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Osc_Waves_Lectures

# Osc_Waves_Lectures - Lectures on Oscillations and Waves...

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Lectures on Oscillations and Waves Michael Fowler, UVa, 6/6/07 FROM A CIRCLING COMPLEX NUMBER TO THE SIMPLE HARMONIC OSCILLATOR. .................... 3 Describing Real Circling Motion in a Complex Way. .......................................................................................... 3 Follow the Shadow: Simple Harmonic Motion. ................................................................................................... 4 OSCILLATIONS. ........................................................................................................................................................ 5 Introduction. ......................................................................................................................................................... 5 Brief Review of Undamped Simple Harmonic Motion . ........................................................................................ 6 Energy. ................................................................................................................................................................. 7 A Heavily Damped Oscillator. ............................................................................................................................. 8 Interpreting the Two Different Exponential Solutions . ........................................................................................ 9 *The Most General Solution for the Highly Damped Oscillator. ....................................................................... 10 *The Principle of Superposition for Linear Differential Equations. .................................................................. 11 A Lightly Damped Oscillator . ............................................................................................................................ 11 The Q Factor. ..................................................................................................................................................... 13 *Critical Damping . ............................................................................................................................................ 13 Shock Absorbers and Critical Damping. ............................................................................................................ 14 A Driven Damped Oscillator: the Equation of Motion . ..................................................................................... 16 The Steady State Solution and Initial Transient Behavior . ................................................................................ 16 Using Complex Numbers to Solve the Steady State Equation Easily. ................................................................ 17 Back to Reality . .................................................................................................................................................. 19 And Now to Work…. ........................................................................................................................................... 21 THE PENDULUM. .................................................................................................................................................... 22 The Simple Pendulum. ........................................................................................................................................ 22 Pendulums of Arbitrary Shape. .......................................................................................................................... 23 Variation of Period of a Pendulum with Amplitude. .......................................................................................... 24 INTRODUCING WAVES: STRINGS AND SPRINGS. ........................................................................................ 25 One-Dimensional Traveling Waves . .................................................................................................................. 25 Transverse and Longitudinal Waves. ................................................................................................................. 26 Traveling and Standing Waves. .......................................................................................................................... 27 ANALYZING WAVES ON A STRING. ................................................................................................................. 27 From Newton’s Laws to the Wave Equation. ..................................................................................................... 27 Solving the Wave Equation . ............................................................................................................................... 29 The Principle of Superposition. .......................................................................................................................... 30 Harmonic Traveling Waves. ............................................................................................................................... 30 Energy and Power in a Traveling Harmonic Wave . .......................................................................................... 31 Standing Waves from Traveling Waves. ............................................................................................................. 33 BOUNDARY CONDITIONS: AT THE END OF THE STRING. ........................................................................ 35 Adding Opposite Pulses . .................................................................................................................................... 35 Pulse Reflection. ................................................................................................................................................. 35 An Experiment on Fixed End Reflection and Free End Reflection . ................................................................... 35 Understanding Sign Change in Pulse Reflection . .............................................................................................. 36 Free End Boundary Condition. .......................................................................................................................... 39 SOUND WAVES. ....................................................................................................................................................... 40 “One-Dimensional” Sound Waves. .................................................................................................................... 40

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2 Relating Pressure Change to How the Displacement Varies. ............................................................................ 41 From F = ma to the Wave Equation . ................................................................................................................ 42 Boundary Conditions for Sound Waves in Pipes. ............................................................................................... 42 Harmonic Standing Waves in Pipes. .................................................................................................................. 43 Traveling Waves: Power and Intensity . ............................................................................................................. 43 WAVES IN TWO AND THREE DIMENSIONS . .................................................................................................. 45 Introduction. ....................................................................................................................................................... 45 The Wave Equation and Superposition in One Dimension . ............................................................................... 45 The Wave Equation and Superposition in More Dimensions. ............................................................................ 45 How Does a Wave Propagate in Two and Three Dimensions? . ........................................................................ 46 Huygen’s Picture of Wave Propagation. ............................................................................................................ 47 Two-Slit Interference: How Young measured the Wavelength of Light . ........................................................... 49 Another Bright Spot . .......................................................................................................................................... 51 THE DOPPLER EFFECT. ....................................................................................................................................... 52 Introduction. ....................................................................................................................................................... 52 Sound Waves from a Source at Rest. .................................................................................................................. 52 Sound Waves from a Moving Source. ................................................................................................................. 53 Stationary Source, Moving Observer. ................................................................................................................ 54 Source and Observer Both Moving Towards Each Other. ................................................................................. 55 Doppler Effect for Light. .................................................................................................................................... 55 Other Possible Motions of Source and Observer. .............................................................................................. 55 APPENDIX: COMPLEX NUMBERS. .................................................................................................................... 56 Real Numbers. .................................................................................................................................................... 56 Solving Quadratic Equations. ............................................................................................................................ 56 Polar Coordinates. ............................................................................................................................................. 59 The Unit Circle. .................................................................................................................................................. 60 COMPLEX EXERCISES. ........................................................................................................................................ 62 OSCILLATIONS AND WAVES HOMEWORK PROBLEMS. ........................................................................... 63 Oscillations . ....................................................................................................................................................... 63 Waves. ................................................................................................................................................................ 69
3 From a Circling Complex Number to the Simple Harmonic Oscillator ( A review of complex numbers is provided in the appendix to these lectures. ) Describing Real Circling Motion in a Complex Way We’ve seen that any complex number can be written in the form i zr e θ = , where r is the distance from the origin, and

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Osc_Waves_Lectures - Lectures on Oscillations and Waves...

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