Series Preface vii Preface ix Acknowledgements xi Chapter 1: Introduction 1 1.1 Brief History of Underwater Sound Transducers..... 2 1.2 Underwater Transducer Applications... 9 1.3 General Description of Linear Electroacoustic Transduction.......... 18 1.4 Transducer Characteristics..... 23 1.4.1 Electromechanical Coupling Coefficient..... 23 1.4.2 Transducer Responses, Directivity Index and Source Level...... 25 1.5 Transducer Arrays........ 27 References.... 28 Chapter 2: Electroacoustic Transduction 31 2.1 Piezoelectric Transducers...... 32 2.1.1 General... 32 2.1.2 The 33-Mode Longitudinal Vibrator... 37 2.1.3 The 31-Mode Longitudinal Vibrator... 40 2.2 Electrostrictive Transducers.... 42 2.3 Magnetostrictive Transducers......... 45 2.4 Electrostatic Transducers...... 48 2.5 Variable Reluctance Transducers....... 51 2.6 Moving Coil Transducers...... 53 2.7 Comparison of Transduction Mechanisms... 55 2.8 Equivalent Circuits... 58 2.8.1 Equivalent Circuit Basics... 58 2.8.2 Circuit Resonance..... 60 2.8.3 Circuit Q and Bandwidth....... 61 2.8.4 PowerFactorandTuning... 63 2.8.5 Power Limits... 67 2.8.6 Efficiency... 69 xiii
xiv Contents 2.8.7 Hydrophone Circuit and Noise........ 71 2.8.8 Extended Equivalent Circuits......... 73 References... 73 Chapter 3: Transducers as Projectors 76 3.1 Principles of Operation... 78 3.2 Ring and Spherical Transducers...... 81 3.2.1 Piezoelectric 31-Mode Ring... 81 3.2.2 Piezoelectric 33-Mode Ring... 86 3.2.3 The Spherical Transducer..... 88 3.2.4 The Magnetostrictive Ring.... 90 3.2.5 Free Flooded Rings...... 91 3.2.6 Multimode Rings........ 95 3.3 Piston Transducers...... 98 3.3.1 TheTonpilzProjector... 98 3.3.2 The Hybrid Transducer... 106 3.4 Transmission Line Transducers... 110 3.4.1 Sandwich Transducers.... 111 3.4.2 Wideband Transmission Line Transducers..... 115 3.4.3 Large Plate Transducers....... 120 3.4.4 Composite Transducers... 122 3.5 Flextensional Transducers....... 126 3.5.1 The Class IV and VII Flextensional Transducers..... 127 3.5.2 The Class I Barrel Stave Flextensional........ 131 3.5.3 The Class V and VI Flextensional Transducers..... 132 3.5.4 The Astroid and X-spring Flextensional Transducers..... 133 3.6 Flexural Transducers..... 136 3.6.1 Bender Bar Transducer... 137 3.6.2 Bender Disc Transducer....... 140 3.6.3 The Slotted Cylinder Transducer...... 142 3.6.4 The Bender Mode X-spring Transducer........ 145 References... 146 Chapter 4: Transducers as Hydrophones 152 4.1 Principles of Operation... 153 4.1.1 Sensitivity... 154 4.1.2 FigureofMerit... 156 4.1.3 Simplified Equivalent Circuit... 157 4.1.4 Other Sensitivity Considerations...... 159 4.2 Cylindrical and Spherical Hydrophones....... 162 4.2.1 Performance with Shielded Ends...... 162 4.2.2 Spherical Hydrophones... 165 4.2.3 Performance with End Caps... 166
xv 4.3 Planar Hydrophones...... 168 4.3.1 Tonpilz Hydrophones.......... 169 4.3.2 The 1-3 Composite Hydrophone..... 170 4.3.3 Flexible Hydrophones......... 173 4.4 Bender Hydrophones..... 174 4.5 Vector Hydrophones...... 176 4.5.1 Dipole Vector Sensors, Baffles, and Images..... 177 4.5.2 Pressure Gradient Vector Sensor..... 182 4.5.3 Velocity Vector Sensor......... 183 4.5.4 Accelerometer Sensitivity...... 184 4.5.5 Multimode Vector Sensor...... 187 4.5.6 Summed Scalar and Vector Sensors......... 189 4.5.7 Intensity Sensors...... 194 4.6 ThePlaneWaveDiffractionConstant... 195 4.7 Hydrophone Thermal Noise... 199 4.7.1 DirectivityandNoise... 200 4.7.2 Low Frequency Hydrophone Noise......... 202 4.7.3 A More General Description of Hydrophone Noise... 202 4.7.4 A Comprehensive Hydrophone Noise Model.... 205 4.7.5 Vector Sensor Internal Noise... 206 4.7.6 Vector Sensor Susceptibility to Local Noise..... 208 References.... 209 Chapter 5: Projector Arrays 213 5.1 Array Directivity Functions.... 217 5.1.1 The Product Theorem......... 217 5.1.2 Line, Rectangular, and Circular Arrays...... 218 5.1.3 Grating Lobes..... 221 5.1.4 Beam Steering and Shaping.... 223 5.1.5 Effects of Random Variations... 229 5.2 Mutual Radiation Impedance and the Array Equations... 230 5.2.1 Solving the Array Equations... 230 5.2.2 VelocityControl... 234 5.2.3 Negative Radiation Resistance...... 235 5.3 Calculation of Mutual Radiation Impedance........ 236 5.3.1 Planar Arrays of Piston Transducers........ 236 5.3.2 Non-Planar Arrays, Nonuniform Velocities... 241 5.4 Arrays of Non-FVD Transducers....... 244 5.4.1 Modal Analysis of Radiation Impedance..... 244 5.4.2 Modal Analysis of Arrays...... 245 5.5 VolumeArrays... 249 5.6 Near Field of a Projector Array... 251 5.7 The Nonlinear Parametric Array....... 253 References.... 259
xvi Contents Chapter 6: Hydrophone Arrays 262 6.1 Hydrophone Array Directional and Wavevector Response.... 264 6.1.1 Directivity Functions..... 264 6.1.2 Beam Steering.... 267 6.1.3 Shading..... 269 6.1.4 Wavevector Response of Arrays....... 274 6.2 ArrayGain... 276 6.3 Sources and Properties of Noise in Arrays..... 279 6.3.1 Ambient Sea Noise... 279 6.3.2 StructuralNoise... 283 6.3.3 FlowNoise... 284 6.4 Reduction of Array Noise........ 285 6.4.1 Reduction of Ambient Noise... 285 6.4.2 Reduction of Structural Noise......... 289 6.4.3 Reduction of Flow Noise...... 294 6.4.4 Summary of Noise Reduction......... 297 6.5 Arrays of Vector Sensors........ 300 6.5.1 Directionality.... 301 6.5.2 Unbaffled Vector Sensor Arrays in Ambient Noise... 303 6.5.3 Hull-Mounted Vector Sensor Arrays in StructuralNoise... 308 References... 317 Chapter 7: Transducer Models 320 7.1 Lumped Parameter Models and Equivalent Circuits.... 321 7.1.1 Mechanical Single Degree-of-Freedom Lumped Equivalent Circuits... 321 7.1.2 Mechanical Lumped Equivalent Circuits for Higher Degrees of Freedom... 324 7.1.3 Piezoelectric Ceramic Lumped-Parameter Equivalent Circuits... 327 7.1.4 Magnetostrictive Lumped-Parameter Equivalent Circuits... 332 7.1.5 Eddy Currents... 336 7.2 Distributed Models...... 338 7.2.1 Distributed Mechanical Model........ 339 7.2.2 Matrix Representation.... 343 7.2.3 Piezoelectric Distributed-Parameter Equivalent Circuit.... 346 7.2.3.1 Segmented 33 Bar.... 346 7.2.3.2 Un-segmented 31 Bar... 350 7.2.3.3 Length Expander Bar......... 351 7.2.3.4 Thickness-Mode Plate........ 353 7.2.3.5 Magnetostrictive Rod......... 354
xvii 7.3 Matrix Models.... 355 7.3.1 Three-Port Matrix Model... 355 7.3.2 Two-Port ABCD Matrix Model..... 358 7.4 Finite Element Models.... 360 7.4.1 A Simple FEM Example.... 360 7.4.2 FEM Matrix Representation.... 361 7.4.3 Inclusion of a Piezoelectric Finite Element... 364 7.4.4 Application of FEM without Water Loading..... 365 7.4.5 Application of FEM with Water Loading.... 368 7.4.6 Water Loading of Large Arrays..... 371 7.4.7 Magnetostrictive FEM......... 372 References.... 373 Chapter 8: Transducer Characteristics 376 8.1 Resonance Frequency..... 376 8.2 The Mechanical Quality Factor........ 379 8.2.1 Definitions... 379 8.2.2 EffectoftheMassoftheBar... 381 8.2.3 The Effect of Frequency-Dependent Resistance.. 383 8.3 Characteristic Mechanical Impedance....... 384 8.4 Electromechanical Coupling Coefficient.... 386 8.4.1 Energy Definitions of Coupling and OtherInterpretations... 386 8.4.1.1 Mason s Energy Definition......... 386 8.4.1.2 The Mutual Energy Definition...... 389 8.4.1.3 Other Features of the Coupling Coefficient... 390 8.4.2 The Effect of Inactive Components on the Coupling Coefficient... 391 8.4.3 The Effect of Dynamic Conditions on the Coupling Coefficient... 396 References.... 400 Chapter 9: Nonlinear Mechanisms and Their Effects 402 9.1 Nonlinear Mechanisms in Lumped-Parameter Transducers....... 403 9.1.1 Piezoelectric Transducers...... 403 9.1.2 Electrostrictive Transducers.... 408 9.1.3 Magnetostrictive Transducers... 409 9.1.4 Electrostatic and Variable Reluctance Transducers........ 412 9.1.5 Moving Coil Transducers...... 413 9.1.6 Other Nonlinear Mechanisms... 415 9.2 Analysis of Nonlinear Effects......... 415 9.2.1 Harmonic Distortion Direct Drive Perturbation Analysis... 416 9.2.2 Harmonic Distortion for Indirect Drive... 424
xviii Contents 9.2.3 Instability in Electrostatic and Variable-Reluctance Transducers...... 425 9.3 Nonlinear Analysis of Distributed- Parameter Transducers... 428 9.4 Nonlinear Effects on the Electromechanical Coupling Coefficient... 435 References... 435 Chapter 10: Acoustic Radiation from Transducers 438 10.1 The Acoustic Radiation Problem...... 438 10.2 Far-Field Acoustic Radiation..... 445 10.2.1 Line Sources..... 445 10.2.2 Flat Sources in a Plane.... 447 10.2.3 Spherical and Cylindrical Sources..... 454 10.3 Near-Field Acoustic Radiation.... 455 10.3.1 Field on the Axis of a Circular Piston... 455 10.3.2 The Effect of the Near Field on Cavitation... 456 10.3.3 Near Field of Circular Sources........ 459 10.4 Radiation Impedance.... 460 10.4.1 Spherical Sources........ 460 10.4.2 Circular Sources in a Plane.... 463 References... 465 Chapter 11: Advanced Acoustic Radiation Calculations 467 11.1 Mutual Radiation Impedance..... 467 11.1.1 Piston Transducers on a Sphere....... 467 11.1.2 Piston Transducers on a Cylinder...... 472 11.1.3 The Hankel Transform... 477 11.1.4 TheHilbertTransform... 479 11.2 Green s Theorem and Acoustic Reciprocity.... 481 11.2.1 Green s Theorem........ 481 11.2.2 Acoustic Reciprocity..... 482 11.2.3 Green s Function Solutions... 483 11.2.4 TheHelmholtzIntegralFormula... 487 11.3 Effects of Acoustic Scattering.... 490 11.3.1 TheDiffractionConstant... 490 11.3.2 Scattering from Cylinders..... 493 11.4 Numerical Methods for Acoustic Calculations........ 495 11.4.1 Mixed Boundary Conditions Collocation..... 495 11.4.2 Boundary Element Methods... 498 References... 501 Chapter 12: Transducer Evaluation and Measurement 505 12.1 Electrical Measurement of Transducers in Air......... 505 12.1.1 Electric Field Transducers..... 506 12.1.2 Magnetic Field Transducers... 509 12.2 Measurement of Transducers in Water........ 511 12.3 Measurement of Transducer Efficiency........ 516
xix 12.4 Acoustic Responses of Transducers.... 517 12.5 Reciprocity Calibration.... 521 12.6 Tuned Responses......... 524 12.6.1 Electric Field Transducers...... 525 12.6.2 Magnetic Field Transducers.... 528 12.7 Near-field Measurements... 530 12.7.1 Distance to the Far-field........ 530 12.7.2 Measurements in Tanks........ 531 12.7.3 Near- to Far-field Extrapolation Small Sources..... 534 12.7.4 Near- to Far-field Extrapolation Large Sources..... 535 12.7.5 Effect of Transducer Housings...... 539 References.... 541 The Future 543 Appendix 546 A.1 ConversionsandConstants... 546 A.2 Transducer Materials Ordered by Impedance, ρc... 547 A.3 Time Averages, Power Factor, Complex Intensity.... 548 A.4 Relationships Between Piezoelectric Coefficients.... 550 A.5 Small-Signal Properties of Piezoelectric Ceramics... 551 A.6 Piezoelectric Ceramic Approximate Frequency Constants...... 554 A.7 Small Signal Properties of Magnetostrictive Materials... 554 A.8 VoltageDividerandTheveninEquivalentCircuit... 555 A.9 Magnetic Circuit Analysis..... 556 A.10NortonCircuitTransformations... 558 A.11IntegralTransformPairs... 559 A.12 Calibrated Transducers.... 560 A.13 Frequently Used Formulas..... 567 A.14 Stress and Field Limits for Piezoelectric Ceramics... 570 A.15 Comprehensive Hydrophone Noise Model.......... 571 A.16 Cables and Transformers... 579 A.17 Thermal Noise and Radiation Resistance.... 582 Glossary of Terms 587 Index 599