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Complex Numbers and Simple Harmonic Oscillation

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Complex Numbers and Simple Harmonic Oscillation 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 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
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