The Foundations of Acoustics
Basic Mathematics and Basic Acoustics
(Sprache: Englisch)
Research and scientific progress are based upqn intuition coordinated with a wide theoretical knowledge, experimental skill, and a realistic sense of the limitations of technology. Only a deep insight into physical phenomena will supply the necessary skills...
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Research and scientific progress are based upqn intuition coordinated with a wide theoretical knowledge, experimental skill, and a realistic sense of the limitations of technology. Only a deep insight into physical phenomena will supply the necessary skills to handle the problems that arise in acoustics. The acoustician today needs to be well acquainted with mathematics, dynamics, hydrodynamics, and physics; he also needs a good knowledge of statistics, signal processing, electrical theory, and of many other specialized subjects. Acquiring this background is a laborious task and would require the study of many different books. It is the goal of this volume to present this background in as thorough and readable a manner as possible so that the reader may turn to specialized publications or chapters of other books for further information without having to start at the preliminaries. In trying to accomplish this goal, mathematics serves only as a tool; the better our understanding of a physical phenomenon, the less mathematics is needed and the shorter and more concise are our computa tions. A word about the choice of subjects for this volume will be helpful to the reader. Even scientists of high standing are frequently not acquainted with the fundamentals needed in the field of acoustics. Chapters I to IX are devoted to these fundamentals. After studying Chapter I, which dis cusses the units and their relationships, the reader should have no difficulty converting from one system of units to any other.
Inhaltsverzeichnis zu „The Foundations of Acoustics “
- Historical IntroductionI. 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.
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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
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
... weniger
Bibliographische Angaben
- Autor: Eugen Skudrzyk
- 2012, Softcover reprint of the original 1st ed. 1971, XXVIII, 790 Seiten, 197 Abbildungen, Maße: 16,9 x 24,5 cm, Kartoniert (TB), Englisch
- Verlag: Springer
- ISBN-10: 3709182573
- ISBN-13: 9783709182574
Sprache:
Englisch
Pressezitat
"This book provides a comprehensive treatment of a number of topics in acoustics and the mathematics of acoustics. It is well organized and clearly written. It clearly leans heavily to the mathematics ... . In summary, I would say that this is an excellent foundation for the mathematics of acoustics and a great reference for the mathematics applicable to particular problems in acoustics." (James K. Thompson, Noise News International, Vol. 24 (4), December, 2016)Kommentar zu "The Foundations of Acoustics"
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