John C. Cochran, PhD, is a Principal Engineering Fellow in the Advanced Technology Department of Raytheon Integrated Defense Systems responsible for sonar system design and operation. He is a subject matter expert in the design and development of advanced sensors, sensor arrays, and undersea sensing systems.

Preface xvii

1 Acoustic Waves and Radiation 1

1.1 Small Signals/Linear Acoustics 1

1.1.1 Compressibility 2

1.1.2 Small Signals/Linear Acoustics 2

1.1.3 Relationship Between Acoustic Pressure and Acoustic Density 2

1.1.4 Condensation 2

1.1.5 Time Derivative Using Eulerian and Lagrangian Description 3

1.2 The Equations of Continuity, Motion, and the Wave Equation in a Fluid Media 3

1.2.1 Equation of Continuity in a Single Dimension 3

1.2.2 The Force Equation in a Single Dimension 4

1.2.3 The Wave Equation in a Single Dimension 5

1.2.4 Generalization of the Wave Equation to Three Dimensions 5

1.2.5 Helmholtz Wave Equation 6

1.2.6 Velocity Potential 6

1.3 Plane Waves 7

1.3.1 Harmonic Plane Waves 7

1.3.2 Plane Waves in an Infinite Media 7

1.3.3 Plane Wave Acoustic Intensity 8

1.3.4 Plane Wave Acoustic Impedance 8

1.4 Radiation from Spheres 8

1.4.1 General Solution to Radiation from Spheres 9

1.4.2 Spherical Wave Acoustic Impedance 11

1.4.3 Axis-Symmetric Radiation from a Sphere – the Spherical Source 11

1.4.4 The Simple Spherical Source 12

1.4.5 Source Strength 12

1.4.6 The General Simple Source 13

1.4.7 Acoustic Reciprocity and Reciprocity Factor 13

1.5 Radiation from Sources on a Cylindrical Surface 14

1.5.1 General Solution to Radiation from Cylinders 15

1.5.2 Radiation from an Infinitely Long Cylinder 18

1.5.3 The Simple, Infinitely Long Cylindrical Source 19

1.5.4 Radiation from an Infinitely Long Strip on an Infinitely Long Cylinder 20

1.5.5 Radiation from a Finite Source on a Cylinder with a Periodic z Dependence 21

1.5.6 Radiation from a Finite Source on a Cylinder with a Uniform z Dependence 22

1.5.7 The Simple Cylindrical Source – Radiation from a Finite Length Cylinder in an Infinitely Long Cylinder Baffle 25

1.6 Integral Formulations 26

1.6.1 The Green’s Function 27

1.6.2 Helmholtz Integral Formulations 28

1.6.3 Far Field Approximation 29

1.6.4 An Application of the Simple Source Integral Formulation – Radiation from a Finite Cylinder 34

1.7 Linear Apertures 36

1.7.1 Far Field Radiation (Beam) Patterns as a Fourier Transform of the Linear Aperture Function – the Directivity Function 36

1.7.2 A Simple Rectangular Aperture Function as an Example of a Linear Aperture 38

1.7.3 The Triangular Window Aperture Function as a Linear Aperture 41

1.7.4 The Cosine Window Aperture Function as a Linear Aperture 43

1.7.5 Other Linear Apertures 45

1.7.6 The Far Field Radiation Pattern of a Linear Aperture on a Cylindrical Surface 45

1.8 Planar Apertures 49

1.8.1 The Green’s Function for Radiation from Planar Apertures Located on a Rigid Plane Baffle 49

1.8.2 Far Field Radiation Patterns as a Fourier Transform of the Planar Aperture Function 50

1.8.3 The Rectangular Piston in an Infinite Plane Baffle 52

1.8.4 The Circular Piston in an Infinite Plane Baffle 54

1.8.5 The Far Field Radiation Pattern of a Circular Annular Ring 59

1.8.6 The Elliptical Piston in an Infinite Plane Baffle 60

1.8.7 Impact of Boundary Impedance on Radiation Patterns from Planar Apertures 60

1.9 Directivity and Directivity Index (DI) 63

1.9.1 Definition of Directivity and Directivity Index (DI) 65

1.9.2 Relationship Between Source Level and Directivity Index 67

1.9.3 The Directivity of Baffled vs. Unbaffled Sources 68

1.9.4 The Directivity Index of a Baffled Circular Piston 68

1.9.5 The Directivity Index of a Baffled Rectangular Piston 70

1.9.6 The Directivity Index of a Line Source 70

1.10 Scattering and Diffraction 72

1.10.1 Scattering and Diffraction from a Rigid Cylinder 72

1.10.1.1 The Incident Wave 72

1.10.1.2 The Scattered Wave 73

1.10.1.3 Matching the Boundary Conditions for the Total Field 73

1.10.1.4 The Scattered Pressure Field in the Far Field 74

1.10.1.5 The Total Pressure Field 74

1.10.1.6 The Average Pressure Exerted on the Cylinder by the Total Pressure Field 74

1.10.2 The Diffraction Constant for a Rigid Cylinder 76

1.10.3 Diffraction Constant for a Strip on a Rigid Cylinder 76

1.10.4 Diffraction of a Cylinder with Variable Boundary Admittance 77

1.10.4.1 The Incident Wave 77

1.10.4.2 The Boundary Admittance 78

1.10.4.3 The Scattered Wave 78

1.10.4.4 Matching the Boundary Conditions 81

1.10.4.5 The Boundary Reflection Coefficient and the Scattered Field 81

1.10.4.6 The Total Field 82

1.10.4.7 The Average Pressure Exerted on the Cylinder With a Variable Boundary Admittance 82

1.10.4.8 The Diffraction Constant for a Cylinder with Variable Boundary Admittance 82

1.10.4.9 The Total Diffracted Field in the Far Field 83

1.10.4.10 The Total Diffracted Field at the Surface of the Cylinder 83

1.10.5 Scattering and Diffraction from a Rigid Sphere 84

1.10.5.1 The Incident Wave 84

1.10.5.2 The Scattered Wave 85

1.10.5.3 Matching the Boundary Conditions for the Total Field 85

1.10.5.4 The Total Pressure Field 85

1.10.5.5 The Scattered Pressure Field in the Far Field 86

1.10.5.6 The Average Pressure Exerted on the Sphere by the Pressure Field 86

1.10.6 The Diffraction Constant for a Rigid Sphere 87

1.10.7 Scattering and Diffraction from a Thin Cylindrical Ring 87

1.11 Radiation Impedance 89

1.11.1 Introduction to Radiation Impedance 89

1.11.2 Units of Acoustic Radiation Impedance 90

1.11.3 What it Means to be ρc Loaded 90

1.11.4 The Relationship Between Resistance and Reactance – The Hilbert Transform 90

1.11.5 The Relationship Between Radiation Resistance, Directivity, and Diffraction Constant 92

1.11.6 The Radiation Impedance of a Spherical Radiator 94

1.11.7 The Radiation Impedance of a Simple Source Radiator 95

1.11.8 The Radiation Impedance of a Circular Piston Radiator in a Plane Baffle 95

1.11.9 The Radiation Impedance of a Circular Piston Radiator at the End of a Tube 97

1.11.10 The Radiation Impedance of a Rectangular Piston Radiator in a Plane Baffle 98

1.11.11 The Radiation Impedance of an Infinitely Long Strip Radiator in a Plane Baffle 100

1.11.12 The Radiation Impedance of a Circular Annular Piston Radiator in a Plane Baffle 101

1.11.13 The Radiation Impedance of an Elliptical Piston Radiator in a Plane Baffle 103

1.11.14 The Radiation Impedance of an Infinitely Long Cylindrical Radiator 103

1.11.15 The Radiation Impedance of a Finite Cylindrical Radiator 104

1.11.16 Mutual Radiation Impedance 105

1.11.17 The Mutual Radiation Impedance Between Spherical Radiators 106

1.11.18 The Mutual Radiation Impedance Between Two Circular Piston Radiators in a Plane Baffle 108

1.11.19 The Mutual Radiation Impedance Between Two Square Piston Radiators in a Plane Baffle 114

1.11.20 The Mutual Radiation Impedance Between a Circular Piston and an Outer Annular Ring 116

1.11.21 The Mutual Radiation Impedance Between Rectangular or Square Pistons Located on a Cylindrical Baffle 118

1.11.22 The Mutual Radiation Impedance Between Bands on a Cylindrical Baffle 124

1.12 Transmission Phenomena 125

1.12.1 Reflection and Transmission of Plane Waves with Normal Incidence at a Boundary 126

1.12.2 Reflection and Transmission of Plane Waves Obliquely Incident at a Plane Boundary 129

1.12.2.1 Snell’s Law 130

1.12.2.2 Reflection and Transmission Factors for Obliquely Incident Plane Waves 131

1.12.2.3 Brewster’s Angle or the Angle of Zero Reflection 131

1.12.2.4 The Critical Angle or the Angle of Complete Reflection 132

1.12.2.5 Evanescent Waves 132

1.13 Absorption and Attenuation of Sound 133

1.13.1 Absorption Phenomena 133

1.13.2 Absorption in Seawater 134

References 135

2 Mechanical/Acoustical Equivalent Circuits 137

2.1 Different Forms of Impedance 138

2.2 Mechanical Equivalent Circuits 139

2.2.1 The Simple Mechanical System 139

2.2.1.1 A Simple Mechanical Oscillator 139

2.2.1.2 Phasor Form of the Solutions to the Equations of Motion 139

2.2.1.3 Damped Oscillations 140

2.2.1.4 Forced Oscillations 141

2.2.1.5 Complete Solution for a Simple Oscillator 142

2.2.1.6 Analogy to Electrical Circuits 142

2.2.1.7 Behavior of the Steady State, Forced, Mechanical Oscillator 143

2.2.1.8 Equivalent Circuit for a Simple Resonator System 144

2.2.2 Introduction to Mobility 145

2.2.2.1 Mechanical Generators 145

2.2.2.2 Combining Impedance and Mobility Elements 145

2.2.2.3 Elements of Mobility and Impedance Analogs 147

2.2.2.4 Examples of Mechanical Systems Described by Mobility Analogs 149

2.2.2.5 An Example of a Gyrator Conversion 150

2.2.2.6 Converting from Mobility to Impedance and Vice Versa 151

2.3 Acoustical Equivalent Circuits 153

2.3.1 Acoustic Circuit Elements 153

2.3.1.1 Acoustic Compliance – the Closed-End Tube 153

2.3.1.2 Acoustic Mass – the Open-Ended Tube 154

2.3.1.3 Acoustic Resistance 156

2.3.1.4 Acoustic Generators 156

2.3.1.5 Pressure Equalization Orifices 156

2.3.1.6 The Thin Acoustic Orifice 159

2.3.1.7 The Narrow Slit 160

2.3.1.8 The Acoustic Mesh or Perforated Sheet 160

2.3.2 Acoustic Equivalent Circuits 161

2.3.2.1 Example of an Acoustic System Described by an Equivalent Circuit 161

2.3.2.2 Another Example of an Acoustic Equivalent Circuit – the Helmholtz Resonator 161

2.4 Combining Mechanical and Acoustical Equivalent Circuits 163

2.5 Introduction to Transduction 165

2.5.1 The Transducer as a Two-Port Equivalent Circuit 165

2.5.2 Reciprocal and Anti-Reciprocal Transducers 166

2.5.3 The Electromechanical Coupling Factor 166

2.5.4 Electromechanical Transformation 167

2.5.5 Transmitters 167

2.5.6 Receivers 169

2.5.7 Relationship Between Transmit and Receive Characteristics 170

References 171

3 Waves in Solid Media 173

3.1 Waves in Homogeneous, Isotropic, Elastic, Solid Media 173

3.1.1 The Components of Stress 173

3.1.2 The Equations of Motion 174

3.1.3 The Components of Strain 175

3.1.4 The Relationship Between Stress and Strain – The Constitutive Equations 177

3.1.4.1 Hooke’s Law – Tensor Form 177

3.1.4.2 Hooke’s Law – Matrix Form 179

3.1.4.3 The Differences Between Tensor and Matrix Forms of the Constitutive Equations 180

3.1.4.4 Lame’s Constants 182

3.1.4.5 Stiffness vs. Compliance Matrices 183

3.1.4.6 Modified Constitutive Equations 184

3.1.5 Acoustic Waves in Isotropic Solids 184

3.1.5.1 The Acoustic Wave Equation for Isotropic Solids 184

3.1.5.2 Waves of Dilatation and Distortion 184

3.1.5.3 Acoustic Plane Waves in Isotropic Solids 186

3.1.6 Longitudinal Waves in Bars 186

3.1.6.1 Vibrations in a Bar with Clamped Boundary Conditions 188

3.1.6.2 Vibrations in a Bar with Free Boundary Conditions 189

3.1.6.3 Equivalent Circuit Representation for Longitudinal Vibrations in a Bar with Arbitrary Boundary Conditions 190

3.1.6.4 A Two-Port Representation of Longitudinal Vibrations Within a Bar 192

3.1.6.5 Impact of Different Load Impedances on the Longitudinal Vibrations Within a Bar 193

3.1.6.6 Equivalent Circuit Representation for a Mass-Loaded Bar with One Free End 194

3.1.6.7 Equivalent Circuit Representation for a Mass-loaded Bar with One End Clamped 196

3.1.6.8 Lumped Parameter Equivalent Circuit for a Longitudinal Resonator 198

3.1.6.9 The Effective Mass of a Spring 200

3.1.7 Equivalent Circuit Representations for Solid Elements 202

3.1.7.1 Longitudinal Vibrations Within a Hollow Cylinder 202

3.1.7.2 Longitudinal Vibrations Within a Conical Section 204

3.1.7.3 Longitudinal Vibrations Within an Exponential Section 206

3.2 Piezo-electricity and Piezo-electric Ceramic Materials 208

3.2.1 The Nature of Piezo-electricity 208

3.2.2 Piezo-electric Ceramic Materials 211

3.2.3 The Piezo-electric Ceramic Constitutive Equations 212

3.2.4 The Meaning of the Piezo-electric Coefficients 214

3.2.5 Piezo-electric, Elastic, and Dielectric Coefficient Nomenclature 215

3.2.6 Piezo-electric Ceramic Material Properties 216

3.2.7 The Electromechanical Coupling Coefficient 219

3.2.8 Further Observations on the Piezo-electric Constitutive Equations 220

3.3 Waves in Non-Homogenous, Piezo-electric Media 222

3.3.1 Vibrations in Rods and Disks 223

3.3.1.1 Constitutive Equations 223

3.3.1.2 Equations of Motion and Strain in Cylindrical Coordinates 224

3.3.1.3 Radial Mode Vibrations in Thin Disks 224

3.3.1.4 Thickness Mode Vibrations in Thin Disks 228

3.3.1.5 The Relationship Between Dielectric Constant and Coupling Factor for Vibrations in Thin Disks 233

3.3.1.6 Length Longitudinal Mode Vibrations in Long, Thin Rods or Bars 235

3.3.1.7 Radial Mode Vibrations in Long, Thin Rods or Bars 237

3.3.1.8 The Relationship Between Dielectric Constant and Coupling Factor for Vibrations in Long, Thin Rods 240

3.3.1.9 Frequency Constants for Vibrations in Rods and Disks 241

3.3.2 Vibrations in Piezo-electric Plates and Parallelepipeds 242

3.3.2.1 Equations of Motion and Strain in Rectangular Coordinates 243

3.3.2.2 Length Expander Bar with Electric Field Perpendicular to Width – The 31 Mode Bar 244

3.3.2.3 Length Expander Bar with Electric Field Parallel to Width – The 33 Mode Bar 248

3.3.2.4 Thickness Mode Vibrations in Thin Piezo-electric Plates with the Electric Field Parallel to the Thickness 250

3.3.2.5 Coupled Mode Vibrations in Parallelepipeds with One Large Dimension 254

3.3.2.6 Coupled Mode Vibrations in Thin Piezo-electric Plates with the Electric Field Perpendicular to the Thickness 256

3.3.2.7 Coupled Mode Vibrations in Thin Piezo-electric Plates with the Electric Field Parallel to the Width 258

3.3.2.8 Coupled Mode Vibrations in Parallelepipeds with Arbitrary Dimensions 259

3.3.3 Vibrations in Piezo-electric Ceramic Cylinders 261

3.3.3.1 Longitudinal Vibrations in Axially Polarized, Piezo-ceramic Cylinders 263

3.3.3.2 Longitudinal Vibrations in Radially Polarized, Piezo-ceramic Cylinders 272

3.3.3.3 Radial Vibrations in Radially Polarized, Piezo-ceramic Cylinders 281

3.3.3.4 Longitudinal Vibrations in Circumferentially Polarized, Segmented, Piezo-ceramic Cylinders 285

3.3.3.5 Radial Vibrations in Circumferentially Polarized, Segmented, Piezo-ceramic Cylinders 294

3.3.4 Vibrations in Radially Polarized Spherical Shells 297

3.3.4.1 Boundary Conditions 297

3.3.4.2 Constitutive Equations 298

3.3.4.3 The Equations of Motion and Strain 298

3.3.4.4 Kinetic Energy and Equivalent Mass 299

3.3.4.5 Internal Energy 299

3.3.4.6 Electromechanical Coupling Coefficient 300

3.3.4.7 In-Air Resonance Frequency of a Spherical Shell 300

3.3.4.8 Equivalent Circuit Model for a Radially Polarized Spherical Shell 300

References 303

4 Sonar Projectors 305

4.1 Tools for Underwater Sonar Projector Design 305

4.1.1 Assembling Circuit Elements 305

4.1.1.1 Two-Port Representations for Non-Piezoelectric Components 305

4.1.1.2 Series Combination of Two-Port Networks for Non-Piezoelectric Components 307

4.1.1.3 Parallel Combinations of Two-Port Networks for Non-Piezoelectric Components 307

4.1.1.4 Two-Port Representations of Piezoelectric Components 308

4.1.1.5 Cascaded Combinations of Two-Port Networks for Piezoelectric Components 309

4.1.1.6 Ladder Network Analysis 311

4.1.2 How to Specify a Projector 312

4.2 Specific Applications in Underwater Sonar Projector Design 313

4.2.1 Frequency Ranges for Different Types of Projectors 313

4.2.2 Spherical Projectors 314

4.2.2.1 The Lossless, Air-Backed Spherical Projector 314

4.2.2.2 The Lossy, Air-Backed Spherical Projector 320

4.2.2.3 Fluid-Filled Spherical Projectors 321

4.2.3 The Radially Polarized Cylindrical Projector 323

4.2.3.1 The Radially Polarized, Air-Backed Cylindrical Projector 323

4.2.3.2 Prestressing for Increased Power-Handling Capability 328

4.2.3.3 The Radially Polarized, Fluid-Filled Cylindrical Projector 330

4.2.3.4 The Radially Polarized, Squirter Projector 332

4.2.3.5 The Radially Polarized, Free-Flooded Cylindrical Projector 339

4.2.3.6 The Free-Flooded Cylindrical Projector with a Reflector Plate 342

4.2.4 Circumferentially Polarized Cylindrical Projectors – The Barrel Stave Projector 343

4.2.4.1 The Circumferentially Polarized, Air-Backed Cylindrical Projector 343

4.2.4.2 The Circumferentially Polarized, Free-Flooded Cylindrical Projector 346

4.2.4.3 The Circumferentially Polarized Striped Cylindrical Projector 348

4.2.5 The Tonpilz Transducer 352

4.2.5.1 The End Mass-Loaded Tonpilz Transducer 353

4.2.5.2 The Nodally Mounted Tonpilz Transducer 355

4.2.6 The Flexural Disk Transducer 355

4.2.6.1 The Trilaminar Flexural Disk Transducer 357

4.2.6.2 The Bilaminar Flexural Disk Transducer 375

4.2.7 Flat Oval Flextensional Projectors 385

4.2.8 Slotted Cylinder Projectors 387

4.2.8.1 Geometry and Description 388

4.2.8.2 Wall Thickness, Radii, and Taper Factors 390

4.2.8.3 Neutral Axis 391

4.2.8.4 Displacement Profiles 392

4.2.8.5 Stress and Strain in the SCP 397

4.2.8.6 Kinetic Energy and Equivalent Mass 398

4.2.8.7 Constitutive Equations for the Piezoceramic Component 398

4.2.8.8 Voltage Across Electrodes and Dielectric Displacement 398

4.2.8.9 Internal Energy 399

4.2.8.10 Flexural Stiffness 400

4.2.8.11 In-Air Resonance Frequency 400

4.2.8.12 Effective Electromechanical Coupling Factor, keff 401

4.2.8.13 In-water Performance 401

4.2.8.14 An SCP Example 407

4.2.9 Moving Coil Transducers 407

4.2.10 The Line-in-Cone Transducer 412

4.2.11 Quarter-Wavelength Resonators 415

4.2.12 Disk Projectors 418

4.2.13 The High-Frequency Line Projector 420

4.3 Special Topics in Underwater Sonar Projector Design 422

4.3.1 Techniques for Increasing Bandwidth 422

4.3.1.1 Bandwidth Increases with Coupling 422

4.3.1.2 Mechanical Tuning with Matching Layers 423

4.3.2 Power Limitations in Sonar Projectors 424

4.3.2.1 Electric Field Limitations 424

4.3.2.2 Loss Tangent Limitations 425

4.3.2.3 Stress Limitations 426

4.3.2.4 Thermal Limitations 427

4.3.2.5 Cavitation Limitations 433

References 436

5 Sonar Hydrophones 439

5.1 Elements of Sonar Hydrophone Design 439

5.1.1 An Equivalent Circuit for a Sonar Hydrophone 440

5.1.2 The Importance of the Piezo-Ceramic g Constant 442

5.1.3 An Equivalent Circuit for a Dielectrically Lossy Sonar Hydrophone 442

5.1.4 The Effect of Cable Capacitance 443

5.1.5 Typical Response of a Sonar Hydrophone 444

5.2 Analysis of Noise in Hydrophone/Preamplifier Systems 445

5.2.1 Ambient Noise 445

5.2.2 Types of Equivalent Noise Sources 446

5.2.3 Ambient Noise Coupling into a Sensor 447

5.2.4 Sensor Self-Noise 448

5.2.5 Sensor Signal to Noise Ratio 450

5.2.6 Preamplifier Noise 450

5.2.7 Combined Sensor and Preamp System Noise, the Equivalent Noise Pressure 452

5.2.8 The Equivalent Noise Pressure at Low Frequencies 453

5.2.9 Comparison of Sensor Noise with Ambient Noise Example 455

5.2.10 Hydrophone Figure of Merit 456

5.2.11 The Effect of Cable Capacitance – Insertion Loss 457

5.3 Specific Applications in Underwater Sonar Hydrophone Design 458

5.3.1 Unidirectional Hydrophone 459

5.3.1.1 Boundary Conditions 460

5.3.1.2 Equation of Motion and Strain 460

5.3.1.3 Constitutive Equations 460

5.3.1.4 Open Circuit Voltage Sensitivity 460

5.3.2 Hydrostatic Hydrophone 461

5.3.3 Spherical Hydrophone 462

5.3.3.1 Boundary Conditions 463

5.3.3.2 Constitutive Equations 464

5.3.3.3 The Equations of Motion and Strain 464

5.3.3.4 Stress Profile in a Spherical Hydrophone 464

5.3.3.5 The Open Circuit Sensitivity of the Spherical Hydrophone 465

5.3.3.6 Spherical Hydrophone Depth Limitations 466

5.3.3.7 The Effect of a Fill Fluid on Hydrophone Performance 467

5.3.4 Cylindrical Hydrophones 468

5.3.4.1 The Radially Polarized Cylindrical Hydrophone 470

5.3.4.2 The Circumferentially Polarized Cylindrical Hydrophone 488

5.3.4.3 The Axially Polarized Cylindrical Hydrophone 493

5.3.5 PVDF Polymer Hydrophones 496

References 497

Appendix 499

Index 509

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