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

2.1.2 Introducing complex numbers

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

4.1.4 Introduction to vector calculus

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

5.3.0 The origin of the signature modes of a violin

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

5.5.4 Testing different banjo necks

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


7.5 Tonal adjustment in the violin

7.5.1 Modelling the violin bridge

7.5.2 Mass compensation and other details


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

9.2 Beyond Helmholtz

9.2.1 Time domain simulation of a bowed string

9.3 How a violinist can go wrong: Schelleng’s diagram

9.3.1 Schelleng’s bow force limits

9.3.2 The Galluzzo bowing machine

9.4 Chasing the wolf

9.4.1 Including a body resonance in a reflection function

9.4.2 Measuring minimum bow force

9.4.3 The tuned-mass damper

9.5 Getting that perfect start: Guettler’s diagram

9.5.1 Raman’s bowed-string model

9.5.2 Bells and whistles for bowed string simulation

9.5.3 Torsional string motion

9.6 Friction and rosin: a sticky problem

9.6.1 The Schumacher experiment

9.6.2 Varieties of thermal friction model

9.7 What about the bow?

9.7.1 The centre of percussion

9.7.2 Simple model of bow bouncing

9.7.3 Simulating a finite-width bow

Chapter 10 Underpinnings V: Getting your hands dirty

10.1 Why to measure?

10.1.1 Dimensional analysis

10.2 Seeing hidden details

10.2.1 Micrographs of wood structure

10.3 Wood: properties and measurement

10.3.1 Measuring stiffness by beam vibration

10.3.2 Measuring stiffness by plate vibration

10.3.3 Material damping and complex moduli

10.4 Measuring frequency response functions

10.4.1 Measuring input impedance of a wind instrument

10.4.2 Frequency response measurement: hints and tips

10.5 Experimental modal analysis

10.5.1 Poles, circles and modal fitting

10.6 Seeing acoustic fields and fluid flows

10.6.1 Nearfield acoustic holography

Chapter 11 Blow, winds

11.1 The world of wind instruments

11.1.1 Input impedance of a cylindrical pipe

11.1.2 Reflection at a single tone-hole

11.1.3 The end correction of a single tone-hole

11.1.4 Bore perturbation and Rayleigh’s principle

11.2 Fluid phenomena

11.2.1 Overview of fluid dynamics

11.3 Reed instruments

11.3.0 Summary of bowed-string behaviour

11.3.1 Modelling the clarinet mouthpiece

11.3.2 Impedance, impulse response and the reflection function

11.3.3 Impedance of a conical tube

11.4 Reeds II: towards the real world

11.4.1 Modal expansion of input impedance

11.4.2 Modal fitting and modal synthesis

11.5 “Brass” instruments

11.5.1 Damping models for tube resonances

11.5.2 Modelling a brass mouthpiece

11.5.3 Improving the lip model

11.6 Free reeds

11.6.1 Fletcher’s model for free-reed instability

11.6.2 Millot’s model for free-reed instability

11.6.3 Modelling the harmonica

11.7 Interlude: how do wind instruments make sound?

11.7.1 Compact sound sources: monopoles, dipoles, quadrupoles

11.8 Air-jet instruments

11.8.1 The jet-drive model for a recorder or flute

11.8.2 Modelling the cork position in a flute

Chapter 12 Making an impact

12.1 The mechanics of bouncing

12.1.1 Simulating impacts

12.1.2 The maximum bandwidth of a bouncing hammer

12.2 Hitting strings: the piano and its relatives

12.2.1 Parameter values for piano simulations

12.2.2 A scaling law for piano hammers

12.2.3 Simulating a clavichord

12.2.4 Reflection at a clavichord tangent

12.3 Getting a buzz

12.3.1 Simulating a tanpura or bray harp


12.4 Ring out the bells

12.4.1 Modelling a church bell

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