GÖKHAN APAYDIN, BSEE, MSEE, PHD, is a freelance electrical and electronics engineer. His research interests include analytical and numerical methods in electromagnetics, especially on electromagnetic computation of wave propagation, diffraction modeling, scattering, and related areas.LEVENT SEVGI, BSEE, MSEE, PHD, is a Professor in the Electrical and Electronics Engineering Department at Okan University, Istanbul, Turkey. His research has focused on propagation in complex environments, EMC/BEM modeling and measurements, analytical and numerical methods in electromagnetics, and radar systems.

• Preface ixAcronyms xiMATLAB Codes xiiiChapter 1 Introduction 11.1 Electromagnetic Problems and Classification 11.2 Maxwell Equations 31.3 Guided Waves and Transverse/Longitudinal Decomposition 41.4 Two Dimensional Helmholtz’s Equation 51.5 Validation, Verification, and Calibration Procedure 61.6 Fourier Transform, DFT and FFT 7Chapter 2 Wave Propagation over Flat Earth 152.1 Flat Earth and GO Two-Ray Model 152.2 Single Knife Edge Problem and Four-Ray Model 162.3 Vertical Linear Refractivity Profile and Mode Summation 19Chapter 3 Parabolic Equation Modeling 233.1 Introduction 233.2 Parabolic Wave Equation Form 243.3 Dirichlet, Neumann, and Cauchy Boundary Conditions 273.4 Antenna/Source Injection 283.5 Split-Step Parabolic Equation (SSPE) Model 293.5.1 Narrow-Angle and Wide-Angle SSPE 303.5.2 A MATLAB-Based Simple SSPE Code 303.6 FEM-Based Parabolic Equation Model 323.7 Atmospheric Refractivity Effects 40Chapter 4 Wave Propagation at Short Ranges 434.1 Introduction 434.2 Accurate Source Modeling 444.3 Wave Propagators in Two Dimensions 474.3.1 Flat Earth and Two-Ray Model 474.3.2 FEM-Based PE Wave Propagator 494.3.3 SSPE-Based PE Wave Propagator 494.3.4 Method of Moments Modeling 494.4 Knife Edge and Four Ray Model 494.5 Canonical Tests and Calibration 50Chapter 5 Pe and Terrain Modeling 535.1 Irregular PEC Terrain 535.2 PE and Impedance Boundary Modeling 545.2.1 Discrete Mixed Fourier Transform (DMFT) 565.3 Numerical Results and Comparison 57Chapter 6 Analytical Exact and Approximate Models 656.1 Wave Propagation in a Parallel Plate Waveguide 656.2 Green’s Function in Terms of Mode Summation 686.3 Mode Summation for a Tilted Gaussian Source 706.4 A Hybrid Ray + Image Method 716.5 Numerical Models 736.5.1 Parabolic Equation Models: SSPE and FEMPE 736.5.2 Method of Moments 75Chapter 7 Wave Propagation Inside Three-dimensional Rectangular Waveguide 797.1 Introduction 797.2 Three-Dimensional Rectangular Waveguide Model 807.3 Three-Dimensional Parabolic Equation Models 817.3.1 SSPE Model 817.3.2 FEMPE Model 827.3.3 ADIPE Model 827.4 Tests and Calibration 83Chapter 8 Two WAY PE MODELS 898.1 Formulation of Two Way FEMPE Method 898.2 Formulation of Two Way SSPE Method 918.3 Flat Earth with Infinite Wall 918.4 Flat Earth with Single and Double Knife Edges 918.5 Two Way Propagation Modeling in Waveguides 968.6 Three-Dimensional Split-Step- and Finite-Element-Based Parabolic Equation Models 968.7 Tests and Calibration 97Chapter 9 Petool Virtual Propagation Package 1019.1 Introduction 1019.2 PETOOL Software 1039.3 Characteristic Examples 107Chapter 10 Femix Virtual Propagation Package 11310.1 Introduction 11310.2 Analytical Surface-Wave Model 11510.2.1 Path Loss 11510.2.2 Norton’s Model 11510.2.3 Wait’s Model 11610.2.4 Millington’s Curve Fitting Approach 11710.3 Numerical Surface-Wave Model 11810.4 FEMIX Package 11910.5 Characteristic Examples 122References 127Index 135

GÖKHAN APAYDIN, BSEE, MSEE, PHD, is a freelance electrical and electronics engineer. His research interests include analytical and numerical methods in electromagnetics, especially on electromagnetic computation of wave propagation, diffraction modeling, scattering, and related areas.LEVENT SEVGI, BSEE, MSEE, PHD, is a Professor in the Electrical and Electronics Engineering Department at Okan University, Istanbul, Turkey. His research has focused on propagation in complex environments, EMC/BEM modeling and measurements, analytical and numerical methods in electromagnetics, and radar systems.

• Preface ixAcronyms xiMATLAB Codes xiiiChapter 1 Introduction 11.1 Electromagnetic Problems and Classification 11.2 Maxwell Equations 31.3 Guided Waves and Transverse/Longitudinal Decomposition 41.4 Two Dimensional Helmholtz’s Equation 51.5 Validation, Verification, and Calibration Procedure 61.6 Fourier Transform, DFT and FFT 7Chapter 2 Wave Propagation over Flat Earth 152.1 Flat Earth and GO Two-Ray Model 152.2 Single Knife Edge Problem and Four-Ray Model 162.3 Vertical Linear Refractivity Profile and Mode Summation 19Chapter 3 Parabolic Equation Modeling 233.1 Introduction 233.2 Parabolic Wave Equation Form 243.3 Dirichlet, Neumann, and Cauchy Boundary Conditions 273.4 Antenna/Source Injection 283.5 Split-Step Parabolic Equation (SSPE) Model 293.5.1 Narrow-Angle and Wide-Angle SSPE 303.5.2 A MATLAB-Based Simple SSPE Code 303.6 FEM-Based Parabolic Equation Model 323.7 Atmospheric Refractivity Effects 40Chapter 4 Wave Propagation at Short Ranges 434.1 Introduction 434.2 Accurate Source Modeling 444.3 Wave Propagators in Two Dimensions 474.3.1 Flat Earth and Two-Ray Model 474.3.2 FEM-Based PE Wave Propagator 494.3.3 SSPE-Based PE Wave Propagator 494.3.4 Method of Moments Modeling 494.4 Knife Edge and Four Ray Model 494.5 Canonical Tests and Calibration 50Chapter 5 Pe and Terrain Modeling 535.1 Irregular PEC Terrain 535.2 PE and Impedance Boundary Modeling 545.2.1 Discrete Mixed Fourier Transform (DMFT) 565.3 Numerical Results and Comparison 57Chapter 6 Analytical Exact and Approximate Models 656.1 Wave Propagation in a Parallel Plate Waveguide 656.2 Green’s Function in Terms of Mode Summation 686.3 Mode Summation for a Tilted Gaussian Source 706.4 A Hybrid Ray + Image Method 716.5 Numerical Models 736.5.1 Parabolic Equation Models: SSPE and FEMPE 736.5.2 Method of Moments 75Chapter 7 Wave Propagation Inside Three-dimensional Rectangular Waveguide 797.1 Introduction 797.2 Three-Dimensional Rectangular Waveguide Model 807.3 Three-Dimensional Parabolic Equation Models 817.3.1 SSPE Model 817.3.2 FEMPE Model 827.3.3 ADIPE Model 827.4 Tests and Calibration 83Chapter 8 Two WAY PE MODELS 898.1 Formulation of Two Way FEMPE Method 898.2 Formulation of Two Way SSPE Method 918.3 Flat Earth with Infinite Wall 918.4 Flat Earth with Single and Double Knife Edges 918.5 Two Way Propagation Modeling in Waveguides 968.6 Three-Dimensional Split-Step- and Finite-Element-Based Parabolic Equation Models 968.7 Tests and Calibration 97Chapter 9 Petool Virtual Propagation Package 1019.1 Introduction 1019.2 PETOOL Software 1039.3 Characteristic Examples 107Chapter 10 Femix Virtual Propagation Package 11310.1 Introduction 11310.2 Analytical Surface-Wave Model 11510.2.1 Path Loss 11510.2.2 Norton’s Model 11510.2.3 Wait’s Model 11610.2.4 Millington’s Curve Fitting Approach 11710.3 Numerical Surface-Wave Model 11810.4 FEMIX Package 11910.5 Characteristic Examples 122References 127Index 135

GÖKHAN APAYDIN, BSEE, MSEE, PHD, is a freelance electrical and electronics engineer. His research interests include analytical and numerical methods in electromagnetics, especially on electromagnetic computation of wave propagation, diffraction modeling, scattering, and related areas.LEVENT SEVGI, BSEE, MSEE, PHD, is a Professor in the Electrical and Electronics Engineering Department at Okan University, Istanbul, Turkey. His research has focused on propagation in complex environments, EMC/BEM modeling and measurements, analytical and numerical methods in electromagnetics, and radar systems.

• Preface ixAcronyms xiMATLAB Codes xiiiChapter 1 Introduction 11.1 Electromagnetic Problems and Classification 11.2 Maxwell Equations 31.3 Guided Waves and Transverse/Longitudinal Decomposition 41.4 Two Dimensional Helmholtz’s Equation 51.5 Validation, Verification, and Calibration Procedure 61.6 Fourier Transform, DFT and FFT 7Chapter 2 Wave Propagation over Flat Earth 152.1 Flat Earth and GO Two-Ray Model 152.2 Single Knife Edge Problem and Four-Ray Model 162.3 Vertical Linear Refractivity Profile and Mode Summation 19Chapter 3 Parabolic Equation Modeling 233.1 Introduction 233.2 Parabolic Wave Equation Form 243.3 Dirichlet, Neumann, and Cauchy Boundary Conditions 273.4 Antenna/Source Injection 283.5 Split-Step Parabolic Equation (SSPE) Model 293.5.1 Narrow-Angle and Wide-Angle SSPE 303.5.2 A MATLAB-Based Simple SSPE Code 303.6 FEM-Based Parabolic Equation Model 323.7 Atmospheric Refractivity Effects 40Chapter 4 Wave Propagation at Short Ranges 434.1 Introduction 434.2 Accurate Source Modeling 444.3 Wave Propagators in Two Dimensions 474.3.1 Flat Earth and Two-Ray Model 474.3.2 FEM-Based PE Wave Propagator 494.3.3 SSPE-Based PE Wave Propagator 494.3.4 Method of Moments Modeling 494.4 Knife Edge and Four Ray Model 494.5 Canonical Tests and Calibration 50Chapter 5 Pe and Terrain Modeling 535.1 Irregular PEC Terrain 535.2 PE and Impedance Boundary Modeling 545.2.1 Discrete Mixed Fourier Transform (DMFT) 565.3 Numerical Results and Comparison 57Chapter 6 Analytical Exact and Approximate Models 656.1 Wave Propagation in a Parallel Plate Waveguide 656.2 Green’s Function in Terms of Mode Summation 686.3 Mode Summation for a Tilted Gaussian Source 706.4 A Hybrid Ray + Image Method 716.5 Numerical Models 736.5.1 Parabolic Equation Models: SSPE and FEMPE 736.5.2 Method of Moments 75Chapter 7 Wave Propagation Inside Three-dimensional Rectangular Waveguide 797.1 Introduction 797.2 Three-Dimensional Rectangular Waveguide Model 807.3 Three-Dimensional Parabolic Equation Models 817.3.1 SSPE Model 817.3.2 FEMPE Model 827.3.3 ADIPE Model 827.4 Tests and Calibration 83Chapter 8 Two WAY PE MODELS 898.1 Formulation of Two Way FEMPE Method 898.2 Formulation of Two Way SSPE Method 918.3 Flat Earth with Infinite Wall 918.4 Flat Earth with Single and Double Knife Edges 918.5 Two Way Propagation Modeling in Waveguides 968.6 Three-Dimensional Split-Step- and Finite-Element-Based Parabolic Equation Models 968.7 Tests and Calibration 97Chapter 9 Petool Virtual Propagation Package 1019.1 Introduction 1019.2 PETOOL Software 1039.3 Characteristic Examples 107Chapter 10 Femix Virtual Propagation Package 11310.1 Introduction 11310.2 Analytical Surface-Wave Model 11510.2.1 Path Loss 11510.2.2 Norton’s Model 11510.2.3 Wait’s Model 11610.2.4 Millington’s Curve Fitting Approach 11710.3 Numerical Surface-Wave Model 11810.4 FEMIX Package 11910.5 Characteristic Examples 122References 127Index 135