Leon Gunther / Леон Гюнтер - The Physics of Music and Color / Физика Музыки и Цвета [2012, PDF, ENG]

Contents
1 Introductory Remarks ..................................................... 1
1.1 The Legend of the Huang Chung .................................. 7
2 The Vibrating String ....................................................... 11
2.1 Waves Along a Stretched String.................................... 11
2.2 A Finite String Can Generate Music! .............................. 13
2.3 Pitch, Loudness, and Timbre ....................................... 16
2.4 The Relation Between Frequency and Pitch ....................... 17
2.5 The Wave Motion of a Stretched Rope ............................ 18
2.6 Modes of Vibration and Harmonics................................ 20
2.7 The Sine Wave ...................................................... 23
2.8 The Simple Harmonic Oscillator................................... 26
2.8.1 The Vibration Frequency of a Simple
Harmonic Oscillator ..................................... 28
2.9 Traveling Sine Waves............................................... 29
2.9.1 Applications.............................................. 31
2.10 Modes of Vibration: Spatial Structure ............................. 32
2.11 The Wave Velocity of a Vibrating String .......................... 34
2.11.1 Application of the Above Relations to the Piano ....... 37
2.12 The Connection Between an SHO and a Vibrating String ........ 38
2.13 Stiffness of a String ................................................. 41
2.14 Resonance ........................................................... 43
2.15 General Vibrations of a String: Fourier’s Theorem ............... 45
2.15.1 Frequency of a Wave with Missing Fundamental ...... 51
2.16 Periodic Waves and Timbre ........................................ 52
2.17 An Application of Fourier’s Theorem to Resonance
Between Strings..................................................... 52
2.18 A Standing Wave as a Sum of Traveling Waves................... 55
2.19 Terms ................................................................ 55
2.20 Important Equations ................................................ 57
2.21 Problems for Chap. 2................................................ 58
xv
xvi Contents
3 The Vibrating Air Column ................................................ 63
3.1 The Air of Our Atmosphere ........................................ 63
3.1.1 Generating a Sound Pulse ............................... 66
3.1.2 Digression on Pushing a Block of Wood ............... 67
3.2 The Nature of Sound Waves in Air................................. 67
3.3 Characterizing a Sound Wave ...................................... 69
3.4 Visualizing a Sound Wave .......................................... 70
3.5 The Velocity of Sound .............................................. 71
3.5.1 Temperature Dependence of Speed
of Sound in Air........................................... 72
3.6 Standing Waves in an Air Column ................................. 73
3.6.1 Standing Waves in a Closed Pipe ....................... 76
3.6.2 End Correction for Modes in a Pipe .................... 79
3.7 Magic in a Cup of Cocoa ........................................ 79
3.8 Terms ................................................................ 80
3.9 Important Equations ................................................ 80
3.10 Problems for Chap. 3................................................ 81
3.10.1 Derivation of the Helmholtz Formula ................... 84
4 Energy ....................................................................... 87
4.1 Forms of Energy and Energy Conservation........................ 88
4.1.1 Fundamental Forms of Energy .......................... 89
4.1.2 “Derived” Forms of Energy ............................. 93
4.1.3 The Energy of Cheerios ................................. 94
4.2 The Principle of Conservation of Energy, Work, and Heat ....... 95
4.3 Energy of Vibrating Systems ....................................... 96
4.3.1 The Simple Harmonic Oscillator........................ 96
4.3.2 Energy in a Vibrating String............................. 98
4.3.3 Energy in a Sound Wave................................. 99
4.4 Power ................................................................ 99
4.5 Intensity ............................................................. 101
4.6 Intensity of a Point Source ......................................... 103
4.7 Sound Level and the Decibel System .............................. 105
4.7.1 Logarithms ............................................... 105
4.7.2 Sound Level .............................................. 107
4.7.3 From Sound Level to Intensity .......................... 108
4.8 Attenuation .......................................................... 110
4.8.1 Attenuation in Time...................................... 110
4.8.2 Resonance in the Presence of Attenuation .............. 113
4.8.3 Attenuation of Travelling Waves:
Attenuation in Space ..................................... 114
4.9 Reverberation Time ................................................. 118
4.10 Terms ................................................................ 120
4.11 Important Equations ................................................ 121
4.12 Problems for Chap. 4................................................ 122
Contents xvii
5 Electricity, Magnetism, and Electromagnetic Waves ................... 127
5.1 The Fundamental Forces of Nature ................................ 127
5.2 The Electric Force .................................................. 129
5.3 Electric Currents in Metal Wires ................................... 130
5.4 The Magnetic Force................................................. 131
5.5 Magnetic Forces Characterized .................................... 133
5.6 Is There a Connection Between Electricity and Magnetism? ..... 135
5.6.1 Action–Reaction Law and Force of Magnet
on Current-Carrying Wire ............................... 138
5.7 The Loudspeaker.................................................... 141
5.8 The Buzzer .......................................................... 141
5.9 The Electric Motor.................................................. 142
5.10 Force Between Two Wires Carrying an Electric Current ......... 143
5.11 The Electromagnetic Force and Michael Faraday ................. 143
5.12 Applications of Faraday’s EMF .................................... 147
5.13 A Final “Twist” ..................................................... 148
5.14 Action-at-a-Distance and Faraday’s Fields ........................ 149
5.15 The Electric Field ................................................... 150
5.16 The Magnetic Field ................................................. 154
5.17 Magnetic Force on a Moving Charge .............................. 157
5.18 Force Between Two Parallel Wires Carrying Currents............ 158
5.19 Generalized Faraday’s Law......................................... 158
5.20 What Do Induced Electric Field Lines Look Like? ............... 163
5.21 Lenz’s Law .......................................................... 164
5.22 The Guitar Pickup................................................... 166
5.23 Maxwell’s Displacement Current .................................. 167
5.24 Electromagnetic Waves ............................................. 169
5.25 What Is the Medium for Electromagnetic Waves? ................ 174
5.26 The Sources of Electromagnetic Waves ........................... 175
5.27 Terms ................................................................ 177
5.28 Important Equations ................................................ 178
5.29 Problems for Chap. 5................................................ 178
6 The Atom as a Source of Light ............................................ 179
6.1 Atomic Spectra ...................................................... 179
6.2 The Hydrogen Spectrum of Visible Lines ......................... 181
6.3 The Bohr Theory of the Hydrogen Atom .......................... 184
6.4 Quantum Theory .................................................... 190
6.5 Complex Scenarios of Absorption and Emission.................. 195
6.5.1 Rayleigh Scattering ...................................... 196
6.5.2 Resonance Fluorescence................................. 196
6.5.3 General Fluorescence .................................... 196
6.5.4 Stimulated Emission ..................................... 197
6.6 Is Light a Stream of Photons or a Wave? .......................... 199
6.7 The Connection Between Temperature and Frequency ........... 200
xviii Contents
6.8 Terms ................................................................ 202
6.9 Important Equations ................................................ 202
6.10 Problems for Chap. 6................................................ 203
7 The Principle of Superposition ............................................ 205
7.1 The Wave Produced by Colliding Pulses .......................... 205
7.2 Superposition of Two Sine Waves of the Same Frequency ....... 207
7.3 Two Source Interference in Space.................................. 209
7.3.1 Sound Level with Many Sources........................ 216
7.3.2 Photons and Two-Slit Interference ...................... 216
7.4 Many-Source Interference .......................................... 217
7.4.1 Gratings .................................................. 217
7.4.2 Diffraction Through a Mesh .......................... 218
7.4.3 X-ray Diffraction of Crystals............................ 220
7.5 Beats................................................................. 221
7.6 Terms ................................................................ 224
7.7 Important Equations ................................................ 224
7.8 Problems for Chap. 7................................................ 225
8 Propagation Phenomena ................................................... 231
8.1 Diffraction ........................................................... 231
8.1.1 Scattering of Waves and Diffraction .................... 238
8.1.2 Why Is the Sky Blue?.................................... 240
8.2 Reflection............................................................ 241
8.2.1 A Complex Surface: A Sand Particle ................... 244
8.3 Reflection and Reflectance ......................................... 245
8.3.1 The Reflectance for a Light Wave....................... 246
8.3.2 The Reflectance for a Sound Wave...................... 248
8.4 Refraction ........................................................... 249
8.5 Total Internal Reflection ............................................ 251
8.6 The Wave Theory of Refraction .................................... 252
8.7 Application to Mirages ............................................. 255
8.8 The Prism ........................................................... 256
8.9 Dispersion ........................................................... 257
8.9.1 Effect of Dispersion on a Prism ......................... 257
8.9.2 Effect of Dispersion on Fiber Optics Communication . 258
8.10 Lenses ............................................................... 259
8.10.1 The Converging Lens .................................... 259
8.10.2 Lens Aberrations......................................... 260
8.10.3 Image Produced by a Converging Lens................. 264
8.10.4 Magnification ............................................ 266
8.10.5 Reversibility of Rays: Interchange
of Object and Image ..................................... 269
8.10.6 The Diverging Lens...................................... 269
8.10.7 Determining the Focal Length
of a Diverging Lens...................................... 271
Contents xix
8.11 The Doppler Effect ................................................. 272
8.11.1 Doppler Effect for Waves in a Medium................. 273
8.11.2 Doppler Effect for Electromagnetic Waves
in Vacuum ................................................ 277
8.11.3 Applications of the Doppler Effect...................... 278
8.12 Polarized Light ................................................... 280
8.12.1 How Can We Obtain a Beam of Polarized Light? ...... 281
8.12.2 Series of Polarizers ...................................... 282
8.12.3 Ideal vs. Real Polarizers ................................. 283
8.12.4 Sample Problems ........................................ 284
8.12.5 Partial Polarization of Reflected Light .................. 285
8.12.6 The Polarization of Scattering Light .................... 286
8.12.7 The Polarizer Eyes of Bees.............................. 286
8.12.8 Using Polarization of EM Radiation
in the Study of the Big Bang ............................ 287
8.12.9 Optical Activity .......................................... 287
8.12.10 Our Chiral Biosphere .................................... 291
8.13 Terms ................................................................ 293
8.14 Important Equations ................................................ 293
8.15 Questions and Problems for Chap. 8 ............................... 294
9 The Ear ...................................................................... 305
9.1 Broad Outline of the Conversion Process.......................... 306
9.2 The Auditory Canal ................................................. 310
9.3 The Eardrum ........................................................ 310
9.4 The Ossicles......................................................... 311
9.5 Improving on the Impedance Mismatch: Details ............... 313
9.6 The Cochlea ......................................................... 315
9.6.1 Summary ................................................. 318
9.7 Pitch Discrimination ................................................ 319
9.7.1 Some Mathematical Details on Pitch vs. the
Peak of the Envelope .................................. 322
9.7.2 Mach’s Law of Simultaneous Contrast in Vision....... 322
9.7.3 Rhythm Theory of Pitch Perception .................... 324
9.8 Terms ................................................................ 325
9.9 Problems for Chap. 9................................................ 326
10 Psychoacoustics ............................................................. 327
10.1 Equal Loudness Curves............................................. 329
10.2 The “Sone Scale” of Expressing Loudness ..................... 331
10.3 Loudness from Many Sources...................................... 334
10.4 Combination Tones and the Nonlinear Response of the Cochlea. 335
10.5 The Blue Color of the Sea and Its Connection
with Combination Tones............................................ 341
10.6 Duration of a Note and Pitch Discrimination ...................... 342
10.7 Fusion of Harmonics: A Marvel of Auditory Processing ......... 344
10.7.1 Mathematica File ........................................ 346
xx Contents
10.8 Additional Psychoacoustic Phenomena ............................ 348
10.9 Terms ................................................................ 349
10.10 Important Equations ................................................ 349
10.11 Problems for Chap. 10 .............................................. 349
11 Tuning, Intonation, and Temperament: Choosing
Frequencies for Musical Notes ............................................ 353
11.1 Musical Scales ...................................................... 355
11.2 The Major Diatonic Scale .......................................... 358
11.3 Comments Regarding Western Music ............................ 360
11.4 Pythagorean Tuning and the Pentatonic Scale ..................... 362
11.5 Just Tuning and the Just Scale...................................... 363
11.6 The Just Chromatic Scale........................................... 365
11.7 Intrinsic Problems with Just Tuning ............................... 367
11.8 Equal Tempered Tuning ............................................ 369
11.9 The Cents System of Expressing Musical Intervals............... 371
11.10 Debussy’s Six-Tone Scale .......................................... 373
11.11 Terms ................................................................ 374
11.12 Important Equations ................................................ 374
11.13 Problems for Chap. 11 .............................................. 375
12 The Eye ...................................................................... 383
12.1 The Cornea and Lens ............................................... 383
12.2 The Iris .............................................................. 386
12.3 The “Humorous” Liquids of the Eye............................... 387
12.4 The Retina ........................................................... 387
12.5 Dark Adaptation .................................................... 391
12.6 Depth Perception .................................................... 391
12.7 Terms ................................................................ 393
12.8 Problems for Chap. 12 .............................................. 393
13 Characterizing Light Sources Color Filters and Pigments ............ 397
13.1 Characterization of a Light Beam .................................. 397
13.1.1 Spectral Intensity vs. Intensity .......................... 402
13.2 Color Filters......................................................... 403
13.2.1 Stacking Filters (Filters in Series)....................... 405
13.3 Pigments............................................................. 409
13.4 Summary Comments on Filters and Pigments..................... 409
13.5 Terms ................................................................ 410
13.6 Important Equations ................................................ 411
13.7 Problems for Chap. 13 .............................................. 411
14 Theory of Color Vision ..................................................... 413
14.1 A Simplified Version of the Three-Primary Theory ............... 414
14.2 Exploration of Color Mixing with a Computer.................... 416
14.3 Introduction to the Chromaticity Diagram......................... 419
14.4 Metamers ............................................................ 420
14.5 A Crude Chromaticity Diagram .................................... 421
Contents xxi
14.6 A Chromaticity Diagram of Practical Use ......................... 423
14.6.1 The Units for the Admixture of the Three Primaries... 424
14.6.2 Tristimulus Values ....................................... 425
14.6.3 Color Coordinates........................................ 426
14.6.4 On the Significance of the Chromaticity Diagram ..... 426
14.7 The Calculation of Color Coordinates ............................. 432
14.7.1 Color Coordinates of Butter ............................. 435
14.8 Using a Different Set of Primaries ................................. 436
14.8.1 General Features of a Different Set of Primaries....... 437
14.9 The Standard Chromaticity Diagram of the C. I. E. ............... 439
14.10 From Computer RGB Values to Color .......................... 443
14.11 How Many Colors Are There? ..................................... 445
14.11.1 Limitations of a Broadened Gamut of a Monitor....... 452
14.12 A Simple Physiological Basis for Color Vision ................... 453
14.13 Color Blindness ..................................................... 458
14.14 After-Images ........................................................ 459
14.14.1 Questions for Consideration............................. 461
14.15 Terms ................................................................ 462
14.16 Important Equations ................................................ 462
14.17 Problems on Chap. 14............................................... 463
A Symbols...................................................................... 473
B Powers of Ten: Prefixes .................................................... 477
C Conversion of Units and Special Constants.............................. 479
D References for The Physics of Music and Color......................... 481
E A Crude Derivation of the Frequency of a Simple
Harmonic Oscillator ..................................................... 485
F Numerical Integration of Newton’s Equation for a SHO ............ 489
G Magnifying Power of an Optical System .............................. 495
G.1 Image with the Naked Eye and with a Magnifying Glass......... 496
G.2 The Microscope ..................................................... 499
G.3 Problems on Magnifying Power.................................... 500
H Threshold of Hearing, Threshold of Aural Pain,
and General Threshold of Physical Pain .............................. 501
I Transformation Between Tables of Color-Matching
Functions for Two Sets of Monochromatic Primaries ............... 507
I.1 Application of the Transformation: Determining
an Ideal Set of Primaries............................................ 509
I.2 Proof of Equations (I.1) and (I.6) .................................. 512
I.3 Problems on the Transformation of TCMFs....................... 518
J Hommage to Pierre-Gilles de Gennes: Art and Science .............. 521
xxii Contents
K MAPPINGS as a Basis for Arriving at a Mutually
Agreed Upon Description of Our Observations
of the World – Establishing ‘Truths’ and ‘Facts’ ....................... 525
K.1 MAPPINGS as Central to Organizing Human Experience ....... 527
K.2 NUMBERS as a Mapping .......................................... 527
K.3 The Concept of TIME as a Mapping............................... 528
K.4 Mappings as the Essential Goal of Physics ........................ 530
Index ............................................................................... 533

1. Why is the sky blue and the setting sun red?
2. How does the rainbow get its colors?
3. How is it that all light is a mixture of the colors of the rainbow? Yet the color brown is not simply a mixture of these colors?
4. How is it that sound can bend around corners?
5. Does light bend around corners?
6. What simple mathematical relationships form the bases of the musical scales of most of the world’s cultures? Are these relationships unique?
7. Are there three primary colors?
8. What are the colors white, black, gray, and brown?
9. How is the eye like a camera?
10. How is it that the ear can perceive two distinct musical tones, yet the eye perceives a mixture of two colors as a single color?
11. How can we get color from purely black and white images?
12. How does the brain determine the direction of a source of sound?
13. What is noise?
14. Why does the trumpet sound different from the violin?
15. What is a mirage?
16. Why do stars seem to twinkle?
17. How do color prints, color slides, and color TV work?
18. Can a soprano really break glass?
19. Why does a flutist have to retune his or her flute a while after having begun playing?
20. How is sound transmitted electrically?
21. How does the ear provide us with a sense of pitch?
22. Can a fish hear a fisherman talking?
23. Why do some automobiles rattle at a speed of about 55 mph?
24. How can we hear sounds which are not in the air? How is this phenomenon related to the blue color of the ocean?
25. How can a person hear a clock ticking at a frequency of one tick per second, while it is said that the lowest frequency that can be heard is about 20 cycles per second?
26. How can we estimate the speed of an overhead propeller-driven airplane from the sound it emits?
27. How does the vibrato of a violin help improve our perception of consonance among groups of notes?
28. Why does it become more difficult to perceive a sense of pitch as we play ever lower-pitched notes on a piano?