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| Figure 1: An example of a transverse wave; a wave induced in a piece of string. |
In most cases, when we hear a sound, the perceived impression is caused by some energy carried to our ears by the air. Our auditory system is sensible to the minute pressure variations which are superimposed on the stationary pressure of the air surrounding us. These pressure deviations from the stationary mean value, comprise the so called "sound pressure variations" ("sound pressure", for short). Our ear is capable of responding to a wide range of sound pressures, from a barely perceptible sound, to the intensity for which the hearing sensation becomes painful.
The pressure variations propagate as longitudinal waves in the air. The actual cause for generating and propagating pressure variations lies in the fact that individual air particles vibrate about their "thermodynamic rest position", and thus collide with neighbouring particles. The velocity with which the particles move, relative to their rest position, is called the particle velocity. Actually, due to the thermal energy, all the air particles are moving randomly at high speed and continuously colliding with other particles, but we can imagine that those collisions keep each particle in a pseudo stable position and keep an stationary pressure of the air. This is the "thermodynamic rest position". The movement due to the sound vibrations and subsequently, the 'sound particle velocity' ('particle velocity' for short) is added to that thermal random movement. While the particle velocity indicates the speed of the air particles relative to their rest position, the speed with which pressure fluctuations spread in the air is the speed of sound.
From the physical point of view, the sound is any mechanical vibration in the air or other elastic medium. Only some types of which are able to cause a sensation of hearing. As part of the physics, the so called physical acoustics deals with the research and description of the elastic vibrations and their behaviour in various media and environments.
An important part of acoustics deals with energy-transfer ("sound transmission"):
Transmission is defined as opposed to absorption or reflection of a wave. Some media transmit sound better than others. The emphasis is usually on the efficiency of energy transfer, an effect which can be quantified with the 'Sound Transmission Coefficient'.
The influence of the environment on the sound propagation, is studied under Sound-Environment Interaction.
There are several classes of wave that can distort a physical medium: transverse waves. longitudinal waves...
In transverse waves, the movement of the elements of the medium move orthogonally (at 90°) to the direction of movement of the wave. A typical example of a transverse wave is a wave pattern on the surface of a body of water (eg. on a pond after a stone has been thrown in or ocean waves before they reach the breaking zone). In such a wave the molecules of water move up and down whilst the wave front moves along the surface of the water.
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| Figure 1: An example of a transverse wave; a wave induced in a piece of string. |
In figure 1 a hand induces a transverse wave in a string by periodically moving up and down. This causes the string to move up and down. This movement propagates through the string producing a series of wavefronts which move towards the fixed wall with a velocity v. Obviously, individual parts of the string only move up and down (as indicated by the vertical arrows).
In longitudinal waves the elements of the medium move back and forth in line with the direction of propagation of the wave fronts.
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| Figure 2: An example of a longitudinal wave; a wave induced in a spring. |
In figure 2 a hand induces a longitudinal wave in a spring by periodically moving back and forth in line with the direction of the spring. This causes the regions of high and low spring compression to move along the spring. This movement propagates through the spring producing a series of wavefronts which move towards the fixed wall with a velocity v. Individual parts of the spring only move backwards and forwards short distances in the direction of wave propagation. This causes the coils to periodically come closer to and further from adjacent coils than would be the case for the spring at rest. A longitudinal wave is a compression wave in which particles move back and forth in the direction of wavefront movement (ie. longitudinally).
A torsional wave is a two-dimensional transverse wave. This type of wave twists along a given medium. For example: if you twist a coil on one end and release it, the "twist" travels in a wave to the other end of the coil.
In seismology, several types of surface waves are encountered. Surface waves, in this mechanical sense, are commonly known as either Love waves (L waves) or Rayleigh waves. A seismic wave is a wave that travels through the Earth, often as the result of an earthquake or explosion. Love waves have transverse motion (movement is perpendicular to the direction of travel, like light waves), whereas Rayleigh waves have both longitudinal (movement parallel to the direction of travel, like sound waves) and transverse motion. Seismic waves are studied by seismologists and measured by a seismograph or seismometer. Surface waves span a wide frequency range, and the period of waves that are most damaging is usually 10 seconds or longer. Surface waves can travel around the globe many times from the largest earthquakes. Surface waves are caused when P waves and S waves come to the surface.
The term "surface wave" can describe waves over an ocean, even when they are approximated by Airy functions and are more properly called creeping waves. Examples are the waves at the surface of water and air (ocean surface waves), or ripples in the sand at the interface with water or air. Another example is internal waves, which can be transmitted along the interface of two water masses of different densities.
Almost all phenomena that we perceive can be explained with a high grade of accuracy by the so called classical physics. It includes the (classical) mechanics, described by the Newton's Laws, and the (classical) electromagnetism described by the Maxwell's Laws. In practice, only the phenomena in the sub particle and cosmological worlds need the accuracy provided by the more complex modern physical theories as the Quantum Mechanics or the Relativity Theory to be explained.
Acoustics is well described by the classical physics. There are a number of important physics concepts that are necessary to deal with real life acoustics situations: mass, velocity, force, work, power... (see physical appendix).
1. Principles of Vibration and Acoustics for Guitar Designers.......................................... 1-1
1.1. Sound and How We Hear................................................................................ 1-2
1.1.1. What is sound? ......................................................................................... 1-2
1.1.2. The ear .................................................................................................... 1-6
1.1.2.1. The anatomy of the ear....................................................................... 1-6
1.1.2.2. Roughness ........................................................................................ 1-9
1.1.2.3. Masking .......................................................................................... 1-12
1.1.3. Consequences for guitar designers of how we hear....................................... 1-15
1.1.4. Section summary .................................................................................... 1-16
1.2. Simple harmonic motion .............................................................................. 1-18
1.2.1. Mathematical definition of simple harmonic motion ...................................... 1-18
1.2.2. Geometrical model of simple harmonic motion............................................. 1-18
1.2.3. The period of simple harmonic motion ........................................................ 1-19
1.2.4. An example of simple harmonic motion ...................................................... 1-19
1.2.5. The energy embodied in simple harmonic motion......................................... 1-22
1.3. Introduction to the mechanics of beams ......................................................... 1-25
1.3.1. Stiffness of simple beams ......................................................................... 1-25
1.3.2. The mass of a simple beam....................................................................... 1-26
1.3.3. Vibrating systems.................................................................................... 1-26
1.4. Resonance and vibration sensitivity ............................................................... 1-28
1.4.1. Simple resonators.................................................................................... 1-28
1.4.2. The Helmholtz resonator........................................................................... 1-30
1.4.3. Damped simple resonators........................................................................ 1-32
1.4.4. Driven, damped simple resonators – strings driving soundboards................... 1-33
1.4.5. The detailed behaviour of a simple driven resonator ..................................... 1-39
1.4.6. The Q-Factor........................................................................................... 1-42
1.4.7. Mechanical Impedance ............................................................................. 1-44
1.4.8. Impedance (mis)matching ........................................................................ 1-45
1.4.9. Impedance, admittance and the high performance guitar.............................. 1-46
1.4.10. Beats ..................................................................................................... 1-48
1.5. Vibrations in strings and the forces they exert................................................. 1-50
1.5.1. Types of waves in strings – frame of reference ............................................ 1-50
1.5.2. Transverse waves in strings ...................................................................... 1-51
1.5.3. Wave partials and harmonicity .................................................................. 1-52
1.5.4. Waves in “ideal” strings............................................................................ 1-53
1.5.5. Forces on the terminations of an ideal string ............................................... 1-54
1.5.6. Real strings ............................................................................................ 1-58
1.5.7. Plucking at places other than the centre ..................................................... 1-60
1.5.8. Forces exerted by a string on a soundboard ................................................ 1-63
1.5.9. Strings with bending stiffness.................................................................... 1-66
1.5.10. Section summary .................................................................................... 1-71
1.6. Introduction to vibrations of bars and plates – strings driving soundboards ......... 1-72
1.6.1. Vibrations of bars .................................................................................... 1-72
1.6.2. Vibrations of plates with free edges............................................................ 1-73
1.6.3. Vibrations of flat plates fixed around their edges.......................................... 1-74
1.6.4. Guitar vibration modes excited by the string forces ...................................... 1-76
1.6.5. Vibrations of curved plates (spherical shells) ............................................... 1-78
1.7. Sound radiation from a guitar ....................................................................... 1-81
1.7.1. How sound is radiated by a guitar.............................................................. 1-81
1.7.2. Monopole sound radiation and specific mobility ............................................ 1-89
1.7.3. Monopole mobility and the frequency response curve ................................... 1-91
1.7.4. The anatomy of a plucked note.................................................................. 1-92
1.7.5. Sound radiation from the guitar cavity at middle frequencies......................... 1-93
1.7.6. Sound radiation and directivity .................................................................. 1-95
1.7.7. Chapter Summary ..................................................................................1-100
2. Analytical Guitar Models and their Use .................................................................... 2-2
2.1. Coupled resonators and simple low frequency guitar models ............................... 2-2
2.2. Coupled resonators........................................................................................ 2-3
2.2.1. Two mass coupled resonator ....................................................................... 2-4
2.3. 2-DOF model of a guitar’s low frequency response............................................. 2-8
2.3.1. Low frequency response of a guitar with a stiff back....................................... 2-8
2.3.2. Two mass coupled resonator model for guitars ............................................ 2-10
2.3.3. Using the 2-DOF model to determine fh...................................................... 2-13
2.3.4. Direct determination of the Helmholtz frequency.......................................... 2-14
2.3.5. Use of the 2-DOF model in the workshop .................................................... 2-15
2.3.6. Using the 2-DOF model for guitars with live backs........................................ 2-17
2.3.7. Some unusual features of guitar shaped cavities.......................................... 2-17
2.3.8. “Live back” guitar low frequency response .................................................. 2-18
2.3.9. Coupled top plate and back plate guitar model ............................................ 2-19
2.3.10. Target values for the main top and main air resonances ............................... 2-24
2.3.11. Investigative modelling – working in the frequency domain ........................... 2-25
2.3.12. Empirical adjustment of plate frequencies ................................................... 2-26
2.3.12.1. Top response to small perturbations in mass........................................ 2-27
2.3.12.2. Guitar response to side mass ............................................................. 2-28
2.4. Simulation of a guitar’s frequency response .................................................... 2-31
2.4.1. A 4-DOF model of low frequency guitar function .......................................... 2-32
2.4.1.1. 4-DOF model sensitivity analysis ........................................................ 2-38
2.4.1.2. 4-DOF model and “live” backs............................................................ 2-40
2.4.2. The behaviour of superimposed simple sound sources .................................. 2-42
2.4.3. Combining the 4-DOF model with superimposed simple sources..................... 2-46
2.5. Coupling of strings to soundboards ................................................................ 2-50
2.6. Chapter Summary ....................................................................................... 2-51
3. The Design Objective - The Perfect Guitar ............................................................... 3-2
3.1. Playability .................................................................................................... 3-2
3.1.1. Plucking hand feel ..................................................................................... 3-3
3.1.2. Fretting hand feel ...................................................................................... 3-4
3.1.2.1. Scale length ...................................................................................... 3-4
3.1.2.2. Tuneability ........................................................................................ 3-5
3.1.2.3. Action............................................................................................... 3-6
3.1.2.4. String height at the nut....................................................................... 3-9
3.1.2.5. Relief................................................................................................ 3-9
3.1.2.6. Fretboard curvature.......................................................................... 3-11
3.1.2.7. Neck profile/nut width....................................................................... 3-12
3.1.2.8. Frets .............................................................................................. 3-13
3.1.2.9. Neck finish ...................................................................................... 3-14
3.1.2.10. Neck to body joint position; cutaways ................................................. 3-14
3.1.2.11. Number of frets ............................................................................... 3-15
3.2. Musicality................................................................................................... 3-15
3.2.1. Consonance and dissonance...................................................................... 3-15
3.2.1.1. Experimental demonstration of consonance and dissonance ................... 3-16
3.2.1.2. The consonance of intervals............................................................... 3-17
3.2.1.3. A brief history of scales..................................................................... 3-19
3.2.1.4. The tuning gap (between what you tune to and what you’d like to hear).. 3-21
3.2.1.5. The tuning error (between what you tune to and what your guitar plays). 3-22
3.2.1.6. Achieving pitch accuracy ................................................................... 3-23
3.2.1.7. Dealing with inharmonicity ................................................................ 3-23
3.2.2. The acoustic response characteristics of a guitar.......................................... 3-23
3.2.2.1. A perfect frequency response? ........................................................... 3-24
3.2.2.2. Guitar performance factors: “Volume and tone”.................................... 3-29
3.2.2.3. Nomenclature .................................................................................. 3-29
3.2.2.4. Tonal qualities of a Dreadnought guitar used for flat picking................... 3-33
3.2.2.5. Tonal qualities of a steel string guitar for finger style playing ................. 3-35
3.2.2.6. Tonal qualities of a classical guitar...................................................... 3-36
3.2.2.7. Tonal qualities of a flamenco guitar (flamenco blanca)........................... 3-38
4. Component Design .............................................................................................. 4-2
4.1. Wood as an organic material........................................................................... 4-2
4.1.1. The composition of wood ............................................................................ 4-3
4.1.2. The annual cycle ....................................................................................... 4-5
4.1.3. Dealing with moisture ................................................................................ 4-5
4.1.4. Dealing with runout ................................................................................... 4-9
4.1.5. Storing wood ...........................................................................................4-11
4.2. Guitar woods ...............................................................................................4-12
4.2.1. Top woods...............................................................................................4-13
4.2.2. Back and side woods.................................................................................4-14
4.2.3. Neck woods .............................................................................................4-15
4.2.4. Brace woods ............................................................................................4-15
4.2.5. Fretboard woods.......................................................................................4-16
4.2.6. Bridge woods ...........................................................................................4-17
4.2.7. Woods for other components......................................................................4-17
4.3. Wood properties that matter..........................................................................4-18
4.3.1. The sound radiation coefficient ...................................................................4-18
4.3.2. The damping factor...................................................................................4-20
4.3.3. Measuring wood properties ........................................................................4-21
4.3.3.1. Measurement of Q.............................................................................4-21
4.3.3.2. Measurement of logarithmic decrement................................................4-23
4.3.3.3. Potential difficulties ...........................................................................4-26
4.3.3.4. Long grain damping vs. cross grain damping.........................................4-28
4.3.3.5. Tap testing for “Q” ............................................................................4-29
4.3.4. Conditioning wood ....................................................................................4-29
4.3.5. Section summary .....................................................................................4-30
4.4. Design of braces and bracing systems.............................................................4-31
4.4.1. Bending stiffness ......................................................................................4-31
4.4.2. Brace design criteria .................................................................................4-32
4.4.3. Calculating the second moment of area of a brace ........................................4-35
4.4.4. Measurement of Young’s modulus E for brace material ..................................4-38
4.4.5. Determination of soundboard flexural rigidity ...............................................4-40
4.4.6. Carbon fibre reinforced braces....................................................................4-45
4.4.6.1. Designing Composite Braces ...............................................................4-46
4.4.6.2. Balancing brace stiffness to panel stiffness ...........................................4-48
4.4.7. Stress limits for braces..............................................................................4-48
4.4.8. Top bracing layout - guidelines...................................................................4-52
4.4.9. Back bracing............................................................................................4-52
4.4.10. Section summary .....................................................................................4-53
4.5. Design of plates ...........................................................................................4-55
4.5.1. Waves in plates........................................................................................4-55
4.5.2. The tap tone method of establishing wood properties ....................................4-57
4.5.3. Establishing the target plate thickness.........................................................4-60
4.5.4. Sensitivity to long and cross grain stiffness variations....................................4-64
4.5.5. Panel design decisions...............................................................................4-64
4.5.6. Chapter summary.....................................................................................4-65
4.6. Body shapes and the design of other components.............................................4-66
4.6.1. Bridge design...........................................................................................4-66
4.6.1.1. The bridge as a distributor of string loading ..........................................4-66
4.6.1.2. Positioning the bridge on X-braced guitars............................................4-66
4.6.1.3. Bridge mass and stiffness...................................................................4-68
4.6.1.4. Steel string bridge planform ...............................................................4-69
4.6.1.5. The bridge plate................................................................................4-71
4.6.1.6. Classical guitar bridges ......................................................................4-71
4.6.2. Linings ....................................................................................................4-73
4.6.3. End blocks...............................................................................................4-73
4.6.4. Cutaways ................................................................................................4-73
4.6.5. Neck joint................................................................................................4-74
4.6.6. Fretboard - fret spacing.............................................................................4-77
4.6.7. Neck cross sectional shape ........................................................................4-77
4.6.8. Truss rods ...............................................................................................4-80
4.6.9. The nut ...................................................................................................4-81
4.6.10. Headstock ...............................................................................................4-82
4.6.11. Body shape..............................................................................................4-84
4.6.11.1. Size and shape .................................................................................4-84
4.6.11.2. Body shape proportions .....................................................................4-87
4.6.11.3. Designing body shapes ......................................................................4-88
4.6.12. Section summary .....................................................................................4-90
4.7. Intonation ...................................................................................................4-91
4.7.1. A first source of intonation error .................................................................4-91
4.7.2. A second source of intonation error.............................................................4-93
4.7.2.1. Fixing intonation errors due to body resonances ....................................4-93
4.7.2.2. Resonance shifting ............................................................................4-93
4.7.2.3. Empirical nut and saddle intonation .....................................................4-95
4.7.2.4. A simple intonation fix .......................................................................4-98
4.7.3. Intonation for a high performance guitar .....................................................4-99
4.7.3.1. Measuring a string’s longitudinal stiffness .............................................4-99
4.7.3.2. Calculating the change in path length as the string is fretted.................4-103
4.7.3.3. The effect of string bending stiffness on intonation error.......................4-111
4.7.4. Compensation without most of the mathematics .........................................4-112
4.7.5. Section summary ...................................................................................4-113
5. Layout Procedure................................................................................................. 5-2
5.1. Drafting the layout of a steel string guitar ........................................................ 5-2
5.2. Drafting the layout of a classical guitar............................................................. 5-5
5.3. Section summary .......................................................................................... 5-6
AI 1. Technical note on collecting spectrographic data............................................... AI 1
AI 2. Using Chladni’s technique to visualise vibration modes ...................................... AI 3
AII 1. Deflection of the soundboard under oscillating longitudinal string forces .............AII 1
AII 2. 4 Degree of Freedom model of low frequency guitar function ............................AII 5
AII 3. Fretboard Curvature .................................................................................. AII 11
AIII 1. Frequency look-up table..............................................................................AIII 1
AIV 1. Bending waves on plates ............................................................................. AIV 1