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

2.3 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 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?

5.3 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