Sound AP Physics Chapter 12. 12.1 Characteristics of Sound Vibration and Waves.
The physics of vibration - GBV
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The physics of vibration A. B. PIPPARD, FRS Emeritus Professor of Physics, University of Cambridge The right of ihr University of Cambridge to print ami sell all niannrt of books №oi gruninl by Henry VIII in IS34. The Unirtrsity has printed and published continuously since ISS4. CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney
Transcript of The physics of vibration - GBV
The physics of vibration
A. B. PIPPARD, FRS Emeritus Professor of Physics, University of Cambridge
The right of ihr University of Cambridge
to print ami sell all niannrt of books
№oi gruninl by Henry VIII in IS34.
The Unirtrsity has printed and published continuously
since ISS4.
CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney
Contents
Preface to Part 1 page xi Preface to Part 2 xii Preface to the combined volume xiii
1 General introduction - author to reader l
PART 1: THE SIMPLE CLASSICAL VIBRATOR
2 The free vibrator
Introduction: the special role of harmonic vibrators 7 Anharmonic simple vibrators 12 Using energy conservation to calculate vibration frequency 21 Systems with more than one mode 24 Dissipation 30 Maintained vibrations 39
3 Applications of complex variables to linear systems
Complex exponential solutions to equations of motion 44 The complex p-plane 46 Complex frequency; conventions 52 The vibration diagram 53 Circuit analysis " 55 Work and power 56 Waves 57 Circular polarization 62
4 Fourier series and integral
Synthesis of waveforms; Fourier series 63 Beats and related phenomena 67 The Fourier integral 69
V
VI Contents
Noise 79 White noise 90 Autocorrelation 93
5 Spectrum analysis Computation of Fourier transforms 97 The ear and the eye; optical spectrometers 100 Linear transducers; response functions 105 Causality 107 Resonant filters: sound spectrograph 113 Mixing and combination tones 121
6 The driven harmonic vibrator Transfer function, compliance, susceptibility, admittance, impedance 128 The form of the resonance curve 134 Low-О resonant peaks 140 Transient response 142 Response of a resonant system to noise 152
7 Waves and resonators Preliminary remarks about one-dimensional waves; characteristic impedance and admittance 161 Analogues of characteristic impedance 169 Resonant lines 171 Finger-holes in woodwind instruments 175 Radiation from an open end 179 Resonators attached to transmission lines 184 Excitation of a resonator by a plane wave 195 Anomalous dispersion 204
8 Velocity-dependent forces Coriolis and Lorentz forces 209 Whirling 212 The gyro-pendulum 216 The gyromagnetic top 218 Nuclear magnetic resonance 223 The physical mechanism of relaxation 230 Cyclotron resonance 237 Helicons 239
Contents VII
9 The driven anharmonic vibrator; subharmonics; stability Slightly anharmonic vibrators 248 An example of subharmonic excitation 253
Behaviour of simple systems near equilibrium 254 Examples of instability 259 The slightly anharmonic vibrator revisited 266 Stability of a system with more than two independent variables 267 Stability testing by discrete sampling (stroboscopic method) 268 The subharmonic generator revisited 271 The cyclotron, and Azbel'-Kaner cyclotron resonance 275 Conditions for subharmonic generation 278
10 Parametric excitation Detailed analysis of parametrically excited systems 289 Parametric amplifiers 301
11 Maintained oscillators The feedback principle 306 Negative resistance and spontaneous oscillation 311 Negative resistance by means of feedback; class С oscillators 314 The pendulum clock, a non-linear class С oscillator 323 Further examples of negative-resistance oscillators 326
(a) A resistive feedback network 326 (b) The tunnel diode 330 (c) Mechanical analogues of negative and non-linear
resistance; Lienard's construction 333 (d) The maser 345
Saturation of negative-resistance oscillators 348 Multivibrator 353 Relaxation oscillators 359 The effect of noise on a class С oscillator 361
12 Coupled vibrators Coupled passive vibrators 366 Energy and normal modes 372 Examples of coupled vibrators 376 Coupled lossy vibrators 379 Impedances and impulse response functions for coupled circuits 381 Coupled active and passive vibrators 385 The Huygens phenomenon (entrainment) 391
viii Contents
Frequency-locking of a multivibrator 405 Locking of multivibrator to a resonant circuit 413 Superconducting weak links (Josephson junctions) 416
PART 2: THE SIMPLE VIBRATOR IN QUANTUM MECHANICS
13 The quantized harmonic vibrator and its classical features 427
Solution of Schrödinger's equation 429 Momentum distribution 433 Non-stationary states 434 The coherent state 435 Potential and kinetic energy 436 Classical behaviour of (£) 438 Impulse-response of a quantized system 439 Critical phenomena 443 The equivalent classical ensemble (^-representation) 445 Energy imparted by an applied force 448 Parametric excitation 449
14 Anharmonic vibrators 452
Isochronous vibrators 452 Arbitrary potential well; the semi-classical method 456 WKB approximation 461
15 Vibrations and cyclotron orbits in two dimensions 467
Fermi resonance: non-linear coupling 470 Cyclotron orbits 475 Quantization in a magnetic field 477 Quantization of non-circular orbits in real metals 479 Magnetic breakdown 483
16 Dissipation, level broadening and radiation 489
Coupled harmonic vibrators 490 Dissipation and level broadening 492 Electromagnetic radiation into free space 497 Spontaneous radiation in quantum mechanics 499 Planck's radiation law; Einstein coefficients 501 Divergences in the theory of dipole radiation; mass enhancement 505
Contents IX
17 The equivalent classical oscillator 510
The/-sum rule 512 Static polarizability 514
18 The two-level system 516
General theory 517 The double-well model 520 Response functions 522 Radiative decay of the two-level system 528 Equations of motion of a damped double-well 532 Stimulated and spontaneous transitions 536 The frequency-dependent susceptibility 538 Dielectric loss in polyethylene - an example of tunnelling relaxation 540 Other examples of double-well systems; ammonia 543
19 Line broadening 550
Phase diffusion 552 Stark broadening and resonance broadening 555 Doppler broadening; saturation spectroscopy 561 Spin echoes and related effects 564
20 The ammonia maser 571
Stimulated emission 572 Dielectric response of a molecular beam 574 Fluctuations in amplitude and phase 578 Quantum mechanics of the resonator-molecule interaction 583 Graphical representation of density matrix 587 The dissipative harmonic oscillator 588 Dissipation and fluctuations in the maser 595 Measurement of maser frequency 599 The maser as frequency standard 600
21 The family of masers: from laser to travelling-wave oscillator 603
Mechanisms of population inversion 605 Classical perspective ort maser processes 609 Electron-cyclotron oscillator 609 Bunching and maser action 613 Acoustic attenuation and amplification in solids 618 Travelling-wave oscillator 620 The klystron 621
X Contents
Epilogue 626
References 627
Index 633
Note: Equations, diagrams and references are numbered serially in each chapter, and the chapter number is omitted if it is something in the same chapter that is referred to. Thus (4.23) in chapter 7 means equation 23 of chapter 4; (23) in chapter 7 means equation 23 in that chapter.
Cross-references are given as page-numbers in the margin.
