Waves and Oscillations: A Prelude to Quantum Mechanics

LEARNING TOOLS USED IN THIS BOOK 1. Simple Harmonic Motion 1.1 SINUSOIDAL OSCILLATIONS ARE EVERYWHERE 1.2 THE PHYSICS AND MATHEMATICS BEHIND SIMPLE HARMONIC MOTION 1.3 IMPORTANT PARAMETERS AND ADJUSTABLE CONSTANT OF SIMPLE HARMONIC MOTION 1.4 MASS ON A SPRING 1.5 ELECTRICAL OSCILLATORS 1.6 REVIEW OF TAYLOR SERIES APPROXIMATIONS 1.7 EULER'S EQUATION 1.8 REVIEW OF COMPLEX NUMBERS 1.9 COMPLEX EXPONENTIAL NOTATION FOR OSCILLATORY MOTION 1.10 THE COMPLEX REPRESENTATION FOR AC CIRCUITS 1.11 ANOTHER IMPORTANT COMPLEX FUNCTION: THE QUANTUM MECHANICAL WAVEFUNCTION 1.12 PURE SINUSOIDAL OSCILLATIONS AND UNCERTAINTY PRINCIPLES CONCEPT AND SKILL INVENTORY PROBLEMS 2. Examples of Simple Harmonic Motion 2.1 REQUIREMENTS FOR HARMONIC OSCILLATION 2.2 PENDULUMS 2.3 ELASTIC DEFORMATIONS AND YOUNG'S MODULUS 2.4 SHEAR 2.5 TORSION AND TORSIONAL OSCILLATORS 2.6 BENDING AND CANTILEVERS CONCEPT AND SKILL INVENTORY PROBLEMS 3. Damped oscillations 3.1 DAMPED MECHANICAL OSCILLATORS 3.2 DAMPED ELECTRICAL OSCILLATORS 3.3 EXPONENTIAL DECAY OF ENERGY 3.4 THE QUALITY FACTOR 3.5 UNDERDAMPED, OVERDAMPED, AND CRITICALLY DAMPED BEHAVIOR 3.6 TYPES OF DAMPING CONCEPT AND SKILL INVENTORY PROBLEMS 4. Driven Oscillations and Resonance 4.1 RESONANCE 4.2 EFFECTS OF DAMPING 4.3 ENERGY FLOW 4.4 LINEAR DIFFERENTIAL EQUATIONS, THE SUPERPOSITION PRINCIPLE FOR DRIVEN SYSTEMS, AND THE RESPONSE TO MULTIPLE DRIVE FORCES 4.5 TRANSIENTS 4.6 ELECTRICAL RESONANCE 4.7 OTHER EXAMPLES OF RESONANCE: MRI AND OTHER SPECTROSCOPIES 4.8 NON-LINEAR OSCILLATORS AND CHAOS CONCEPT AND SKILL INVENTORY PROBLEMS 5. Symmetric coupled oscillators and Hilbert space 5.1 BEATS: AN ASIDE? 5.2 TWO SYMMETRIC COUPLED OSCILLATORS: EQUATIONS OF MOTION 5.3 NORMAL MODES 5.4 SUPERPOSING NORMAL MODES 5.5 NORMAL MODE ANALYSIS, AND NORMAL MODES AS AN ALTERNATE DESCRIPTION OF REALITY 5.6 HILBERT SPACE AND BRA-KET NOTATION 5.7 THE ANALOGY BETWEEN COUPLED OSCILLATORS AND MOLECULAR ENERGY LEVELS 5.8 NON-ZERO INITIAL VELOCITIES 5.9 DAMPED, DRIVEN COUPLED OSCILLATORS CONCEPT AND SKILL INVENTORY PROBLEMS 6. Asymmetric coupled oscillators and the eigenvalue equation 6.1 MATRIX MATH 6.2 EQUATIONS OF MOTION AND THE EIGENVALUE EQUATION 6.3 PROCEDURE FOR SOLVING THE EIGENVALUE EQUATION 6.4 SYSTEMS WITH MORE THAN TWO OBJECTS 6.5 NORMAL MODE ANALYSIS FOR MULIT-OBJECT, ASYMMETRICAL SYSTEMS 6.6 MORE MATRIX MATH 6.7 ORTHOGONALITY OF NORMAL MODES, NORMAL MODE COORDINATES, DEGENERACY, AND SCALING OF HILBERT SPACE FOR UNEQUAL MASSES CONCEPT AND SKILL INVENTORY PROBLEMS 7. String theory 7.1 THE BEADED STRING 7.2 STANDING WAVE GUESS: BOUNDARY CONDITIONS QUANTIZE THE ALLOWED FREQUENCIES 7.3 THE HIGHEST POSSIBLE FREQUENCY CONNECTION TO WAVES IN A CRYSTALLINE SOLID 7.4 NORMAL MODE ANALYSIS FOR THE BEADED STRING 7.5 LONGITUDINAL OSCILLATIONS 7.6 THE CONTINUOUS STRING 7.7 NORMAL MODE ANALYSIS FOR CONTINUOUS SYSTEMS 7.8 K-SPACE CONCEPT AND SKILL INVENTORY PROBLEMS 8. Fourier analysis 8.1 INTRODUCTION 8.2 THE FOURIER EXPANSION 8.3 EXPANSIONS USING NON-NORMALIZED ORTHOGONAL BASIS FUNCTIONS 8.4 FINDING THE COEFFICIENTS IN THE FOURIER EXPANSION 8.5 FOURIER TRANSFORMS AND THE MEANING OF NEGATIVE FREQUENCY 8.6 THE DISCRETE FOURIER TRANSFORM (DFT) 8.7 SOME APPLICATIONS OF FOURIER ANALYSIS CONCEPT AND SKILL INVENTORY PROBLEMS 9. Traveling waves 9.1 INTRODUCTION 9.2 THE WAVE EQUATION 9.3 TRAVELING SINUSOIDAL WAVES 9.4 THE SUPERPOSITION PRINCIPLE FOR TRAVELING WAVES 9.5 ELECTROMAGNETIC WAVES IN VACUUM 9.6 ELECTROMAGNETIC WAVES IN MATTER 9.7 WAVES ON TRANSMISSION LINES 9.8 SOUND WAVES 9.9 MUSICAL INSTRUMENTS BASED ON TUBES 9.10 POWER CARRIED BY ROPE AND ELECTROMAGNETIC WAVES RMS AMPLITUDES 9.11 INTENSITY OF SOUND WAVES DECIBELS 9.12 DISPERSION RELATIONS AND GROUP VELOCITY CONCEPT AND SKILL INVENTORY PROBLEMS 10. Waves at interfaces 10.1 REFLECTIONS AND THE IDEA OF BOUNDARY CONDITIONS 10.2 TRANSMITTED WAVES 10.3 CHARACTERISTIC IMPEDANCES FOR MECHANICAL SYSTEMS 10.4 "UNIVERSAL" EXPRESSIONS FOR TRANSMISSION AND REFLECTION 10.5 REFLECTED AND TRANSMITTED WAVES FOR TRANSMISSION LINES 10.6 REFLECTION AND TRANSMISSION FOR ELECTROMAGNETIC WAVES IN MATTER: NORMAL INCIDENCE 10.7 REFLECTION AND TRANSMISSION FOR SOUND WAVES, AND SUMMARY OF ISOMORPHISMS 10.8 SNELL'S LAW 10.9 TOTAL INTERNAL REFLECTION AND EVANESCENT WAVES CONCEPT AND SKILL INVENTORY PROBLEMS APPENDIX A: GROUP VELOCITY FOR AN ARBITRARY ENVELOPE FUNCTION INDEX