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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

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