1. Statement of the Problem

1.1 Introduction

1.2 Assumptions

1.3 Formulation of the Structural Response

1.4 Formulation of the Acoustic Pressure Field

1.5 The Integral Equation of the Structure-Fluid Interaction

1.6 Historical Development of Structural Acoustics

2. The Wave Equation and Its Elementary Solutions

2.1 Coupled Space and Time Dependence of Sound

2.2 The One-Dimensional Wave Equation for Plane Waves

2.3 Harmonic Time Variation

2.4 Steady-State Plane Waves

2.5 The Three-Dimensional Wave Equation

2.6 The Uniformly Pulsating Spherical Source

2.7 Specific Acoustic Impedance of Spherical Waves

2.8 The Pressure Field Inside a Fluid-Filled Pulsating Sphere

2.9 An Elementary Interaction Problem: Liquid-Filled Elastic Waveguides

2.10 Cylindrical Waveguides: An Introduction to Two-Dimensional Pressure
Fields

3. Applications of the Elementary Acoustic Solutions

3.1 Introduction

3.2 The Point Source

3.3 Rigid and "Pressure-Release" Boundaries

3.4 The Image Source Simulating a Rigid Boundary; Directivity

3.5 The "Pressure-Release" Boundary

3.6 Linear Arrays of Point Sources

3.7 Far-Field Conditions

3.8 Acoustic Power and Intensity

3.9 The Pulsating Gas Bubble

3.10 Dispersion and Attenuation: Sound Propagation in Bubble Swarms

4. The Pressure Field of Arbitrary Source Configurations

4.1 General Formulation of the Radiation Problem

4.2 The Free-Space Green's Function

4.3 The Helmholtz Integral Equation

4.4 The Sommerfeld Radiation Condition

4.5 Physical Interpretation of the Helmholtz Integral

4.6 Approaches to the Solution of the Helmholtz Integral Equation

4.7 Analytical Solution of the Helmholtz Integral Equation

4.8 Rayleigh's Formula for Planar Sources

4.9 The Scattered Field

5. Planar Sound Radiators

5.1 Source Geometry and Analytical Formulations

5.2 Pressure Field of the Circular Piston Using Rayleigh's Formulation

5.3 The Transform Formulation of Axisymmetric Pressure Fields

5.4 The Transform Solution of the Circular Piston

5.5 The Far-Field of Rectangular Radiators Evaluated by Rayleigh's Formula; the
Rigid Piston

5.6 The Transform Solution of Rectangular Sound Radiators

5.7 Equivalence of Rayleigh's and the Stationary-Phase Formulations of the
Far-Field

5.8 The Far-Field of Rectangular Radiators Displaying a Sinusoidal Acceleration
Distribution

5.9 Physical Interpretation of This Solution; Coincidence

5.10 Other Standing-Wave Configurations of Rectangular Radiators

5.11 The Infinite Planar Radiator with a Sinusoidal Acceleration Distribution

5.12 The Radiating Strip of Infinite Length and Finite Width

5.13 Acoustic Resistance of Circular Pistons and of Surface-Radiating Rectangular
Source Configurations with Sinusoidal Acceleration Distributions

5.14 Acoustic Resistance of Edge-Radiating Rectangular Source Configurations
with Sinusoidal
Acceleration Distributions

5.15 The Resistance of Corner-Radiating Rectangular Source Configurations with
Sinusoidal Source Configurations

6. Convex Sound Radiators

6.1 Characteristics of Convex Boundaries

6.2 The Green's Function for the Spherical Radiator

6.3 The Pressure Field of a Spherical Radiator

6.4 Circular Piston and Point Sources on a Spherical Baffle

6.5 The Radiation Loading of Spherical Radiators

6.6 Concentrated Force Applied to the Acoustic Medium

6.7 Cylindrical Radiators with Spatially Periodic Configurations

6.8 Radiation Loading of Infinite Cylinders with Standing-Wave Configurations

6.9 Transform Formulation of the Pressure Field of Cylindrical Radiators

6.10 Stationary-Phase Approximation to the Far-Field of Cylindrical Radiators

6.11 Piston in a Cylindrical Baffle

6.12 Far-Field of Cylinders with Standing-Wave Configurations of Finite Axial
Extent

6.13 Comparison of Planar and Cylindrical Standing-Wave Radiators; Specific
Acoustic Resistance

6.14 Nodal Planes and Acoustic Intensity in a Three-Dimensional Pressure Field

6.15 Far-Field of Slender Bodies of Revolution

7. Vibration of Beams, Plates, and Shells

7.1 Introduction

7.2 Longitudinal Vibrations of an Elastic Bar

7.3 Flexural Vibrations in an Elastic Bar

7.4 Group Velocity

7.5 Rotatory Inertia and Transverse Shear Effects: Timoshenko Beam Equation

7.6 Forced Vibrations of an Infinite Elastic Beam

7.7 Vibrations of a Finite Elastic Beam

7.8 Flexural Vibrations of Thin Elastic Plates

7.9 Point Excitation of an Infinite Plate

7.10 Flexural Vibrations of Finite Elastic Plates

7.11 Thick-Plate Theory; Timoshenko-Mindlin Plate Theory

7.12 Introduction to the in Vacuo Vibration of Shells

7.13 Equations of Motion for Cylindrical Shells

7.14 Planar Vibrations of a Thin Cylindrical Shell

7.15 Forced Planar Vibrations of a Cylindrical Shell

7.16 Nonplanar Vibrations of a Cylindrical Shell

7.17 Spherical Shells; Equations of Motion

7.18 Free Axisymmetric Nontorsional Vibrations of a Spherical Shell

7.19 Forced Vibrations of a Spherical Shell

8. Sound Radiation from Submerged Plates

8.1 Coincidence Frequency

8.2 Phase Velocity of Flexural Waves in a Submerged Plate

8.3 Effectively Infinite Locally Excited Plates

8.4 Infinite Line-Driven Elastic Plate

8.5 Pressure and Power Radiated by an Infinite Plate Driven by Distributed Loads

8.6 Power Radiated by Plates

8.7 Sound Radiation from Rectangular Plates

8.8 Low-impedance Layers

9. Sound Radiation by Shells at Low and Middle Frequencies

9.1 Introduction

9.2 Characteristic Equation of the Submerged Spherical Shell

9.3 Natural Frequencies, Modal Configurations, and Radiation Damping of
Submerged Spherical Shells

9.4 Response and Pressure Field of Point-Excited Submerged Spherical Shells

9.5 Normal Modes of Fluid-Filled Spherical Shells

9.6 Normal Modes of the Infinite, Submerged Cylindrical Shell

9.7 Natural Frequencies of Infinite Submerged Cylindrical Shells

9.8 Intermodal Fluid Coupling in the Submerged Finite Cylindrical Shell

9.9 Approximations to the Radiation Loading of Finite Cylindrical Shells

9.10 The Far-Field of Point-Excited Cylindrical Shells

9.11 The Effect of Structural Damping on Sound Radiation

9.12 Uncoupled Modes in a Submerged Structure

10. Scattering of Sound by Rigid Boundaries

10.1 Scattering and Echo Formation

10.2 Formulation of the Scattering Problem

10.3 The Infinite Plane Reflector

10.4 The Spherical Scatterer

10.5 The Infinite Cylindrical Scatterer

10.6 The Cylindrical Scatterer of Finite Length

10.7 Asymptotic Formulation of the Scattered Field of Slender Bodies of
Revolution

10.8 Nature of the Kirchhoff Approximation; Surface Pressure

10.9 Kirchhoff Scattering from Cylinders and Spheres; Fresnel Zones

10.10 Reflection form a Rectangular Baffle

10.11 The Helmholtz Reciprocity Principle

11. Elastic Scatterers and Waveguides

11.1 The Effect of Scatterer Elasticity

11.2 Sound Reflection by an Infinite Elastic Plate

11.3 Sound Transmission through an Infinite Elastic Plate

11.4 Sound Transmission through Finite Plates; Reciprocity

11.5 The Spherical Shell as an Acoustic Scatterer

11.6 The Scattering Action of the "Pressure-Release" Sphere

11.7 Structure-Acoustic Medium Reciprocity Relation Illustrated for the Spherical
Shell

11.8 Rayleigh Scattering by Compressible, Movable Spheres

11.9 Extension of the Rayleigh Scattering Formulation to Slender Bodies of
Revolution

11.10 The Cylindrical Shell as a Scatterer

11.11 Sound Sources Located on an Elastic Baffle

11.12 Sound Propagation in Fluid-Filled Elastic Waveguides

12. High-Frequency Formulation of Acoustic and Structural Vibration Problems

12.1 Watson's Creeping Wave Formulation of the Diffracted Field

12.2 Point Source on a Rigid Spherical Baffle

12.3 High-Frequency Response of a Spherical Shell

12.4 The Point-Excited Spherical Shell in Vacuo

12.5 The Submerged Spherical Shell

12.6 The Point Source on a Cylindrical Surface

12.7 Cylindrical Shells

12.8 Pressure Radiated by a Point-Excited Cylindrical Shell

12.9 Spherical Shell Radiated Field

Glossary

Index

Errata

Preface to the Second Edition

Like the 1972 (first) edition, this text is intended for the applied physicist and engineer acquainted with the mathematical tools found in graduate textbooks. A familiarity with elementary theory of vibrations and strength of materials is desirable. No prior acquaintance with acoustics is expected from the reader.

The primary difference between this book and more familiar texts is the space assigned to the
effect of radiation loading exerted by the ambient fluid on the vibrations of elastic structures and
the resulting modification of radiated and scattered pressures. Unlike the standard modern
acoustic texts, this book returns to the tradition of Raleigh's *Theory of Sound* by covering
the vibrations of elastic shells. The presentation of plate vibrations includes the
Timoshenko-Mindlin correction required to generate meaningful high frequency results. The
chapters dealing with acoustics are self-contained. They address primarily sound radiation and
scattering, to the exclusion of numerical solutions, statistical techniques, and consequently
flow-related phenomena and other broad-band excitations.

Even though the original title has been retained as being still appropriate to the material covered, there are substantial differences from the 1972 edition. To retain the manageable size of the original edition, the theories of plates and shells have been combined into a single chapter and the chapter dealing with acoustic transients has been dropped. There is an increased emphasis on asymptotic solutions. Acoustics in the first edition was limited to rigorously tractable geometries: the plane, the cylinder, and the sphere. Had we wanted to discuss radiation and scattering by slender bodies of revolution, we would have had to use prolate spheroidal wave harmonics where applicable, and for nonspheroidal geometries we would have referred readers to papers using numerical methods. These configurations are covered in this new edition, but in preference to rigorous formulations, the pressure fields are computed asymptotically by means of simple mathematical models that are solvable in terms of familiar cylinder functions. The chapter on sound radiation by submerged plates has been extensively rewritten to incorporate some of the new results in this area developed over the past decade--in particular, a closer examination of the near-field, the effect of stiffeners and compliant layers, and the relation of load distribution to far-field directivity and acoustic power. The more concise analytical treatment of sound radiation by simply supported cylindrical shells has been supplemented with a study of low-frequency radiation by free-floating, not necessarily cylindrical shells of revolution. Other new subjects covered in this second edition are the acoustics of bubble swarms, the propagation of sound waves in elastic pipes, and the insertion loss of finite panels. Both Rayleigh and Kirchhoff scattering receive more extensive treatment. Sound radiation by a source placed in a planar elastic baffle is used to illustrate the reciprocity principle, which is then used to analyze the far-field of sources located on elastic spherical and cylindrical baffles. The introductory chapter has been supplemented with a historical review of the development of structural acoustics.

Except for the extensive bibliography associated with that historical section, references listed at the end of each chapter are intended to supplement the material in this text either by providing the point of departure for the analysis presented here or by extending the analysis to areas not covered. Since, with the exception of the mathematical foundation, the development is relatively self-contained (the required knowledge of acoustics and theory of structures being derived or restated in the text), the references cited at the outset of an analysis are primarily mathematical in nature, thus sparing the reader the task of correlating the notations used in different texts on acoustics and plate and shell theory.

While our main goal is to present the underlying theories, we illustrate their application by means of problems selected for their practical interest. We hope to provide readers with the analytical tools for studying practical problems of interest to them. If an apology is needed for not having included those particular problems, we gladly accept the reproach that Shakespeare has Hamlet address to Horatio: "There are more things on heaven and earth, Horatio, than are dreamt of in your philosophy."

We are happy to acknowledge the moral and financial support of individuals and agencies within the U.S. Navy that enabled us to generate much of the material that is not part of the acoustician's stock in trade. Finally, it is with pleasure that we acknowledge the consistent helpfulness and patience displayed by our respective coworkers: J.M. Garrelick, J.E. Cole, III, and Rudolph Martinez at Cambridge Acoustical Associates, Inc., and numerous staff members at the David W. Taylor Naval Ship Research and Development Center.

Miguel C. Junger, Cambridge, Massachusetts

David Feit, Bethesda, Maryland

September 1985

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