Acoustics for engineers : Troy lectures

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 De¯nition of Three Basic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Specialized Areas within Acoustics . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 About the History of Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Relevant Quantities in Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Some Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Levels and Logarithmic Frequency Intervals . . . . . . . . . . . . . . . . 8 1.7 Double-Logarithmic Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Mechanic and Acoustic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Basic Elements of Linear, Oscillating, Mechanic Systems . . . . . 14 2.2 Parallel Mechanic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Free Oscillations of Parallel Mechanic Oscillators . . . . . . . . . . . . 17 2.4 Forced Oscillation of Parallel Mechanic Oscillators . . . . . . . . . . . 19 2.5 Energies and Dissipation Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Basic Elements of Linear, Oscillating, Acoustic Systems . . . . . . 24 2.7 The Helmholtz Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Electromechanic and Electroacoustic Analogies . . . . . . . . . . . . 27 3.1 The Electromechanic Analogies. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 The Electroacoustic Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Levers and Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Rules for Deriving Analogous Electric Circuits . . . . . . . . . . . . . . 31 3.5 Synopsis of Electric Analogies of Simple Oscillators . . . . . . . . . 33 3.6 Circuit Fidelity, Impedance Fidelity and Duality . . . . . . . . . . . . 33 3.7 Examples of Mechanic and Acoustic Oscillators . . . . . . . . . . . . . 34 4 Electromechanic and Electroacoustic Transduction . . . . . . . . . 37 4.1 Electromechanic Couplers as Two- or Three-Port Elements . . . 38 4.2 The Carbon Microphone { A Controlled Coupler . . . . . . . . . . . . 39 4.3 Fundamental Equations of Electroacoustic Transducers . . . . . . . 40 4.4 Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.5 Coupling of Electroacoustic Transducers to the Sound Field . . . 44 4.6 Pressure and Pressure-Gradient Receivers . . . . . . . . . . . . . . . . . . 46 4.7 Further Directional Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 49 4.8 Absolute Calibration of Transducers . . . . . . . . . . . . . . . . . . . . . . . 52 5 Magnetic-Field Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.1 The Magnetodynamic Transduction Principle . . . . . . . . . . . . . . . 57 5.2 Magnetodynamic Sound Emitters and Receivers . . . . . . . . . . . . . 59 5.3 The Electromagnetic Transduction Principle . . . . . . . . . . . . . . . . 65 5.4 Electromagnetic Sound Emitters and Receivers . . . . . . . . . . . . . . 67 5.5 The Magnetostrictive Transduction Principle . . . . . . . . . . . . . . . . 68 5.6 Magnetostrictive Sound Transmitters and Receivers . . . . . . . . . . 69 6 Electric-Field Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.1 The Piezoelectric Transduction Principle . . . . . . . . . . . . . . . . . . . 71 6.2 Piezoelectric Sound Emitters and Receivers . . . . . . . . . . . . . . . . . 74 6.3 The Electrostrictive Transduction Principle . . . . . . . . . . . . . . . . . 78 6.4 Electrostrictive Sound Emitters and Receivers . . . . . . . . . . . . . . . 79 6.5 The Dielectric Transduction Principle . . . . . . . . . . . . . . . . . . . . . . 80 6.6 Dielectric Sound Emitters and Receivers . . . . . . . . . . . . . . . . . . . . 81 6.7 Further Transducer and Coupler Principles . . . . . . . . . . . . . . . . . 85 7 The Wave Equation in Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.1 Derivation of the One-Dimensional Wave Equation . . . . . . . . . . 89 7.2 Three-Dimensional Wave Equation in Cartesian Coordinates . . 94 7.3 Solutions of the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.4 Field Impedance and Power Transport in Plane Waves . . . . . . . 96 7.5 Transmission-Line Equations and Re°ectance . . . . . . . . . . . . . . . 97 7.6 The Acoustic Measuring Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8 Horns and Stepped Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8.1 Webster's Di®erential Equation { the Horn Equation . . . . . . . . . 104 8.2 Conical Horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.3 Exponential Horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.4 Radiation Impedances and Sound Radiation . . . . . . . . . . . . . . . . 110 8.5 Steps in the Area Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.6 Stepped Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9 Spherical Sound Sources and Line Arrays . . . . . . . . . . . . . . . . . 117 9.1 Spherical Sound Sources of 0th Order . . . . . . . . . . . . . . . . . . . . . . 118 9.2 Spherical Sound Sources of 1st Order . . . . . . . . . . . . . . . . . . . . . . 122 9.3 Higher-Order Spherical Sound Sources . . . . . . . . . . . . . . . . . . . . . 124 9.4 Line Arrays of Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9.5 Analogy to Fourier Transforms as Used in Signal Theory . . . . . 127 9.6 Directional Equivalence of Sound Emitters and Receivers . . . . . 130 10 Piston Membranes, Di®raction and Scattering . . . . . . . . . . . . 133 10.1 The Rayleigh Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.2 Fraunhofer's Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.3 The Far Field of Piston Membranes . . . . . . . . . . . . . . . . . . . . . . . . 136 10.4 The Near Field of Piston Membranes . . . . . . . . . . . . . . . . . . . . . . 138 10.5 General Remarks on Di®raction and Scattering . . . . . . . . . . . . . . 142 11 Dissipation, Re°ection, Refraction, and Absorption . . . . . . . . 145 11.1 Dissipation During Sound Propagation in Air . . . . . . . . . . . . . . . 147 11.2 Sound Propagation in Porous Media . . . . . . . . . . . . . . . . . . . . . . . 148 11.3 Re°ection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 11.4 Wall Impedance and Degree of Absorption . . . . . . . . . . . . . . . . . . 152 11.5 Porous Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 11.6 Resonance Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 12 Geometric Acoustics and Di®use Sound Fields . . . . . . . . . . . . 161 12.1 Mirror Sound Sources and Ray Tracing . . . . . . . . . . . . . . . . . . . . . 162 12.2 Flutter Echoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 12.3 Impulse Responses of Rectangular Rooms . . . . . . . . . . . . . . . . . . 167 12.4 Di®use Sound Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 12.5 Reverberation-Time Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 12.6 Application of Di®use Sound Fields . . . . . . . . . . . . . . . . . . . . . . . . 173 13 Isolation of Air- and Structure-Borne Sound . . . . . . . . . . . . . . 177 13.1 Sound in Solids { Structure-Borne Sound . . . . . . . . . . . . . . . . . . . 177 13.2 Radiation of Airborne Sound by Bending Waves . . . . . . . . . . . . . 179 13.3 Sound-Transmission Loss of Single-Leaf Walls . . . . . . . . . . . . . . . 181 13.4 Sound-Transmission Loss of Double-Leaf Walls . . . . . . . . . . . . . . 184 13.5 The Weighted Sound-Reduction Index . . . . . . . . . . . . . . . . . . . . . 186 13.6 Isolation of Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 13.7 Isolation of Floors with Regard to Impact Sounds . . . . . . . . . . . 192 14 Noise Control { A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 14.1 Origins of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 14.2 Radiation of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 14.3 Noise Reduction as a System Problem . . . . . . . . . . . . . . . . . . . . . . 200 14.4 Noise Reduction at the Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 14.5 Noise Reduction Along the Propagation Paths . . . . . . . . . . . . . . 204 14.6 Noise Reduction at the Receiver's End . . . . . . . . . . . . . . . . . . . . . 208 15 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 15.1 Complex Notation for Sinusoidal Signals . . . . . . . . . . . . . . . . . . . . 211 15.2 Complex Notation for Power and Intensity . . . . . . . . . . . . . . . . . . 212 15.3 Supplementary Textbooks for Self Study . . . . . . . . . . . . . . . . . . . 214 15.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 15.5 Letter Symbols, Notations and Units . . . . . . . . . . . . . . . . . . . . . . . 234 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239