The Physics of Musical Instruments

Transcription

1 Neville H. Fletcher Thomas D. Rossing The Physics of Musical Instruments Second Edition With 485 Illustrations Springer

2 Contents Preface Preface to the First Edition v vii I. Vibrating Systems 1. Free and Forced Vibrations of Simple Systems Simple Harmonie Motion in One Dimension 1.2. Complex Amplitudes 1.3. Superposition of Two Harmonie 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 Harmonie Oscillator Graphical Representations of Vibrations: Lissajous Figures Normal Modes of Two-Mass Systems Nonlinearity Appendix References 2. Continuous Systems in One Dimension: Strings and Bars Linear Array of Oscillators Transverse Wave Equation for a String General Solution of the Wave Equation: Traveling Waves Reflection at Fixed and Free Ends Simple Harmonie Solutions to the Wave Equation

3 xii Contents 2.6. Standing Waves Energy of a Vibrating String Plucked String: Time and Frequency Analyses Struck String Bowed String Driven String: Impedance Motion of the End Supports Damping Longitudinal Vibrations of a String or Thin Bar Bending Waves in a Bar Bars with Fixed and Free Ends Vibrations of Thick Bars: Rotary Inertia and Shear Deformation Vibrations of a Stiff String Dispersion in Stiff and Loaded Strings: Cutoff Frequency Torsional Vibrations of a Bar 66 References Two-Dimensional Systems: Membranes, Plates, and Shells Wave Equation for a Rectangular Membrane Square Membranes: Degeneracy Circular Membranes Real Membranes: Stiffness and Air Loading Waves in a Thin Plate Circular Plates Elliptical Plates Rectangular Plates Square Plates Square and Rectangular Plates with Clamped Edges Rectangular Wood Plates Bending Stiffness in a Membrane Vibration of Shells Driving Point Impedance 96 References Coupled Vibrating Systems Coupling Between Two Identical Vibrators Normal Modes Weak and Streng Coupling Forced Vibrations Coupled Electrical Circuits Forced Vibration of a Two-Mass System Systems with Many Masses 116

4 Contents xiii 4.8. Graphical Representation of Prequency Response Functions Vibrating String Coupled to a Soundboard Two Strings Coupled by a Bridge 120 APPENDIX 125 References Nonlinear Systems A General Method of Solution The Nonlinear Oscillator The Self-Excited Oscillator Multimode Systems Mode Locking in Self-Excited Systems Nonlinear Effects in Strings Nonlinear Effects in Plates and Shells 148 References 150 II. Sound Waves 6. Sound Waves in Air Plane Waves Spherical Waves Sound Pressure Level and Intensity Reflection, Diffraction, and Absorption Normal Modes in Cavities 167 References Sound Radiation Simple Multipole Sources Pairs of Point Sources Arrays of Point Sources Radiation from a Spherical Source Line Sources Radiation from a Plane Source in a Baffle Unbaffled Radiators Radiation from Large Plates 186 References Pipes, Horns and Cavities Infinite Cylindrical Pipes Wall Losses Finite Cylindrical Pipes Radiation from a Pipe 201

5 xiv Contents 8.5. Impedance Curves Horns Finite Conical Horns Bessel Horns Compound Horns Perturbations Numerical Calculations Curved Horns Measurement of Acoustic Impedance The Time Domain Network Analogs 227 References 232 III. String Instruments 9. Guitars and Lutes Design and Construction of Guitars The Guitar as a System of Coupled Vibrators Force Exerted by the String Modes of Vibration of Component Parts Coupling of the Top Plate to the Air Cavity: Two-Oscillator Model Coupling to the Back Plate: Three-Oscillator Model Resonances of a Guitar Body Response to String Forces Sound Radiation Resonances, Radiated Sound, and Quality A Family of Scaled Guitars Use of Synthetic Materials Electric Guitars Frets and Compensation Lutes Other Plucked String Instruments One-Sided Bridge Constraints 268 References Bowed String Instruments A Brief History Research on Violin Acoustics Construction of the Violin Motion of Bowed Strings Violin Body Vibrations Transient Wave Response of the Violin Body Soundpost and Bass Bar 295

6 Contents xv The Bridge Sound Radiation The Bow Wolf Notes and Playability Tonal Quality of Violins Viola, Cello, and Double Bass Viols A New Violin Family 322 References Harps, Harpsichords, Clavichords, and Dulcimers Traditional Instruments The Harp The Harpsichord Harpsichord Design Considerations Harpsichord Characteristics The Clavichord 347 References The Piano General Design of Pianos Piano Action Piano Strings Piano Hammers The Soundboard Sound Decay: Interaction of Strings, Bridge, and Soundboard Scaling and Tuning Tuning and Inharmonicity Timbre Electric Pianos 396 References 396 IV. Wind Instruments 13. Sound Generation by Reed and Lip Vibrations Pressure-Controlled Valves Quasi-Static Model Generator Behavior at Playing Prequency Free Reeds Generators Coupled to Horns Large-Amplitude Behavior Nonlinear Analysis 422

7 xvi Contents Numerical Simulation 424 References Lip-Driven Brass Instruments Historical Development of Brass Instruments HornProfiles Mouthpieces Radiation Slides and Valves Small-Amplitude Nonlinearity Large-Amplitude Nonlinearity Input Impedance Curves Transients Acoustic Spectra Mutes Performance Technique 455 References Woodwind Reed Instruments Woodwind Bore Shapes Finger Holes Impedance Curves Reed and Air Column Interaction Directionality Performance Technique Acoustic Efficiency The Limiting Spectrum The Clarinet The Oboe The Bassoon The Saxophone Capped Reed Instruments 497 References Flutes and Flue Organ Pipes Dynamics of an Air Jet Disturbance of an Air Jet Jet-Resonator Interaction The Regenerative Excitation Mechanism Rigorous Fluid-Dynamics Approaches Nonlinearity and Harmonie Generation Transients and Mode Transitions Aerodynamic Noise 528

8 Contents xvii Simple Flute-Type Instruments The Recorder The Flute 537 References Pipe Organs General Design Principles Organ Pipe Ranks Flue Pipe Ranks Characteristic Flue Pipes Mixtures and Mutations Tuning and Temperament Sound Radiation from Flue Pipes Transients in Flue Pipes Flue Pipe Voicing Effect of Pipe Material Reed Pipe Ranks Analysis of Timbre Tonal Architecture 577 References 578 V. Percussion Instruments 18. Drums Kettledrums Bass Drums Snare Drums Tom-Toms Indian Drums Japanese Drums Indonesian Drums Latin American Drums Tambourines 620 References Mallet Percussion Instruments Glockenspiel The Marimba Tuning the Bars Resonators The Xylophone 636

9 xviii Contents Vibes Mallets Chimes Triangles and Pentangles Gamelan Instruments Tubaphones and Gamelan Chimes 645 References Cymbals, Gongs, Plates, and Steel Drums Cymbals Tam-Tams Gongs Crotales Bell Plates Flexed Plates: The Musical Saw Steel Pans References 21. Beils Modes of Vibration of Church Beils Tuning and Temperament The Strike Note Major-Third Beils Sound Decay and Warble Scaling of Beils Modes of Vibration of Handbeils Timbre and Tuning of Handbeils Sound Decay and Warble in Handbeils Scaling of Handbeils Sound Radiation Bass Handbeils Clappers Ancient Chinese Two-Tone Beils Temple Beils of China, Korea, and Japan 701 References 705 Part VI. Materials 22. Materials for Musical Instruments Mechanical Properties of Materials Materials for Wind Instruments Wood 719

10 Contents xix Plastics and Composite Materials Metals Conclusion 732 References 733 Name Index 735 Subject Index 743