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Acoustic High-Frequency Diffraction Theory

Book Description

Covering analytical research in aerial and underwater acoustics, this new scholarly work treats the interaction of acoustic waves with obstacles which may be rigid, soft, elastic, or characterized by an impedance boundary condition. The approach is founded on asymptotic high frequency methods which are based on the concept of rays. For despite the progress in numerical methods for diffraction problems, ray methods still remain the most useful method of approximation for analyzing wave motions. They provide not only considerable physical insight and understanding of diffraction mechanisms but they are also able to treat objects which are still too large in terms of wavelength to fall in the realm of numerical analysis.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Contents
  5. List of Figures
  6. List of Tables
  7. About the Author
  8. Preface
  9. Chapter 1  Introduction to the Geometrical Theory of Diffraction
    1. 1.1.    General Overview and Basic Concepts
    2. 1.2.    The Fermat Principle
      1. 1.2.1.    Conditions for a Path to Be a Ray
      2. 1.2.2.    Application of Conditions (1.5) and (1.6) to Specific Problems
        1.    Segments in a Fluid with Constant Sound Velocity
        2.    Reflection from a Smooth Surface
        3.    Transmission through a Smooth Interface between Two Different Homogeneous Fluids
        4.    Excitation of Elastic Waves at a Smooth Interface between a Homogeneous Fluid and an Isotropic Homogeneous Elastic Body
        5.    Diffraction by an Edge in a Homogeneous Fluid
        6.    Surface Rays
    3. 1.3.    Extension of Fermat’s Principle to Surface Waves
      1. 1.3.1.    Planar Interface
      2. 1.3.2.    Curved Interface
    4. 1.4.    Fundamentals of Asymptotic Expansions
      1. 1.4.1.    Asymptotic Sequence
      2. 1.4.2.    Compatible Asymptotic Sequence
      3. 1.4.3.    Properties of an Asymptotic Expansion
    5. 1.5.    Asymptotic Solution of the Wave Equation in a Source-Free, Unbounded Medium
      1. 1.5.1.    Derivation of the Asymptotic Expansion of the Solution
      2. 1.5.2.    Resolution of the Eikonal Equation
      3. 1.5.3.    Properties of the Characteristic Curves
      4. 1.5.4.    Resolution of the Transport Equation
    6. 1.6.    Acoustic Field Reflected by a Smooth, Rigid, Soft, or Impedance Surface
    7. 1.7.    Reflected and Transmitted Waves at a Smooth Interface between a Fluid and an Elastic Medium
    8. 1.8.    Acoustic Field Diffracted by the Edge of an Impenetrable Wedge
    9. 1.9.    Acoustic Field in the Shadow Zone of a Smooth Convex Object
    10. References
  10. Chapter 2  Canonical Problems and Nonuniform Asymptotic Theory of Acoustic Wave Diffraction
    1. 2.1.    Introduction
    2. 2.2.    The Wedge
      1. 2.2.1.    Hard and Soft Straight Wedge at Normal and Oblique Incidence
        1.    Line Source Excitation
        2.    Plane Wave Incidence
      2. 2.2.2.    Impedance Wedge at Normal and Oblique Incidence: Maliuzhinets’s Solution
        1.    Analytical Details for the Justification of the General Form of the Solution
        2.    Derivation of the Solution
        3.    Asymptotic Evaluation of the Solution
        4.    Generalization to Oblique Incidence
      3. 2.2.3.    Elastic Wedge-Shaped Shell
        1.    Derivation of the Solution
        2.    Asymptotic Evaluation of the Solution
        3.    Generalization of the Elastic Wedge-Shaped Shell Solution to Oblique Incidence
        4.    Typical Examples
    3. 2.3.    The Circular Cylinder
      1. 2.3.1.    Circular Cylinder with Hard, Soft, or Impedance Boundary Conditions
        1.    General Solution to the Problem
        2.    Asymptotic Expansion and Physical Interpretation of the Solution
        3.    Expression of the Diffraction Coefficient
        4.    Noncircular Convex Cylinder
        5.    Field at an Observation Point Located on the Surface of a Circular Cylinder
        6.    Position of the Poles in the Complex ν-Plane
        7.    Oblique Incidence
      2. 2.3.2.    Three-Dimensional Convex Surface with Hard, Soft, or Impedance Boundary Conditions
      3. 2.3.3.    Hollow Elastic Circular Cylinder
        1.    General Solution for Normal Incidence
        2.    Watson Transformation
        3.    Asymptotic Expansion and Physical Interpretation of the Solution
        4.    Field Diffracted in the Shadow Region by Creeping Waves
        5.    Noncircular Convex Cylindrical Shell
        6.    Oblique Incidence for Creeping Waves
        7.    Elastic Surface Waves
        8.    Oblique Incidence for Elastic Surface Waves
        9.    Noncircular Cylindrical Shell at Normal and Oblique Incidence
      4. 2.3.4.    Three-Dimensional Convex Shell
        1.    General Solution for Creeping Waves
        2.    General Solution for Elastic Surface Waves
    4. 2.4.    Concave Surface
      1. 2.4.1.    Introduction
      2. 2.4.2.    Solution of the Canonical Problem of a Line Source Parallel to the Generatrix of a Concave Circular Cylinder
        1.    Derivation of the Solution
        2.    Extraction of the GA Contributions
        3.    Expression of the Remainder Integral
      3. 2.4.3.    Extension of the Solution to More Generalized Situations
    5. References
  11. Chapter 3  Uniform Asymptotic Theory of Acoustic Wave Diffraction
    1. 3.1.    Introduction
    2. 3.2.    The Three-Dimensional Convex Wedge
      1. 3.2.1.    Uniform Solution for the Acoustic Field Scattered by a Convex Three-Dimensional Hard or Soft Wedge
        1.    Uniform Asymptotic Solution of a Hard or Soft Wedge with Planar Faces Submitted to a Planar Sound Wave
        2.    Uniform Asymptotic Solutions for a Soft or Hard Wedge with Curved Faces
      2. 3.2.2.    Uniform Asymptotic Solutions for the Acoustic Field Scattered by a Convex Impedance Wedge
        1.    Uniform Asymptotic Solution of an Impedance Wedge with Planar Faces
        2.    Uniform Asymptotic Solution for a General Three-Dimensional Impedance Wedge
      3. 3.2.3.    Uniform Asymptotic Solutions for the Acoustic Field Scattered by a Convex Wedge-Shaped Elastic Shell
        1.    Uniform Asymptotic Solution for a Wedge-Shaped Shell with Planar Faces
        2.    Uniform Asymptotic Solution for a Three-Dimensional Wedge-Shaped Shell
    3. 3.3.    The Three-Dimensional Smooth Convex Surface
      1. 3.3.1.    Uniform Asymptotic Solution through the Shadow Boundary of a Smooth Convex Two-Dimensional Surface
        1.    Uniform Asymptotic Solution for a Circular Impedance Cylinder
        2.    Uniform Asymptotic Solution for a Smooth Convex Impedance Cylinder
        3.    Field in the Boundary Layer of a Smooth Convex Impedance Cylinder
      2. 3.3.2.    Uniform Asymptotic Solution through the Shadow Boundary of a Smooth Convex Three-Dimensional Surface
    4. 3.4.    The Three-Dimensional Smooth Convex Shell
      1. 3.4.1.    Uniform Solution for a Hollow Elastic Cylinder
        1.    Uniform Solution for the GA Field Associated with the Creeping Wave Field
        2.    Uniform Solution for the Reflected Field Associated with the Elastic Surface Wave Field
        3.    Oblique Incidence
        4.    Noncircular Cylindrical Shell
      2. 3.4.2.    Uniform Solution for a Three-Dimensional Convex Shell
        1.    Uniform Solution for Creeping Waves
        2.    Uniform Solution for Elastic Surface Waves
    5. 3.5.    Numerical Results
      1. 3.5.1.    Results Concerning the Wedge
      2. 3.5.2.    Results Concerning the Circular Cylinder
    6. References
  12. Chapter 4  Wave Field Near a Caustic
    1. 4.1.    Introduction
    2. 4.2.    Techniques for Calculating the Field on a Caustic and in Its Neighborhood
      1. 4.2.1.    Observation Point Located Close to a Regular Caustic
        1.    Canonical Problem for a Regular Caustic
        2.    Uniform Asymptotic Expansion for a Regular Three-Dimensional Caustic
        3.    Caustic of Rays Reflected by a Smooth Surface
        4.    Caustic of Rays Diffracted by an Edge
        5.    Caustic of Rays Diffracted by a Smooth Surface (Creeping Rays)
        6.    Comments on the Solutions for a Regular Caustic
      2. 4.2.2.    Observation Point Located Close to a Line Cusp of a Caustic
    3. References
  13. Chapter 5  Hybrid Diffraction Coefficients
    1. 5.1.    Introduction
    2. 5.2.    Edge-Excited and Edge-Diffracted Creeping Waves
      1. 5.2.1.    Spectral Representation of the Fock Field on a Smooth Convex Cylindrical Surface
      2. 5.2.2.    Hybrid Diffraction Coefficients for Creeping Waves on a Curved Wedge
        1.    Two-Dimensional Wedge
        2.    Three-Dimensional Wedge
        3.    Solution Valid at Grazing Incidence and Grazing Observation
    3. 5.3.    Edge-Excited and Edge-Diffracted Surface Waves
      1. 5.3.1.    Impedance Wedge
      2. 5.3.2.    Elastic Wedge-Shaped Shell
      3. 5.3.3.    Elastic Surface Wave on a Curved Wedge
    4. 5.4.    Edge-Excited and Edge-Diffracted Whispering Gallery Waves
    5. References
  14. Appendix A    A Brief Presentation of the Governing Equations for Wave Processes in Fluids
    1. A.1.    Wave Propagation in Ideal Fluids
    2. A.2.    Sound Waves
    3. A.3.    Boundary Conditions
      1. A.3.1.    Absolutely Rigid Boundary
      2. A.3.2.    Absolutely Soft Boundary
      3. A.3.3.    Interface between Tw o Fluids at Rest
    4. A.4.    Harmonic Waves
    5. A.5.    Reflection at a Boundary
      1. A.5.1.    Absolutely Rigid Boundary
      2. A.5.2.    Absolutely Soft Boundary
      3. A.5.3.    Impedance Boundary
    6. A.6.    Reflection and Refraction at the Interface of Two Homogeneous Fluids
    7. References
  15. Appendix B    A Brief Presentation of the Governing Equations of Linearized Elasticity
    1. B.1.    Deformation
    2. B.2.    Linear Momentum and Stress Tensor
    3. B.3.    Hooke’s Law
    4. B.4.    Waves in an Elastic Medium
    5. B.5.    Boundary Conditions at Interfaces
    6. B.6.    Elastic Waves in Homogeneous Isotropic Solids
    7. B.7.    Harmonic Waves
    8. B.8.    Reflection and Refraction at a Plane Interface
    9. B.9.    Reflection on an Elastic Plate Separating a Fluid from Vacuum
      1. B.9.1.    Application of the Boundary Conditions
      2. B.9.2.    Solution of the System of Linear Equations
    10. References
  16. Appendix C    Surface Waves
    1. C.1.    Introduction
    2. C.2.    Rayleigh Waves
    3. C.3.    Surface Waves at Fluid–Solid Interfaces
    4. C.4.    Leaky Waves
    5. References
  17. Appendix D    General Formulas for the Principal Radii of Curvature of the Reflected Wave Front on a Three-Dimensional Surface
    1. References
  18. Appendix E    Symmetric Form of the Maliuzhinets Diffraction Coefficient
  19. Appendix F    Elements of the Determinant of the Boundary Conditions for a Circular Elastic Shell in a Fluid
    1. F.1.    Normal Incidence
    2. F.2.    Oblique Incidence
  20. Index
  21. Back Cover