Vibrations and Waves

Professor George C. King, Department of Physics & Astronomy, University of Manchester, Manchester, UK.

• Editors' Preface to the Manchester Physics Series xiAuthor's Preface xiii1 SIMPLE HARMONIC MOTION 11.1 Physical Characteristics of Simple Harmonic Oscillators 11.2 A Mass on a Spring 21.2.1 A mass on a horizontal spring 21.2.2 A mass on a vertical spring 51.2.3 Displacement, velocity and acceleration in simple harmonic motion 51.2.4 General solutions for simple harmonic motion and the phase angle φ 71.2.5 The energy of a simple harmonic oscillator 101.2.6 The physics of small vibrations 121.3 The Pendulum 171.3.1 The simple pendulum 171.3.2 The energy of a simple pendulum 191.3.3 The physical pendulum 221.3.4 Numerical solution of simple harmonic motion3 241.4 Oscillations in Electrical Circuits: Similarities in Physics 271.4.1 The LC circuit 271.4.2 Similarities in physics 29PROBLEMS 1 292 THE DAMPED HARMONIC OSCILLATOR 332.1 Physical Characteristics of the Damped Harmonic Oscillator 332.2 The Equation of Motion for a Damped Harmonic Oscillator 342.2.1 Light damping 352.2.2 Heavy damping 372.2.3 Critical damping 382.3 Rate of Energy Loss in a Damped Harmonic Oscillator 412.3.1 The quality factor Q of a damped harmonic oscillator 432.4 Damped Electrical Oscillations 46PROBLEMS 2 473 FORCED OSCILLATIONS 493.1 Physical Characteristics of Forced Harmonic Motion 503.2 The Equation of Motion of a Forced Harmonic Oscillator 503.2.1 Undamped forced oscillations 503.2.2 Forced oscillations with damping 543.3 Power Absorbed During Forced Oscillations 603.4 Resonance in Electrical Circuits 643.5 Transient Phenomena 663.6 The Complex Representation of Oscillatory Motion 683.6.1 Complex numbers 683.6.2 The use of complex numbers to represent physical quantities 713.6.3 Use of the complex representation for forced oscillations with damping 74PROBLEMS 3 744 COUPLED OSCILLATORS 774.1 Physical Characteristics of Coupled Oscillators 774.2 Normal Modes of Oscillation 784.3 Superposition of Normal Modes 814.4 Oscillating Masses Coupled by Springs 874.5 Forced Oscillations of Coupled Oscillators 934.6 Transverse Oscillations 96PROBLEMS 4 995 TRAVELLING WAVES 1055.1 Physical Characteristics of Waves 1065.2 Travelling Waves 1065.2.1 Travelling sinusoidal waves 1095.3 The Wave Equation 1125.4 The Equation of a Vibrating String 1145.5 The Energy in a Wave 1165.6 The Transport of Energy by a Wave 1195.7 Waves at Discontinuities 1215.8 Waves in Two and Three Dimensions 1265.8.1 Waves of circular or spherical symmetry 130PROBLEMS 5 1336 STANDING WAVES 1376.1 Standing Waves on a String 1376.2 Standing Waves as the Superposition of Two Travelling Waves 1446.3 The Energy in a Standing Wave 1476.4 Standing Waves as Normal Modes of a Vibrating String 1496.4.1 The superposition principle 1496.4.2 The superposition of normal modes 1506.4.3 The amplitudes of normal modes and Fourier analysis 1536.4.4 The energy of vibration of a string 156PROBLEMS 6 1587 INTERFERENCE AND DIFFRACTION OF WAVES 1617.1 Interference and Huygen’s Principle 1617.1.1 Young’s double-slit experiment 1637.1.2 Michelson spectral interferometer 1707.2 Diffraction 1727.2.1 Diffraction at a single slit 1727.2.2 Circular apertures and angular resolving power 1777.2.3 Double slits of finite width 179PROBLEMS 7 1818 THE DISPERSION OF WAVES 1838.1 The Superposition of Waves in Non-Dispersive Media 1838.1.1 Beats 1848.1.2 Amplitude modulation of a radio wave 1868.2 The Dispersion of Waves 1878.2.1 Phase and group velocities 1888.3 The Dispersion Relation 1928.4 Wave Packets 1958.4.1 Formation of a wave packet 197PROBLEMS 8 201APPENDIX: SOLUTIONS TO PROBLEMS 205Index 223

"Each chapter is accompanied by a set of problems that form an important part of the book. The book could be used by undergraduate students taking a course in oscillation or wave physics." (Zentralblatt MATH, 2010) "The text concisely describes vibrations and waves through mathematical equations with an emphasis on their physical meaning." (Outrider, January 2010)