Preface
Acknowledgements
Contents
Symbols (for Chaps. 2–6)
1 Introduction
1.1 Motivation
1.2 Rough Overview and Structure of the Book
1.3 Historical and Scientific Development of Complex Sources and Their Application
2 Complex Monopoles and the Helmholtz Equation in Cartesian Coordinates
3 Complex Monopoles in Oblate Spheroidal Coordinates
4 The Driving Source of the Complex Monopole
4.1 The Driving Source of the Complex Monopole in One Dimension (1D)
4.2 The Sommerfeld Integral for Real and Complex Source Locations
4.3 The Driving Source of the Complex Monopole in Two Dimensions (2D)
4.4 The Driving Source of the Complex Monopole in Three Dimensions (3D)
5 Application of Complex Sources for Half-Space Green’s Functions Above an Impedance Plane
5.1 General Impedance Formula
5.2 One-Dimensional Green’s Function on a Half-Line with Impedance Boundary Condition and Its Driving Sources
5.2.1 Half-Line Green’s Function
5.2.2 Driving Sources of the Half-Line Green’s Function
5.3 Two-Dimensional Green’s Function on a Half-Plane with Impedance Boundary Condition and Its Driving Sources
5.3.1 Half-Plane Green’s Function
5.3.2 Driving Sources of the Half-Plane Green’s Function
5.4 Three-Dimensional Green’s Function on a Half-Space with Impedance Boundary Condition and Its Driving Sources
5.4.1 Half-Space Green’s Function
5.4.2 Driving Sources of the Half-Space Green’s Function
5.5 Overview About the Sources and a Generalized Formula for 1D, 2D, and 3D
6 New and Old Formulas from the Helmholtz Equation with Half-Space Driving Sources
6.1 Solutions Obtained by Using the Helmholtz Huygens Integral
6.2 Solutions by Fourier Transform in z Followed by Fourier or Hankel Transform
6.2.1 2D Case: Solution by Fourier Transform in z Followed by Fourier Transform in x
6.2.2 3D Case: Solution by Fourier Transform in z Followed by Hankel Transform in (x, y)
6.3 Solutions Represented as an Integral Over Horizontal Plane Waves
6.3.1 Solutions Represented as an Integral Over Horizontal Plane Waves in 2D
6.3.2 Solutions Represented as an Integral Over Horizontal Plane Waves in 3D
7 Branch Cuts of the Square Root with Complex Argument
7.1 Complex Source Position
7.1.1 “Source” Branch
7.1.2 “Beam” Branch Cut
7.1.3 “Open Duct” Branch
7.2 Sommerfeld Branches
7.2.1 Branches of the Square Root µ= sqrtλ2 - k2 with Real k
7.2.2 Branches of the Square Root µ= sqrtλ2 - k2 with Complex k
8 Realization of Complex Sources
8.1 Sound Focusing—Beamwidth
8.2 Integral Representation of a Monopole
8.3 Integral Representation of a Complex Monopole
8.4 Physical Realization of a Complex Monopole
8.4.1 The 2D-Case
8.4.2 The 3D-Case
8.5 Complex Multipoles
8.6 Physical Realization of Complex Multipoles
9 Simulation of Vibrating and Scattering Objects with ESM/CESM
9.1 Conventional Equivalent Source Method (ESM)
9.1.1 Sound Radiation of a Rectangular Radiator
9.1.2 Sound Scattering of a Non-convex Cat’s Eye
9.2 Complex Equivalent Source Method (CESM)
9.2.1 Sound Radiation of a Spherical Cap
9.2.2 Sound Scattering of a Cat's Eye
9.2.3 Sound Radiation of a Non-convex Cat's Eye
9.2.4 Sound Radiation of a Vibrating Cone Embedded in a Baffle
10 Green’s Function Above Homogeneous Ground
10.1 Two-Dimensional Problem
10.1.1 Impulsive Time Dependence
10.1.2 Sound Field of a Line Source with Arbitrary Time Dependence
10.2 Three-Dimensional Problem
10.2.1 Sommerfeld Approach
10.2.2 Exact Image Theory (EIT)
10.2.3 Comparison Between the Sommerfeld Approach and the EIT
11 Boundary Element Techniques for Sound Propagation Above Impedance Planes
11.1 Basic Concepts of the BEM for Half-Space Problems
11.1.1 Direct Formulation
11.1.2 Indirect Formulation
11.2 Discretization
11.3 Non-uniqueness at Certain Frequencies
11.4 Green’s Function of Rigid (Zp → ∞) and Soft (Zp = 0) Infinite Plane
11.5 The General Impedance Boundary Condition at the Infinite Plane
11.5.1 Classical Formula with Cylindrical Waves (Sommerfeld)
11.5.2 Formula with Reflected Plane Waves (Thomasson)
11.5.3 Formula with Complex Image Sources
11.5.4 Approximated Formula
11.5.5 Comparison of the Performance of the Half-Space Formulas in a BEM Formulation
11.6 Example of Application: Acoustic Thin Barrier Above Impedance Ground
11.6.1 Numerical Model
11.6.2 IL for Rigid and Soft Ground
11.6.3 IL for Grounds with General Type of Impedance
12 Final Remarks and Outlook
Appendix A Different Expressions for the Driving Source of a Complex 3D Monopole
Appendix B Proof of the Expression for the Driving Sources of the Half-Plane Problem
Appendix C Proof of the Expression for the Driving Sources of the Half-Space Problem
Appendix D Exact Image Theory
Reflection Coefficient
Source Density r(q)
Transmission Coefficient w
Derivative of F(α,B,q)
Derivative of tildeF(α,B,q)
Appendix E Formula of Thomasson
Derivatives of Rs
Derivatives of δ
Derivatives of W
Derivatives of IW
Derivatives of R0
Appendix F Weyl–Van der Pol Approximation
Derivatives of Rg
Derivatives of Complex Distance ε
Derivatives of F(ε)
References
Index