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Principles of Vibration and Sound

Principles of Vibration and Sound

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Some years ago we set out to write a detailed book about the basic physics of musical instruments. There have been many admirable books published about the history of the development of musical instruments, about their construction as a master craft, and about their employment in musical perfor mance; several excellent books have treated the acoustics of musical instru ments in a semiquantitative way; but none to our knowledge had then at tempted to assemble the hard acoustic information available in the research literature and to make it available to a wider readership. Our book The Physics of Musical Instruments, published by Springer-Verlag in 1991 and subsequently reprinted several times with only minor corrections, was the outcome of our labor. Because it was our aim to make our discussion of musical instruments as complete and rigorous as possible, our book began with a careful introduction to vibrating and radiating systems important in that field. We treated simple linear oscillators, both in isolation and coupled together, and extended that to a discussion of some aspects of driven and autonomous nonlinear oscilla tors. Because musical instruments are necessarily extended structures, we then went on to discuss the vibrations of strings, bars, membranes, plates, and shells, paying particular attention to the mode structures and characteristic frequencies, for it is these that are musically important.
I Vibrating Systems.- 1 Free and Forced Vibrations of Simple Systems.- 1.1. Simple Harmonic Motion in One Dimension.- 1.2. Complex Amplitudes.- 1.3. Superposition of Two Harmonic Motions in One Dimension.- 1.4. Energy.- 1.5. Damped Oscillations.- 1.6. Other Simple Vibrating Systems.- 1.7. Forced Oscillations.- 1.8. Transient Response of an Oscillator.- 1.9. Two-Dimensional Harmonic Oscillator.- 1.10. Graphical Representations of Vibrations: Lissajous Figures.- 1.11. Normal Modes of Two-Mass Systems.- 1.12. Nonlinear Vibrations of a Simple System.- A.1. Alternative Ways of Expressing Harmonic Motion.- A.2. Equivalent Electrical Circuit for a Simple Oscillator.- References.- 2 Continuous Systems in One Dimension: Strings and Bars.- 2.1. Linear Array of Oscillators.- 2.2. Transverse Wave Equation for a String.- 2.3. General Solution of the Wave Equation: Traveling Waves.- 2.4. Reflection at Fixed and Free Ends.- 2.5. Simple Harmonic Solutions to the Wave Equation.- 2.6. Standing Waves.- 2.7. Energy of a Vibrating String.- 2.8. Plucked String: Time and Frequency Analyses.- 2.9. Struck String.- 2.10. Bowed String.- 2.11. Driven String Impedance.- 2.12. Motion of the End Supports.- 2.13. Damping.- 2.14. Longitudinal Vibrations of a String or Thin Bar.- 2.15. Bending Waves in a Bar.- 2.16. Bars with Fixed and Free Ends.- 2.17. Vibrations of Thick Bars: Rotary Inertia and Shear Deformation.- 2.18. Vibrations of a Stiff String.- 2.19. Dispersion in Stiff and Loaded Strings: Cutoff Frequency.- 2.20. Torsional Vibrations of a Bar.- References.- 3 Two-Dimensional Systems: Membranes and Plates.- 3.1. Wave Equation for a Rectangular Membrane.- 3.2. Square Membranes: Degeneracy.- 3.3. Circular Membranes.- 3.4. Real Membranes: Stiffness and Air Loading.- 3.5. Waves in a Thin Plate.- 3.6. Circular Plates.- 3.7. Elliptical Plates.- 3.8. Rectangular Plates.- 3.9. Square Plates.- 3.10. Square and Rectangular Plates with Clamped Edges.- 3.11. Rectangular Wood Plates.- 3.12. Bending Stiffness in a Membrane.- 3.13. Shallow Spherical Shells.- 3.14. Nonlinear Vibrations in Plates and Shallow Shells.- 3.15. Driving Point Impedance.- References.- 4 Coupled Vibrating Systems.- 4.1. Coupling Between Two Identical Vibrators.- 4.2. Normal Modes.- 4.3. Weak and Strong Coupling.- 4.4. Forced Vibrations.- 4.5. Coupled Electrical Circuits.- 4.6. Forced Vibration of a Two-Mass System.- 4.7. Systems with Many Masses.- 4.8. Graphical Representation of Frequency Response Functions.- 4.9. Vibrating String Coupled to a Soundboard.- 4.10. Two Strings Coupled by a Bridge.- A.1. Structural Dynamics and Frequency Response Functions.- A.2. Modal Analysis.- A.3. Finite Element Analysis.- References.- 5 Nonlinear Systems.- 5.1. A General Method of Solution.- 5.2. Illustrative Examples.- 5.3. The Self-Excited Oscillator.- 5.4. Multimode Systems.- 5.5. Mode Locking in Self-Excited Systems.- References.- II Sound Waves.- 6 Sound Waves in Air.- 6.1. Plane Waves.- 6.2. Spherical Waves.- 6.3. Sound Pressure Level and Intensity.- 6.4. Reflection and Transmission.- 6.5. Absorption.- 6.6. Normal Modes in Cavities.- References.- 7 Sound Radiation.- 7.1. Simple Multipole Sources.- 7.2. Pairs of Point Sources.- 7.3. Arrays of Point Sources.- 7.4. Radiation from a Spherical Source.- 7.5. Line Sources.- 7.6. Radiation from a Plane Source in a Baffle.- 7.7. Unbaffled Radiators.- 7.8. Radiation from Large Plates.- References.- 8 Pipes and Horns.- 8.1. Infinite Cylindrical Pipes.- 8.2. Wall Losses.- 8.3. Finite Cylindrical Pipes.- 8.4. Radiation from a Pipe.- 8.5. Impedance Curves.- 8.6. Horns.- 8.7. Finite Conical and Exponential Horns.- 8.8. Bessel Horns.- 8.9. Compound Horns.- 8.10. Perturbations.- 8.11. Numerical Calculations.- 8.12. The Time Domain.- References.- 9 Acoustic Systems.- 9.1. Low-Frequency Components and Systems.- 9.2. High-Frequency Components and Systems.- 9.3. Finite Horns.- 9.4. Coupled Mechanical Components.- 9.5. Multi-Port Systems.- 9.6. Conclusion.- References.- Selected Bibliography.- Problems.- Answers to Selected Problems.- Name Index.
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