Waves and Oscillations: A Prelude to Quantum Mechanics Book

Learning Tools Used in This Book ix

1 Simple Harmonic Motion 1

1.1 Sinusoidal oscillations are everywhere 1

1.2 The physics and mathematics behind simple sinusoidal motion 3

1.3 Important parameters and adjustable constants of simple harmonic motion 5

1.4 Mass on a spring 8

1.5 Electrical oscillators 10

1.6 Review of Taylor series approximations 12

1.7 Euler's equation 13

1.8 Review of complex numbers 14

1.9 Complex exponential notation for oscillatory motion 16

1.10 The complex representation for AC circuits 18

1.11 Another important complex function: The quantum mechanical wavefunction 24

1.12 Pure sinusoidal oscillations and uncertainty principles 26

Concept and skill inventory 29

Problems 31

2 Examples of Simple Harmonic Motion 39

2.1 Requirements for harmonic oscillation 39

2.2 Pendulums 40

2.3 Elastic deformations and Young's modulus 42

2.4 Shear 47

2.5 Torsion and torsional oscillators 49

2.6 Bending and Cantilevers 52

Concept and skill inventory 56

Problems 58

3 Damped Oscillations 64

3.1 Damped mechanical oscillators 64

3.2 Damped electrical oscillators 68

3.3 Exponential decay of energy 69

3.4 The quality factor 70

3.5 Underdamped, overdamped, and critically damped behavior 72

3.6 Types of damping 74

Concept and skill inventory 76

Problems 77

4 Driven Oscillations and Resonance 84

4.1 Resonance 84

4.2 Effects of damping 91

4.3 Energy flow 95

4.4 Linear differential equations the superposition principle for driven systems, and the response to multiple drive forces 99

4.5 Transients 101

4.6 Electrical resonance 104

4.7 Other examples of resonance: MRT and other spectroscopies 107

4.8 Nonlinear oscillators and chaos 114

Concept and skill inventory 128

Problems 129

5 Symmetric Coupled Oscillators and Hilbert Space 137

5.1 Beats: An aside? 137

5.2 Two symmetric coupled oscillators: Equations of motion 139

5.3 Normal modes 142

5.4 Superposing normal modes 146

5.5 Normal mode analysis, and normal modes as an alternate description of reality 149

5.6 Hilbert space and bra-ket notation 153

5.7 The analogy between coupled oscillators and molecular energy levels 163

5.8 Nonzero initial velocities 165

5.9 Damped, driven coupled oscillators 166

Concept and skill inventory 168

Problems 170

6 Asymmetric Coupled Oscillators and the Eigenvalue Equation 179

6.1 Matrix math 179

6.2 Equations of motion and the eigenvalue equation 182

6.3 Procedure for solving the eigenvalue equation 186

6.4 Systems with more than two objects 191

6.5 Normal mode analysis for multi-object, asymmetrical systems 194

6.6 More matrix math 198

6.7 Orthogonality of normal modes, normal mode coordinates, degeneracy, and scaling of Hilbert space for unequal masses 201

Concept and skill inventory 208

Problems 210

7 String Theory 216

7.1 The beaded string 216

7.2 Standing wave guess: Boundary conditions quantize the allowed frequencies 219

7.3 The highest possible frequency; connection to waves in a crystalline solid 222

7.4 Normal mode analysis for the beaded string 226

7.5 Longitudinal oscillations 227

7.6 The continuous string 230

7.7 Normal mode analysis for continuous systems 231

7.8 k-space 234

Concept and skill inventor 236

Problems 236

8 Fourier Analysis 246

8.1 Introduction 246

8.2 The Fourier Expansion 247

8.3 Expansions using nonnormalized orthogonal basis functions 250

8.4 Finding the coefficients in the Fourier series expansion 251

8.5 Fourier Transforms and the meaning of negative frequency 254

8.6 The Discrete Fourier Transform (DFT) 258

8.7 Some applications of Fourier Analysis 265

Concept and skill inventory 267

Problems 268

9 Traveling Waves 280

9.1 Introduction 280

9.2 The wave equation 280

9.3 Traveling sinusoidal waves 284

9.4 The superposition principle for traveling waves 285

9.5 Electromagnetic waves in vacuum 287

9.6 Electromagnetic waves in matter 296

9.7 Waves on transmission lines 301

9.8 Sound waves 305

9.9 Musical instruments based on tubes 314

9.10 Power carried by rope and electromagnetic waves; RMS amplitudes 316

9.11 Intensity of sound waves; decibels 320

9.12 Dispersion relations and group velocity 323

Concept and skill inventory 332

Problems 334

10 Waves at Interfaces 343

10.1 Reflections and the idea of boundary conditions 343

10.2 Transmitted waves 349

10.3 Characteristic impedances for mechanical systems 352

10.4 "Universal" expressions for transmission and reflection 356

10.5 Reflected and transmitted waves for transmission lines 359

10.6 Reflection and transmission for electromagnetic waves in matter: Normal incidence 364

10.7 Reflection and transmission for sound waves, and summary of isomorphisms 367

10.8 Snell's Law 368

10.9 Total internal reflection and evanescent waves 371

Concept and skill inventory 378

Problems 379

Appendix A Group Velocity for an Arbitrary Envelope Function 388

Index 393