The Physics of Music and Color : Sound and Light

This undergraduate textbook aids readers in studying music and color, which involve nearly the entire gamut of the fundamental laws of classical as well as atomic physics. The objective bases for these two subjects are, respectively, sound and light. Their corresponding underlying physical principles overlap greatly: Both music and color are manifestations of wave phenomena. As a result, commonalities exist as to the production, transmission, and detection of sound and light. Whereas traditional introductory physics textbooks are styled so that the basic principles are introduced first and are then applied, this book is based on a motivational approach: It introduces a subject with a set of related phenomena, challenging readers by calling for a physical basis for what is observed. A novel topic in the first edition and this second edition is a non-mathematical study of electric and magnetic fields and how they provide the basis for the propagation of electromagnetic waves, of light in particular. The book provides details for the calculation of color coordinates and luminosity from the spectral intensity of a beam of light as well as the relationship between these coordinates and the color coordinates of a color monitor. The second edition contains corrections to the first edition, the addition of more than ten new topics, new color figures, as well as more than forty new sample problems and end-of-chapter problems. The most notable additional topics are: the identification of two distinct spectral intensities and how they are related, beats in the sound from a Tibetan bell, AM and FM radio, the spectrogram, the short-time Fourier transform and its relation to the perception of a changing pitch, a detailed analysis of the transmittance of polarized light by a Polaroid sheet, brightness and luminosity, and the mysterious behavior of the photon. The Physics of Music and Color is written at a level suitable for college students without any scientific background, requiring only simple algebra and a passing familiarity with trigonometry. The numerous problems at the end of each chapter help the reader to fully grasp the subject. Preface to the Second Edition Preface to the First Edition Note on Problems and Questions Questions Discussed in This Book Contents 1 Introductory Remarks 1.1 Two Contrary Attitudes About Science 1.2 A Scene vs. a Painting 1.2.1 The Joy of a Physicist in Looking at a Mathematical Equation 1.3 Two Views on the Creation of a Musical Composition vs. the Discovery of a Law of Physics The Legend of the Huang Chung: The Ancient Recognition of the Connection Between Numbers and Music 1.4 Outline of the Book 1.5 What About the Sentient Perception of Musicand Color? 1.6 Questions and Problems for Chap. 1 2 The Vibrating String 2.1 Waves Along a Stretched String 2.2 A Finite String Can Generate Music! 2.3 Pitch, Loudness, and Timbre 2.4 The Relation Between Frequency and Pitch 2.5 The Wave Motion of a Stretched Rope 2.6 Modes of Vibration and Harmonics 2.7 The Sine Wave 2.8 The Simple Harmonic Oscillator 2.8.1 The Physical Basis for the Oscillation of a Simple Harmonic Oscillator 2.8.2 The Vibration Frequency of a Simple Harmonic Oscillator 2.9 Traveling Sine Waves Applications 2.10 Modes of Vibration—Spatial Structure 2.11 The Wave Velocity of a Vibrating String Application of the Above Relations to the Piano 2.12 SHO vs. Vibrating String 2.13 Stiffness of a String 2.14 Resonance Digression on the Modes of Two Coupled SHOs 2.15 General Vibrations of a String 2.15.1 Frequency of a Wave with MissingFundamental 2.16 Periodic Waves and Timbre 2.17 An Application of Fourier's Theorem to Resonance Between Strings Home Exercise with a Piano 2.18 A Standing Wave as a Sum of Traveling Waves 2.19 Terms 2.20 Important Equations 2.21 Problems for Chap. 2 3 The Nature of Sound 3.1 The Air of Our Atmosphere 3.1.1 Pressure 3.1.2 Generating a Sound Pulse 3.1.3 Digression on Pushing a Block of Wood 3.2 The Nature of Sound Waves in Air 3.3 Characterizing a Sound Wave 3.4 Visualizing a Sound Wave 3.5 The Velocity of Sound 3.5.1 Temperature Dependence of Speed of Sound in Air 3.6 Standing Waves in an Air Column Standing Waves in a Closed Pipe End Correction for Modes in a Pipe 3.7 Magic in a Cup of Cocoa 3.8 The Helmholtz Resonator 3.8.1 The Physical Basis for the Helmholtz Resonator Formula for the Frequency of Helmholtz Resonator Derivation of the Helmholtz Formula 3.8.2 Flutter of Air Through the Window of a Speeding Automobile 3.9 Terms 3.10 Important Equations 3.11 Problems for Chap. 3 4 Energy 4.1 Forms of Energy and Energy Conservation 4.1.1 Fundamental Forms of Energy 4.1.2 ``Derived'' Forms of Energy 4.1.3 The Energy of Cheerios 4.2 The Principle of Conservation of Energy, Work,and Heat 4.2.1 Series of Changes of Forms of Energy 4.3 Energy of Vibrating Systems 4.3.1 The Simple Harmonic Oscillator 4.3.2 Energy in a Vibrating String 4.3.3 Energy in a Sound Wave 4.4 Power Power of Various Sources of Sound 4.5 Intensity 4.6 Intensity of a Point Source 4.7 Spectral Intensity with Respect to Frequency 4.7.1 White Noise and Pink Noise 4.8 Sound Level and the Decibel System 4.8.1 Logarithms 4.8.2 Sound Level 4.8.3 From Sound Level to Intensity 4.9 Attenuation 4.9.1 Attenuation in Time 4.9.2 Response and Resonance in the Presence of Attenuation 4.9.3 Attenuation of Traveling Waves—Attenuation in Space 4.9.4 Attenuation of Light in a TransparentMedium 4.10 Reverberation Time 4.11 Terms 4.12 Important Equations 4.13 Problems for Chap. 4 5 Electricity and Magnetism 5.1 The Fundamental Forces of Nature 5.2 The Electric Force 5.3 Electric Currents in Metal Wires 5.4 The Magnetic Force 5.5 Characterization of Magnetic Forces 5.6 Is There a Connection Between Electricity and Magnetism? 5.6.1 Action–Reaction Law and Force of Magnet on Current-Carrying Wire 5.7 The Loudspeaker 5.8 The Buzzer 5.9 The Electric Motor 5.10 Force Between Two Wires Carryingan Electric Current 5.11 The Electromagnetic Force and Michael Faraday 5.12 Applications of Faraday's EMF 5.13 A Final ``Twist'' 5.14 Action-at-a-Distance and Faraday's Fields 5.15 The Electric Field 5.16 The Magnetic Field 5.17 Magnetic Force on a Moving Charge 5.18 Force Between Two Parallel Wires Carrying Currents 5.19 Generalized Faraday's Law 5.20 What Do Induced Electric Field Lines Look Like? 5.21 Lenz's Law 5.22 The Guitar Pickup 5.23 Maxwell's Displacement Current 5.24 Electromagnetic Waves 5.25 What Is the Medium for Electromagnetic Waves? 5.26 The Sources of Electromagnetic Waves SUMMARY of Electricity and Magnetism 5.27 Terms 5.28 Important Equations 5.29 Problems for Chap. 5 6 The Atom as a Source of Light 6.1 Atomic Spectra 6.2 The Hydrogen Spectrum of Visible Lines 6.3 The Bohr Theory of the Hydrogen Atom 6.4 Quantum Theory 6.5 Line Width 6.6 Complex Scenarios of Absorption and Emission 6.6.1 Rayleigh Scattering 6.6.2 Resonant Fluorescence 6.6.3 General Fluorescence 6.6.4 Stimulated Emission 6.7 Is Light a Stream of Photons or a Wave? 6.8 The Connection Between Temperature and Frequency 6.9 Terms 6.10 Important Equations 6.11 Problems for Chap. 6 7 The Principle of Superposition 7.1 The Wave Produced by Colliding Pulses 7.2 Superposition of Two Sine Waves of theSame Frequency 7.3 Two-Source Interference in Space 7.3.1 Detecting the Epicenter of an Earthquake or the Location of a Gunshot 7.3.2 Sound Level with Many Sources 7.3.3 Photons and Two-Slit Interference 7.4 Many-Source Interference 7.4.1 Gratings 7.4.2 Diffraction Through a Mesh 7.4.3 X-ray Diffraction Off Crystals 7.5 Terms 7.6 Important Equations 7.7 Problems for Chap. 7 8 Complex Waves 8.1 Beats 8.1.1 Beats of a Tibetan Bell 8.2 AM and FM Transmission 8.3 AM Transmission 8.3.1 Sidebands 8.3.2 AM Demodulation 8.4 FM Transmission 8.4.1 FM Demodulation 8.4.2 Bandwidth Limitation Imposed on FM Radio Frequency Separations 8.5 Spectrogram 8.5.1 Understanding the Content of a Spectrogram 8.5.2 The Short Time Fourier Spectrum 8.5.3 The Relationship of Spectrograms with Measurement and Perception The Measurement of Frequency The Perception of Pitch 8.5.4 Spectrogram of the Gravitational Wave (GW) from a Collapsing Binary Neutron Star 8.6 Polarized Light 8.6.1 How Can We Obtain a Beam of PolarizedLight? Ideal Polarizer 8.6.2 Calcite 8.6.3 Calcite Loop 8.6.4 Polaroid 8.6.5 Series of Ideal Polarizers 8.6.6 Sample Problems 8.6.7 Partial Polarization of Reflected Light 8.6.8 The Polarization of Scattering Light 8.6.9 The Polarizer Eyes of Bees 8.6.10 Using Polarization of EM Radiation in the Study of the Big Bang 8.6.11 Optical Activity 8.6.12 Our Chiral Biosphere 8.7 Terms 8.8 Important Equations 8.9 Questions and Problems for Chap. 8 9 Propagation Phenomena 9.1 Diffraction 9.1.1 Scattering of Waves and Diffraction 9.1.2 Why Is the Sky Blue? 9.2 Reflection 9.3 Reflection and Refractance 9.3.1 The Reflectance for a Light Wave 9.3.2 The Reflectance for a Sound Wave 9.4 Refraction 9.5 Total Internal Reflection 9.6 The Wave Theory of Refraction 9.7 Application to Mirages 9.8 The Prism 9.9 Dispersion 9.9.1 Effect of Dispersion on a Prism 9.9.2 Effect of Dispersion on Fiber Optics Communication 9.10 Lenses 9.10.1 The Converging Lens 9.10.2 Lens Aberrations 9.10.3 Image Produced by a Converging Lens 9.10.4 Magnification The Real Image of a Converging Lens as a Secondary Object 9.10.5 The Compound Lens 9.10.6 Reversibility of Rays—Interchange of Object and Image 9.10.7 The Diopter 9.10.8 The Diverging Lens 9.10.9 Determining the Focal Length of a Diverging Lens 9.11 The Doppler Effect 9.11.1 Doppler Effect for Waves in a Medium Case (i): The Source Is at Rest with Respect to the Medium, While the Observer Is Moving with Respect to the Medium Case (ii): The Source Is Moving with Respect to the Medium, While the Observer Is at Rest with Respect to the Medium 9.11.2 Doppler Effect for Electromagnetic Waves in Vacuum 9.11.3 Applications of the Doppler Effect 9.12 Terms 9.13 Important Equations 9.14 Questions and Problems for Chap. 9 10 The Ear 10.1 Broad Outline of the Conversion Process 10.2 The Auditory Canal 10.3 The Eardrum 10.4 The Ossicles 10.5 Improving on the Impedance Mismatch: Details 10.6 The Cochlea 10.7 Pitch Discrimination 10.7.1 Some Mathematical Details on Pitch vs. the Peak of the Envelope 10.7.2 Mach's Law of Simultaneous Contrastin Vision 10.7.3 Rhythm Theory of Pitch Perception 10.8 Terms 10.9 Problems for Chap. 10 11 Psychoacoustics 11.1 Equal Loudness Curves 11.2 The ``Sone Scale'' of Expressing Loudness 11.3 Loudness from Many Sources 11.4 Combination Tones and the Non-Linear Response of the Cochlea 11.5 The Blue Color of the Sea and Its Connection with Combination Tones 11.6 Duration of a Note and Pitch Discrimination 11.7 Fusion of Harmonics—A Marvel of AuditoryProcessing 11.8 Additional Psychoacoustic Phenomena 11.9 Terms 11.10 Important Equations 11.11 Problems for Chap. 11 12 Tuning, Intonation, and Temperament—Choosing Frequencies for Musical Notes 12.1 Musical Scales 12.2 The Major Diatonic Scale 12.3 Comments Regarding Western Music 12.4 Pythagorean Tuning and the Pentatonic Scale 12.5 Just Tuning and the Just Scale 12.6 The Just Chromatic Scale 12.7 Intrinsic Problems with Just Tuning 12.8 Equal Tempered Tuning 12.9 The Cents System of Expressing Musical Intervals 12.10 Debussy's 6-Tone Scale 12.11 String-Harmonics 12.12 Terms 12.13 Important Equations 12.14 Problems for Chap. 12 13 The Eye 13.1 The Cornea and Lens 13.2 The Iris 13.3 The ``Humourous'' Liquids of the Eye 13.4 The Retina 13.5 Dark Adaptation 13.6 Depth Perception 13.7 Terms 13.8 Problems for Chap. 13 14 Characterizing Light Sources, Color Filters, and Pigments 14.1 Characterization of a Light Beam 14.2 Spectral Intensity 14.2.1 Measurement of Spectral Intensity 14.2.2 Examples of Spectral Intensities 14.2.3 Determining an Intensity from the Spectral Intensity 14.3 Comments on the Two Spectral Intensities 14.4 White Noise 14.5 Color Filters Stacking Filters (Filters in Series) 14.6 Reflectance 14.7 Pigments 14.8 Summary Comments on Filters and Pigments 14.9 Terms 14.10 Important Equations 14.11 Problems for Chap. 14 15 Theory of Color Vision 15.1 The Three Primary Colors Theory of Color Vision 15.2 Metamers 15.3 Simplification with Just Six Hues 15.4 Exploration of Color Mixing with a Computer 15.5 Introduction to the Chromaticity Diagram 15.5.1 A Crude Chromaticity Diagram—Color Coördinates 15.6 A Standard Chromaticity Diagram 15.6.1 The Calculation of Color Coördinates 15.6.2 Color Coördinates of Butter 15.6.3 Chromaticity Diagram 15.6.4 Primary Units 15.6.5 Mixing Primaries to Produce a Color 15.6.6 Properties of a Chromaticity Diagram 15.6.7 Mixing Two Incoherent Sources of Light 15.7 Using a Different Set of Primaries 15.7.1 General Features of a Different Setof Primaries 15.8 Comments About Tables of Color Matching Functions 15.9 Brightness and the Luminous Efficiency 15.9.1 Luminous Efficacy and Brightness 15.10 The Standard Chromaticity Diagram of the Commission Internationale de l'Éclairage (C. I. E.) 15.10.1 sRGB Primaries 15.11 From Computer RGB Values to Color 15.12 How Many Colors Are There? 15.12.1 Limitations of a Broadened Gamutof a Monitor 15.13 A Simple Physiological Basis for Color Vision 15.14 Color-Blindness 15.15 After-Images 15.15.1 Questions for Consideration 15.16 Terms 15.17 Important Equations 15.18 Problems on Chap.15 Correction to: The Physics of Music and Color: Sound and Light Correction to:L. 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