Table of contents

Chapter 1 Introduction: “Is that a real subject?”

Chapter 2 Underpinnings I: Good vibrations

2.1 Linear and non-linear

2.1.1 Linearity and sine waves

2.2 Frequency analysis and modes

2.2.1 Fourier series

2.2.2 The undamped harmonic oscillator

2.2.3 Linearity for small vibration

2.2.4 Degenerate modes of a drum

2.2.5 Vibration frequency response

2.2.6 Frequency spectrum of a hammer tap

2.2.7 Vibration damping

2.2.8 Impulse response and convolution

2.3 Frequency and pitch

2.4 Images of vibration

Chapter 3 When does a structure become a musical instrument?

3.1 Harmonics and non-harmonics

3.1.1 Vibration of an ideal stretched string

3.2 Building blocks: beams, plates and shells

3.2.1 Bending beams and free-free modes

3.2.2 Synthesising percussion sounds

3.2.3 Plate vibration

3.2.4 The modal density of a vibrating plate

3.2.5 Rayleigh’s principle

3.3 Marimbas and xylophones

3.3.1 Rayleigh’s principle and tuning a marimba bar

3.4 Church bells

3.5 Steel pans and the musical saw

3.5.1 Time-average holography

3.5.2 Waveguide reflection: the beam on an elastic foundation

3.6 Tuned drums

3.6.1 Vibration modes of a circular drum

Chapter 4 Underpinnings II: Making waves

4.1 Sound waves

4.1.1 The wave equation

4.1.2 The sound field of a pulsating sphere

4.1.3 Energy, intensity and impedance

4.2 Acoustic resonators

4.2.1 The Helmholtz resonator

4.2.2 Coupling of a Helmholtz resonator and a body mode

4.2.3 The Webster horn equation

4.2.4 Weinreich’s formula for modal density

4.3 Sound radiation

4.3.1 Monopoles and dipoles

4.3.2 The Rayleigh integral and the baffled piston

4.3.3 The critical frequency of a vibrating plate

Chapter 5 Strings, mainly plucked

5.1 Stringed instrument overview

5.1.1 Averaging and the coherence function

5.1.2 Coupling a string to the instrument body

5.2 “Can you make it louder?

5.2.1 Merit index for soundboard material selection

5.3 Signature modes and formants

5.3.1 The bridge hill: a resonator near the driving point

5.3.2 Skudrzyk’s method

5.4 Synthesising plucked string sounds

5.4.1 Motion of a plucked string as a modal sum

5.4.2 Waves on a string: d’Alembert’s solution

5.4.3 Natural frequencies of a stiff string

5.4.4 Energy loss in a string: internal damping

5.4.5 Energy loss in a string: air damping

5.4.6 Frequency responses for string synthesis

5.5 An extreme case: the banjo

5.5.1 The “square banjo” model

5.5.2 Housekeeping variables in the banjo synthesis models

5.5.3 Testing different bridges

Chapter 6 Underpinnings III: Hearing things

6.1 “It’s science, but not as we know it”

6.2 The hardware of hearing

6.3 Loudness

6.4 Pitch, timbre and excitation patterns

6.5 Making a difference

6.6 Testing received wisdom

Chapter 7 Variations for strings

7.1 The world of struck and plucked strings

7.2 Choosing strings

7.2.1 Selection chart algebra

7.2.2 The damping criterion for string selection

7.3 Multiple strings and double decays

7.3.1 Coupling of damped systems and the first-order method

7.4 Add a touch of nonlinearity…

7.4.1 Nonlinearity and longitudinal string motion

Chapter 8 Underpinnings IV: Nonlinearity

8.1 “It’s not a bug, it’s a feature”

8.2 Sources of nonlinearity: smooth and non-smooth

8.2.1 The simple pendulum

8.2.2 Duffing’s equation and harmonic balance

8.2.3 The friction damper

8.3 The view from phase space

8.3.1 Stability of equilibrium for the pendulum

8.3.2 Singular points and their phase portraits

8.3.3 Singular points of Duffing’s equation

8.4 Chaos!

8.4.1 The Lorenz equations

8.4.2 The double pendulum

8.5 Self-excited vibration

8.5.1 The Van der Pol equation

8.5.2 Harmonic balance for a simplified reed instrument

8.5.3 Time-domain simulation of the clarinet


Chapter 9 Bowed strings

9.1 On the shoulders of giants: Helmholtz and Raman

9.1.1 Bridge-force sensor for bowed strings

9.1.2 Raman’s argument for bowed-string waveforms

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