The Foundation of Acoustics. Basic Mathematics and Basic Acoustics. - gebrauchtes Buch

EAN (ISBN-13): 9783211809884
ISBN (ISBN-10): 3211809880
Gebundene Ausgabe
Taschenbuch
Erscheinungsjahr: 1972
Herausgeber: Springer

Buch in der Datenbank seit 2008-12-09T01:52:24+01:00 (Berlin)
Detailseite zuletzt geändert am 2022-03-22T21:39:03+01:00 (Berlin)
ISBN/EAN: 3211809880

ISBN - alternative Schreibweisen:
3-211-80988-0, 978-3-211-80988-4
Alternative Schreibweisen und verwandte Suchbegriffe:
Autor des Buches: skudrzyk
Titel des Buches: basic mathematics, foundations mathematics, the foundations acoustics, basic mathematic, acoustic

---

Daten vom Verlag:

Autor/in: Eugen Skudrzyk
Titel: The Foundations of Acoustics - Basic Mathematics and Basic Acoustics
Verlag: Springer; Springer Wien
790 Seiten
Erscheinungsjahr: 1971-12-07
Vienna; AT
Gewicht: 1,970 kg
Sprache: Englisch
85,55 € (DE)
87,95 € (AT)
106,60 CHF (CH)
Not available, publisher indicates OP

BB; Book; Hardcover, Softcover / Physik, Astronomie/Allgemeines, Lexika; Physik; Verstehen; acoustics; C; Physics, general; Physics and Astronomy; BC; EA

Historical Introduction.- I. Equations and Units.- 1.1. Dimensional and Numerical Equations.- 1.2. The kg-m-sec-amp System of Units.- 1.3. The Definition of the Unit of Electric Current, the Ampere [A].- 1.4. Derived Electrical Units.- 1.5. The Practical (Physical) Units for Electrical Quantities.- 1.6. The Fundamental Electrical Laws.- 1.7. Transformation of Units.- II. Complex Notation and Symbolic Methods.- 2.1. Complex Notation and Rotating Vectors.- 2.2. Computations with Complex Vectors.- 2.2.1. Definitions.- 2.2.2. Addition.- 2.2.3. Subtraction.- 2.2.4. Multiplication.- 2.2.5. Division.- 2.2.6. Logarithm of a Complex Number.- 2.2.7. Raising a Complex Number to a Given Power.- 2.2.8. Differentiation and Integration.- 2.3. Conjugate Complex Vectors and Their Applications.- 2.4. Addition of Harmonic Functions of the Same Frequency.- 2.5. Symbolic Method for Solving Linear Differential Equations.- 2.6. Complex Solution and Boundary Conditions.- 2.7. Computation of Power.- 2.8. Basic Theory of Internal Friction.- III. Analytic Functions: Their Integration and the Delta Function.- 3.1. Analytic Functions.- 3.2. Representation of an Analytic Function by a Power Series.- 3.3. Cauchy’s Formula.- 3.4. The Cauchy Integral Formula.- 3.5. Residues.- 3.6. Examples.- 3.6.1. Evaluation of Integrals of the type $$ \\int\\limits_0^{2\\pi } {R\\left( {\\cos \\theta ,\\sin\\theta } \\right)d\\theta } $$.- 3.6.2. Summation of a Series by Contour Integration.- 3.7. Evaluation of Integrals of the Form $$ \\int\\limits_{ - \\infty }^\\infty {Q\\left( x \\right)dx} $$.- 3.7.1. Integrals Involving Sines and Cosines.- 3.8. Contour Integrals for Hankel and Bessel Functions.- 3.9. Jordan’s Lemma.- 3.10. Integrals Through Poles, Principal Value of Integrals.- 3.11. Multivalued Functions.- 3.12. Contour Integrals in Vector Notation.- 3.13. Determination of the Real and the Imaginary Parts of an Analytic Function (the Hilbert Transform).- 3.14. Debye’s Saddle-Point Method.- 3.15. Method of Stationary Phase.- 3.16. Example: Stirling’s Formula.- 3.17. Double Integrals.- 3.18. Differentiation and Integration of Integrals with Respect to Parameters.- 3.19. The Delta Function.- 3.20. Transformation of the Variables in Integrals.- 3.21. Singular Points, Integral-, Rational-, and Meromorphic Functions.- 3.22. Singularities of a Second Order Differential Equation.- 3.22.1. The Wronskian Determinant.- 3.22.2. Second Independent Solution of Second Order Differential Equation.- 3.22.3. Ordinary Points and Regular Singular Points.- 3.22.4. Essential Singularity and Branch Point.- 3.22.5. Irregular Singular Point.- 3.22.6. How to Solve a Differential Equation.- IV. Fourier Analysis.- 4.1. The Fourier Series.- 4.2. Examples.- 4.2.1. The Fourier Spectrum of a Periodic Series of Short Pulses.- 4.2.2. The Fourier, Spectrum of a Periodically Repeated Saw-Tooth Curve.- 4.2.3. Fourier Spectrum of a Warble Tone.- 4.3. Fourier Analysis in Terms of Rotating Vectors.- 4.4. Completeness of the Fourier Series and Parseval’s Theorem.- 4.5. Fourier Analysis with the Aid of Filters.- 4.6. Transition to the Fourier Integral.- 4.7. Example: A Point-Mass Compliance System Excited by a Pulse of Very Short Duration.- 4.8. Relation Between Fourier Transform and Fourier Coefficient.- V. Advanced Fourier Analysis.- 5 1 Important Relations in Fourier Analysis.- 5.1.1. Requirements for the Existence of a Fourier Transform.- 5.1.2. Degree of Convergence of a Fourier Series.- 5.1.3. Spectral Amplitude at Low Frequencies: Theorem I.- 5.1.4. Translation of Origin of Time: Theorem II.- 5.1.5. Translation of the Origin in Frequency Space: Theorem III.- 5.1.6. Similarity Theorem: Theorem IV, Compression of Frequency Scale.- 5.1.7. Amplitude Modulation: Theorem V.- 5.1.8. Convolution: Theorem VI.- 5.1.9. Partial-Fraction Development: Theorem VII.- 5.2. Enforced Convergence of Fourier Integrals by Assuming Infinitely Small Damping of the Time Function.- 5.3. Enforcing Convergence by Assuming the High Frequency Components of the Spectrum to be Dissipated.- 5.4. The Shape of an Impulse and Its Spectrum.- 5.5. Examples.- 5.5.1. The Step Function ?0 (t) and Its Spectrum.- 5.5.2. The Rectangular Pulse and the Impulse Function ?i (t).- 5.5.3. Switching on of a Sinusoidal Vibration.- 5.5.4. The Spectrum of a Sinusoidal Vibration of Finite Duration.- 5.5.5. Frequency Modulated Pulse with Sliding Modulation Frequency (“FM Slide”).- VI. The Laplace Transform.- 6.1. One-Sided Time Functions and Enforced Convergence.- 6.2. Computation Rules.- 6.3. Example: The Vibrating Point Mass-Spring.- VII. Integral Transforms and the Fourier Bessel Series.- 7.1. The Fourier Transform.- 7.2. The Laplace Transform.- 7.3. The Infinite Hilbert Transform.- 7.4. The Finite Hilbert Transform.- 7.5. The Mellin Transform.- 7.6. The Infinite Hankel Transform.- 7.7. The Finite Hankel Transform and the Fourier Bessel Series.- VIII. Correlation Analysis.- 8.1. Power Spectrum and Correlation Function.- 8.2. Cross-Spectral Density and Cross-Correlation Function.- 8.3. Running Fourier Transform and Instantaneous Power Spectrum.- 8.4. Running Autocorrelation Function.- 8.5. Derivatives of the Correlation Function.- 8.6. Convolution Integral and Power Spectrum.- IX. Wiener’s Generalized Harmonic Analysis.- X. Transmission Factor, Filters, and Transients (“Küpfmüller’s Theory”).- 10.1. The Transients of Mechanical and Electrical Systems.- 10.2. The Transmission Factor.- 10.3. Relations Between Real Part and Imaginary Part of Transmission Factor.- 10.4. Relation Between Amplitude Response and Phase Response or Time Delay.- 10.5. Frequency Curve and Acoustic Quality.- 10.6. Exact Computation of the Transients by Convolution of Signal with Response for Step or Impulse Function.- 10 7 Exact Computation of the Transient for the Impulse Function.- 10.8. Two Causes for Transients: Phase Distortion and Amplitude Distortion.- 10.9. Phase Distortion and Group Delay.- 10.10 Examples.- 10.10.1. The Series Resonant Circuit.- 10.10.2. The Time Delay of a Wave Traveling in a Rod and Reflected at the Driving Piezoelectric Crystal.- 10.10.3. Time Delay in Low-Pass and High-Pass Filters.- 10.11. Transients Caused by Frequency-Dependent Amplitude Response.- 10.12. Response for Impulse and Step Function.- 10.12.1. The Ideal Low-Pass Filter.- 10.12.2. Low-Pass Filter with Discontinuities of the Transmission Factor.- 10.12.3. Periodic Fluctuations of the Frequency Curve of the Filter in the Pass Range.- 10.12.4. Transients for a Low-Pass Filter with Arbitrary Frequency Transmission.- 10.12.5. The High-Pass.- 10.12.6. Band-Pass.- 10.12.7. Bandpass of Arbitrary Transmission Factor.- 10.13. Transients for Sinusoidal Oscillations as Input Functions.- 10.14. Transients Generated by Phase Distortion.- 10.14.1. Phase Distortion Alone.- 10.14.2. Ideal Low-Pass with Phase Distortion.- 10.14.3. Symmetric Band-Pass with Phase Distortion.- 10.15. Search-Tone Analysis.- XI. Probability Theory, Statistics, and Noise.- 11.1. Basic Concepts of Probability Theory and Statistics.- 11.1.1. Statistical, Random, or Stochastic Variable.- 11.1.2. Set of Functions.- 11.1.3. Ensemble of Functions.- 11.1.4. Stationary Random Function.- 11.1.5. Variations with Time and from Sample to Sample.- 11.2. Ergodic Hypothesis.- 11.3. Statistical Independence.- 11.4. The Probability Distribution for the Sum of Two Independent Random Variables.- 11.5. Probability Density of the Values of a Function of a Stochastic Variable.- 11.6. Mean, Variance, Standard Deviation, and Moments.- 11.7. Characteristic Function.- 11.8. Central Limit Theorem.- 11.9. The Binomial Distribution.- 11.10. The Poisson Distribution.- 11.11. The Rayleigh Distribution.- 11.12. The Normal or Gaussian Distribution.- 11.13. Multidimensional Normal Distribution.- 11.14. Chi-Square Distribution.- 11.15. Standard Deviation, Skewness, and Flatness of a Distribution.- 11.16. Relationship Between Binomial, Poisson, and Normal Distribution.- 11.17. White Noise.- 11.18. Thermal Noise.- 11.19. Measurements with Gaussian Noise.- 11.20. Appendix: Unbiased Estimate of Variance of Small Sets of Samples.- XII. Signals and Signal Processing.- 12.1. Beats and Signals.- 12.2. Resolution in Time and Frequency Domain.- 12.3. Sampling Theorem in Time Domain.- 12.4. Sampling Theorem in Frequency Domain.- 12.5. Derivation of the Sampling Theorems by Convolution Method.- 12.6. Sampling and Scatter of Mean Values for Samples of Limited Dimensions.- 12.7. Detection of a Periodic Signal of Infinite Duration in Noise.- 12.7.1. By Autocorrelation.- 12.7.2. Gain in Detection of Periodic Signal by Cross Correlation.- 12.8. Determination of Periodic Component in Random Wave.- 12.9. Rectifier with an RC Filter.- 12.10. Square-Law Detector.- 12.11. Ideal Correlator.- 12.12. Two-Channel Correlator.- 12.13. Sign Correlator.- 12.14. Two-Channel Sign Correlator.- 12.15. Comparison of the Systems.- 12.16. The Variable Reference Level Correlator.- 12.17. Practical Correlators.- 12.17.1. Analog and Sampling Correlator.- 12.17.2. Dynamic Reference Correlator.- 12.17.3. Delay Line or Deltic Correlator.- 12.17.4. Static Reference Correlators.- 12.18. Signal-to-Noise Ratio and Optimum Processing.- 12.19. Matched-Filter System.- 12.19.1. Constant Frequency Pulses.- 12.19.2. Monotonic Frequency-Modulated Pulses.- XIII. Sound.- 13.1. Definition of Sound.- 13.2. The Sound Variables.- 13.3. The State Equation.- 13.4. Examples.- 13.4.1. Relationship Between Sound Velocity c and Bulk Modulus ?K of a Fluid.- 13.4.2. The Sound Velocity of an Ideal Adiabatic Gas.- 13.5. The Euler Equation.- 13.6. The Continuity Equation.- 13.7. The Wave Equation.- 13.8. The Velocity Potential.- 13.9. The Wave Equation for Forced Vibrations.- 13.10. The Physical Significance of the Velocity Potential.- 13.11. Wave Equation for an Inhomogeneous Medium.- 13.12. The Effect of Viscosity.- XIV. The One-Dimensional Wave Equation and Its Solutions.- 14.1. Plane Sound Waves.- 14.2. Progressive Wave Solution.- 14.3. Standing Wave Solution.- 14.4. The Relation Between the Standing-Wave and the Progressive-Wave Solution.- 14.5. Pressure and Particle Velocity in a Plane Progressive Wave.- 14.6. Radiation Resistance in the Plane Wave.- XV. Reflection and Transmission of Plane Waves at Normal Incidence.- 15.1. Reflection at a Rigid Surface.- 15.2. Reflection at Resilient Surface.- 15.3. Reflection at the Interface Between Two Media and the Coefficient of Absorption.- 15.4. Acoustic Point Impedance.- 15.5. Reflection and Absorption at an Interface Whose Properties Are Represented by an Acoustic Point Impedance.- 15.6. Graphical Procedure to Construct the Reflection and Absorption Factor for Any Acoustical Impedance.- 15.7. Sound Field in Front of an Absorbent Reflector at Normal Incidence.- 15.8. Measurement of Acoustic Impedances for Normal Incidence by the Standing Wave Method.- 15.9. Description of the Sound Field in Front of an Absorbing Surface in Terms of Complex Harmonic Functions.- 15.10. Reflection Factor and Time Delay.- 15.11. Reflection Factor Relative to an Arbitrarily Selected Plane Parallel to the Plane of the Reflector.- XVI. Plane Waves in Three Dimensions.- 16.1. Plane Waves in Three-Dimensional Space.- 16.2. Reflection of a Plane Wave at Oblique Incidence.- 16.2.1. Rigid Reflecting Surface.- 16.2.2. Resilient Reflector.- 16.2.3. Reflecting Medium Described by Its Acoustic Impedance.- 16.2.4. Reflecting Medium Infinitely Extended; Refraction and Snell’s Law.- 16.3. Sound Radiation of an Infinite Plate Excited to a Sinusoidal Vibration Pattern.- 16.3.1. Nodal Line Pattern Independent of Frequency.- 16.3.2. Nodal Line Pattern Not Fixed, but Due to Bending Vibrations of a Plate of Constant Thickness.- XVII. Sound Propagation in Ideal Channels and Tubes.- 17.1. The Solution of the Wave Equation, Sound Velocity, Phase Velocity, and Group Velocity.- 17.2. Propagating Waves and Distortion Fields.- 17.3. Sound Propagation in Channels and Tubes Below Their Radial Resonant Frequency.- 17.3.1. Both Terminations Rigid.- 17.3.2. Tube Terminations Resilient (Open Ends).- 17.3.3. One End of Tube Resiliently Terminated, the Other Rigidly Closed.- 17.3.4. Tube with an Abrupt Change of Cross Section.- 17.4. Change of Cross Section as Acoustic Transformer.- 17.5. Sound Propagation in Infinitely Long Horns.- 17.6. Sound Propagation in Channels and Tubes with Non-Plane Rigid Terminations Below the First Non-Axial Resonance.- 17.7. Examples.- 17.7.1. Tube Terminated by a Conical Horn.- 17.7.2. Tube Terminated by Tube of Different Diameter.- 17.7.3. Rectangular Tube Terminated by an Oblique Wall.- 17.8. The Natural Frequencies of Pipes with Different Terminations.- XVIII. Spherical Waves, Sources, and Multipoles.- 18.1. The Wave Equation for Centrally Symmetric Spherical Propagation and its Solution.- 18.2. Farfield and Nearfield.- 18.3. Sound Pressure and Volume Flow.- 18.4. Spherical Wave Impedance and Radiation of Small Sound Sources.- 18.5. The Sound Power Generated by a Pulsating Sphere.- 18.6. Radiation Resistance and Effective (Acoustic) Mass of a Small Pulsating Source and the Equivalent Sphere.- 18.7. Radiation Resistance Referred to Volume Flow.- 18.8. Standing Spherical. Waves of Zero Order.- 18.9. Acoustic Dipoles and Oscillating Rigid Bodies.- 18.10. The Radiation Resistance of a Small Oscillating Rigid Body.- 18.11. The Effective (Acoustic) Mass for a Small Oscillating Body of Any Shape.- 18.12. Examples.- 18.12.1. The Effective (Acoustic) Mass for an Oscillating Sphere.- 18.12.2. The Sound Radiation of a Piston Membrane that is Not Enclosed in a Baffle.- 18.13. The Motion of a Small Rigid Sphere or a Solid Particle in a Sound Wave.- 18.14. Quadrupole Radiators.- 18.15. Sound Radiation at High Frequencies.- 18.16. Reflection of a Spherical Wave at a Plane Boundary.- 18.17. Interaction Between Sound Sources and Between Sound Sources and Their Images.- 18.17.1. (a) Interaction Between Sound Sources of Zero Order.- 18.17.2. (b) Interaction Between Dipoles.- 18.17.3. Interaction Between Quadrupoles.- 18.18. Radiation from Nonperiodic Sources, Dipoles, and Quadrupoles.- XIX. Solution of the Wave Equation in General Spherical Coordinates.- 19.1. The Wave Equation in General Spherical Coordinates.- 19.2. Solution of the Wave Equation.- 19.3. The Surface Harmonics or Laplace Functions.- 19.4. Radial Part of the Solution.- 19.4.1. The Stokes Functions.- 19.4.2. Bessel Function Solution and the Spherical Bessel Functions.- 19.5. Radiation Impedance of a Sphere Vibrating in a Spherical Harmonic.- XX. Problems of Practical Interest in General Spherical Coordinates.- 20.1. Development of a Power of into Legendre Polynomials.- 20.2. Radiation from a Sphere Vibrating with Axial Symmetry.- 20.3. Point Source on Sphere, Shielding of Radiation by Sphere.- 20.4. The Pressure at the Surface of a Scattering Sphere.- 20.5. Sound Radiation of a Radially Vibrating Spherical Cap Set in a Sphere.- 20.6. Axially Vibrating Cap Set in a Rigid Sphere.- 20.7. Acoustic Radiation from Plane Circular Piston Set in a Rigid Sphere.- 20.7.1. The Minimum Error Method.- 20.7.2. Application to the Plane Piston Set in a Sphere.- 20.8. Representation of a Plane Wave by a Series of Concentric Spherical Waves.- 20.9. Reflection and Refraction of a Plane Wave at a Rigid Sphere.- 20.10. Reflection at Compressible Sphere or at Sphere Covered with Acoustic Absorbent.- 20.11. Spherical Liquid Lens.- 20.12. The Cavity Resonator.- 20.13. Relation Between Multipoles and Wave Functions.- XXI. The Wave Equation in Cylindrical Coordinates and Its Applications.- 21.1. Derivation of the Wave Equation in Cylindrical Coordinates for the Pulsating Cylinder.- 21.2. The Radially Symmetric Wave Equation and the Structure of Its Basic Solutions.- 21.3. The Wave Equation in General Cylindrical Coordinates.- 21.4. The Solution of the Wave Equation—General Cylindrical Coordinates.- 21.5. Sound Propagation in Circular Tubes.- 21.6. Progressive Cylindrical Waves.- 21.7. Rotating Modes.- 21.8. Standing Cylindrical Waves.- 21.9. Infinitely Long Cylinder Excited in a Single Vibrational Mode.- 21.10. Radiation Impedance of a Vibrating Cylinder.- 21.11. The Power Radiated Per Unit Area of the Cylinder.- 21.12. The Pulsating Cylinder.- 21.13. Sound Radiation of an Infinitely Long String.- 21.14. The Cylindrical Quadrupole.- 21.15. Reaction Between Two Parallel Cylindrical Sources of Zero Order.- 21.16. Scattering of Normally Incident Plane Wave at a Rigid Cylinder.- 21.17. Cylinder with End Caps.- XXII. The Wave Equation in Spheroidal Coordinates and Its Solutions.- 22.1. Prolate Spheroidal Coordinates.- 22.2. The Wave Equation in Spheroidal Coordinates.- 22.3. The Angle Functions.- 22.4. The Radial Functions.- 22.5. Modal Velocities and the Weighted Modal Velocities, Sound Pressure and Particle Velocity in Spheroidal Coordinates.- 22.6. Sound Pressure and Particle Velocity in Spheroidal Coordinates.- 22.7. Integrated or Total Modal Radiation Impedance.- 22.8. Approximations for Thin and Long Spheroids.- 22.9. Examples.- 22.9.1. Sound Pressure at Arbitrary Distance ? on Polar Axis (? = 1) Due to a Thin Spheroid Vibrating in the (01) Mode.- 22.9.2. Numerical Example, Sound Pressure Generated by a Thin Spheroid in (00) and (01) Mode on Polar Axis.- 22.10. The Integrated Modal Impedance for a Thin Spheroid.- 22.11. Radiation by Rigid Body Axial Vibration.- 22.12. Radiation by “Accordion” Vibration Mode.- 22.13. Oblate Spheroidal Coordinates.- 22.14. Example: Pressure Generated by a Circular Piston That is Not in a Baffle.- 22.15. Tables on Spheroidal Wave Functions.- 22.16. Appendix: Curvilinear Coordinates.- 22.16.1. Coordinate Transformations and the Metric Tensor.- 22.16.2. Fundamental, Differential Operators in Curvilinear Coordinates.- XXIII. The Helmholtz Huygens Integral.- 23.1. Green’s Integral Formula and Gauss’ Theorem.- 23.2. Helmholtz Huygens Radiation Integral.- 23.2.1. The Integration Surface Surrounds the Field Point and Separates It from Sources.- 23.2.2. Field Point and Sources Outside Surface of Integration.- 23.2.3. Surface of Integration Encloses Field Point and Sources. The Sommerfeld Infinity Condition.- 23.2.4. The Helmholtz Huygens Integral for any Surface of Integration.- 23.3. Field Point and One Source Inside Surface of Integration, Other Sources Outside.- 23.4. The Helmholtz Huygens Integral with Internal Sources and Forces.- 23.5. The Simplified Diffraction Formulae and the Green’s Function.- 23.5.1. Transition from the Helmholtz Huygens Radiation Integral to Huygens Theorem for Plane Radiators and Screens.- 23.5.2. Helmholtz Huygens Integral for the Pressure.- 23.6. Physical Meaning of the Helmholtz Huygens Integral.- 23.7. The Many-Valuedness of the Source and Dipole Distributions in the Helmholtz Huygens Integral.- 23.8. The Helmholtz Huygens Integral as a Solution of a Discontinuity Problem.- 23.9. Examples.- 23.9.1. The Sound Field Scattered at a Small Incompressible Particle or Generated by a Small Oscillating Particle.- 23.9.2. Scattering by Inhomogeneities of the Medium.- 23.10. Other Forms of the Radiation or Diffraction Integral.- 23.10.1. Axially Symmetric Field.- 23.10.2. King’s Diffraction and Radiation Integral.- 23.11. The Helmholtz Huygens Integral for Unsteady Phenomena.- 23.12. Poisson’s Wave Formula.- XXIV. Huygens Principle and the Rubinowicz—Kirchhoff Theory of Diffraction.- 24.1. The Huygens—Rayleigh Integral.- 24.2. Huygens Zone Construction.- 24.3. Examples.- 24.3.1. The Plane Sound Wave.- 24.3.2. The Sound Field Along the Central Axis of a Piston Membrane (or Circular Aperture) as a Function of the Distance. Ray Region and Region of Spherical Propagation.- 24.4. Kirchhoff Theory of Diffraction.- 24.5. Babinet’s Principle.- 24.6. The Diffraction Integral of Rubinowicz.- 24.7. The Edge Wave.- 24.7.1. The Edge Wave at High Frequencies and at a Great Distance from the Screen or Vibrator and Far Away from the Shadow Boundary.- 24.7.2. Near the Shadow Boundary.- 24.8. Application of the Theory.- 24.8.1. Piston Membrane.- 24.8.2. Series Developments and Approximate Solutions for Diffraction at Circular Disc or Radiation by Piston Membrane for the Vicinity of the Disc or Piston.- 24.8.3. Plane Wave Diffracted at Semi Infinite Plane.- 24.9. Spherical Wave Diffracted at Edge of a Semi Infinite Plane.- 24.10. Analytic Continuation of the Kirchhoff Integral.- 24.11. Non Plane Screens.- 24.12. Phase Anomaly Near Focus.- 24.13. Comparison of the Kirchhoff Assumptions and the Results of the Kirchhoff Theory with the Results of Accurate Computations.- 24.14. Appendix: Series and Asymptotic Development of the Fresnel Integral.- XXV. The Sommerfeld Theory of Diffraction.- 25.1. The Properties of the Sommerfeld Function $$ \\omega \\left( {r,\\varphi ,z,r{}_0,{\\varphi _0},{z_0};{2_\\chi }} \\right) $$ for the Straight Edge and Wedge for a Plane Incident Wave.- 25.2. The Derivation of the Sommerfeld Function w.- 25.3. The Sound Field Inside a Wedge of Angular Opening 2?/n.- 25.4. The General Multivalued Solution.- 25.5. The Straight Edge (p = 2).- 25.6. Approximations to the Sommerfeld Functions.- 25.7. Approximate Evaluation of the Sommerfeld Solution for the Straight Edge.- 25.8. Spherical Incident Wave.- 25.9. Black Screens.- 25.10. The Wedge.- 25.11. The Concept of Riemann Spaces.- 25.12. The Generalized Babinet Principle.- 25.13. Approximate Treatment of Diffraction by Screens and by Three-Dimensional Objects; J. B. Keller’s Method.- 25.13.1. Keller Approximation for Plane Screens.- 25.13.2. Examples.- 25.13.3. Keller Approximation for Three-Dimensional Diffractors.- 25.13.4. The Shadowing Effect of a Hemisphere and of Three-Dimensional Screens.- XXVI. Sound Radiation of Arrays and Membranes.- 26.1. Basic Definitions: Hydrophone Sensitivity, Directivity Function, Directivity Factor, and Directivity Index.- 26.2. The Fraunhofer Integral and the Directivity Function.- 26.3. Examples for Arrays with Point Sources of Constant Strength.- 26.3.1. Two Point Sources of Equal Volume Flow at x = 0 and x = d, Respectively.- 26.3.2. Point Sources Equally Spaced Along a Line.- 26.4. Major and Minor Lobes, Repetition of Directivity Pattern of Linear Array.- 26.5. The Densely Packed Linear Array.- 26.6. Circular Ring Densely Packed with Transducers.- 26.7. Transducers at Constant Intervals Along a Circular Ring.- 26.8. The Circular Piston Membrane in a Baffle and the Circular Aperture.- 26.9. The Rectangular Piston Membrane in a Baffle.- 26.10. Comparison of the Directivity Functions of Various Arrays.- 26.11. Variable Velocity Distributions.- 26.12. Rectangular Membrane.- 26.12.1. Rectangular Membrane Supported at Two Edges.- 26.12.2. Rectangular Membrane With Free Edges.- 26.12.3. Comparison of the Directivity Patterns of Rectangular Membranes in Their Fundamental Mode.- 26.12.4. Circular Membrane, Rigidly Supported at Its Circumference.- 26.12.5. Circular Membrane; Azimuthal and Radial Nodal Lines.- 26.12.6. Directivity Function of Compound Arrays.- 26.13. Shaded Arrays.- 26.14. Binomial Group.- 26.15. Sound Sources at the Corner Points of a Two-Dimensional Grating and the Rectangular Piston Membrane.- 26.16. The Sharpness of the Directivity Pattern.- 26.17. Chebyshev Shaded Array.- 26.18. Chebyshev Polynomials.- 26.18.1. Example.- 26.18.2. Spacing of Transducer Elements.- 26.19. Sum and Difference Patterns.- 26.20. Synthesis of the Difference Pattern.- 26.20.1. Example: Difference Pattern of an Element Array.- 26.21. Directivity Function and Radiation Resistance.- 26.22. Examples.- 26.22.1. Two Sources a Distance d Apart.- 26.22.2. The Rectangular Piston Membrane.- 26.22.3. Membrane or Thin Plate, Rigidly Supported at Its Circumference.- 26.23. The Sound Field in the Proximity of the Radiator: The Fresnel Approximation.- 26.24. Examples.- 26.24.1. Diffraction at a Straight Edge.- 26.24.2. Circular Piston Membrane in an Infinite Baffle.- 26.24.3. The Far Sound Field Generated by a Piston Membrane.- 26.24.4. Application to the Loudspeaker.- 26.25. The Loudspeaker in a Finite Baffle or Without a Baffle.- 26.25.1. Fraunhofer Approximation.- 26.25.2. Fresnel Approximation.- 26.25.3. The Loudspeaker in a Room and Multi-Unit Speakers in Small Baffle and Box.- 26.26. H. Stenzel’s Exact Computation for the Sound Field Generated by a Piston Membrane.- XXVII. The Green’s Functions of the Helmholtz Equation and Their Applications.- 27.1. Definitions.- 27.2. Reciprocity Theorem.- 27.3. The Nature of the Singularity of the Green’s Function.- 27.4. Solution for Finite Space in Terms of the Infinite Space Green’s Function.- 27.5. The Impulse Function and the Time Dependent Solution of the Wave Equation.- 27.6. Expansion of the Green’s Function in Natural Functions.- 27 7 Infinite Space Green’s Function and Complex Natural Functions.- 27.8. Continuous Eigenvalue Spectrum.- 27.9. Examples in Two Dimensions.- 27.9.1. Plane Waves.- 27.9.2. The Axially Symmetric Green’s Function for the Infinite Two-Dimensional Space.- 27.9.3. Cylindrical Waves.- 27.9.4. The Infinite Space Green’s Function in Polar Coordinates in Two Dimensions.- 27.10. Examples in Three Dimensions.- 27.11. The Green’s Function in Spherical Harmonics.- 27.11.1. The Green’s Function in Cylindrical Coordinates.- 27.12. The Green’s Function for Bounded Spaces.- 27.12.1. Perfectly Rigid or Perfectly Resilient Boundary.- 27.12.2. Reflection of a Spherical Wave at an Acoustical Impedance.- XXVIII. Self and Mutual Radiation Impedance.- 28.1. Rayleigh Computation of the Acoustic Impedance of the Piston Membrane in an Infinite Baffle.- 28.2. Computation of the Acoustic Impedance of a Piston Membrane with the Aid of the Green’s Function in Cylindrical Coordinates.- 28.3. The Acoustic Impedance of a Membrane Whose Velocity Varies Over Its Surface.- 28.4. Self and Mutual Radiation Impedance.- 28.5. Example: Mutual Radiation Impedance of Two Rigid Circular Disks.- 28.6. Appendix: Pritchard’s Integrals, Evaluation of an Important Radiation Integral.- Tables.- I. Elementary Functions.- II. Trigonometric Functions.- III. Hyperbolic Functions.- IV. Harmonic and Hyperbolic Functions of Complex Argument.- V. The Inverse Harmonic and Hyperbolic Functions.- VI. Legendre Polynomials and Surface Harmonics.- VII. The Solutions of the Wave Equation.- VIII. Properties of the Bessel Functions.- IX. Spheroidal Functions.- X. The Gamma Function.- XI. The Lommel Functions of Two Variables.- References.- 1. Early History of Acoustics.- 1. Equations and Units.- 2. Complex Notation and Symbolic Methods.- 3. Analytic Functions; Their Integration and the Delta Function.- Chapters 4, 5. Fourier Analysis.- Chapters 6, 7. The Laplace Transform and Transform Theory.- 8. Correlation and Correlation Analysis.- 9. Wiener’s Generalized Harmonic Analysis.- 10. Transmission Factors, Filters, and Transients.- 11. Probability, Theory, Statistics, and Noise.- 12. Signals and Signal Processing.- Chapters 13 to 17. Sound and Simple Sound Fields; Transmission and Reflection; Channels.- 16. Channels and Ducts. (See also Literature Chapters 20, 21.).- 17. Acoustic Impedances and Their Measurement.- 17. Horns.- 17. (Supplementary Literature.) Plates..- Chapters 18, 28. Radiation Impedance.- 18. Simple Spherical Sound Propagation, Sources, Dipoles and Quadrupoles.- 19. The Wave Equation in Spherical Coordinates and Its Solutions, Applications of the Theory.- Chapters 20, 21. The Wave Equation in Cylindrical Coordinates and Its Applications (See also Literature Chapter 17.).- 22. The Wave Equation in Spheroidal Coordinates and Its Solutions.- 23. The Helmholtz-Huygens Integral (See also Literature Chapters 24, 25.).- Chapters 24, 25. Diffraction.- 26. Sound Radiation of Arrays and Membranes. (See also Literature Chapters 24, 25.).- 27. The Green’s Function and Its Application. (See also Literature Chapters 17, 20, 21, 22.).- 28. Radiation Impedance. (See Literature Chapter 18.).- List of Symbols.