysis of the Field of Rays
1.2.2 The Parabolic Equation Method
1.2.3 Asymptotic Expansion in the Fock Domain
1.2.4 Generalizations to 3D and Electromagnetic Problems
1.2.5 The Higher Order Corrections
1.3 Accuracy of the Fock Approximation
1.4 The Transverse Curvature Effects
1.4.1 Transverse Curvature in the Classical Fock Asymptotic Approximation
1.4.2 Moderately Elongated Bodies
1.4.3 Strongly Elongated Bodies
1.5 Conclusion
References
2 Diffraction by an Elliptic Cylinder
2.1 Introduction
2.2 Stretched Coordinates and Separation of Variables
2.3 The Forward Wave
2.3.1 Asymptotic Representation
2.3.2 Integral Representation of the Incident Plane Wave
2.3.3 Integral Representation of the Line Source Field
2.4 The Backward Wave
2.4.1 Integral Representation for the Backward Wave
2.4.2 The Reflection Coefficients
2.5 The Induced Currents
2.5.1 Integral Representations for the Currents
2.5.2 The Case of Plane Wave Incidence
2.5.3 Reduction to Fock Asymptotics in the Case of χtoinfty
2.5.4 The Case of Line Source Field Incidence
2.5.5 Accuracy of Approximation and Test Examples
2.5.6 Results for the Line Source Field Diffraction
2.6 The Far Field
2.6.1 Far Field in the Forward Cone
2.6.2 Numerical Examples and Tests for the Far Field
2.6.3 Far Field in the Backward Cone
2.7 Conclusion
References
3 Acoustic Wave Diffraction by Spheroid
3.1 Introduction
3.2 Problem Formulation and Assumptions on the Parameters
3.3 Stretched Coordinates and Separation of Variables
3.4 The Forward Wave
3.4.1 Integral Representation
3.4.2 The Incident Plane Wave Representation
3.4.3 The Incident Spherical Wave Representation
3.4.4 The Currents
3.5 Vicinities of the End-points and the Backward Diffracted Wave
3.6 Numerical Results and Validations for the Currents
3.7 The Far Field
3.7.1 Kirchhoff Integral
3.7.2 The Far Field Approximation in the Forward Cone
3.7.3 The Far Field Approximation in the Backward Cone
3.8 Conclusion and Discussion
References
4 Electromagnetic Wave Diffraction by Spheroid
4.1 Introduction
4.2 The Problem Formulation and the Boundary-Layer Coordinates
4.3 Representation for the Forward Wave
4.4 Matching with the Incident Wave
4.4.1 The Plane Wave Incidence
4.4.2 Point Dipole Source Field Incidence
4.5 Backward Wave
4.5.1 Representation of the Field Near the Trailing Tip
4.5.2 Determining the Amplitudes λn,j and νn,j
4.5.3 Representation for the Backward Wave
4.6 Induced Currents
4.7 Validation Tests
4.8 The Far Field
4.9 Test Examples
References
5 Other Strongly Elongated Shapes
5.1 Introduction
5.2 The Boundary-Layer Coordinates
5.3 Surfaces Which Allow Variable Separation
5.4 Strongly Elongated Two-Sheeted Hyperboloid
5.4.1 Boundary-Layer Coordinates and the Parabolic Equation
5.4.2 Axial Incidence
5.4.3 Skew Incidence
5.5 Strongly Elongated Paraboloid
5.6 One-Sheeted Hyperboloid
5.7 Narrow Cone
5.7.1 Problem of Diffraction
5.7.2 Diffraction of the Plane Acoustic Wave
5.7.3 Diffraction of the Plane Electromagnetic Wave
5.8 Conclusion
References
Appendix A Airy and Whittaker Functions
A.1 Airy Functions
A.2 Whittaker Functions
A.2.1 Integral Representations and Functional Relations
A.2.2 The Coulomb Wave Functions
A.2.3 Asymptotic Expansions
A.3 Solutions of Dispersion Equations
A.4 The Integral Transform
A.5 Integrals Containing Whittaker Functions
References
Index
