The Physics of Oscillations and Waves: With Applications in Electricity and Mechanics PDF
Preview The Physics of Oscillations and Waves: With Applications in Electricity and Mechanics
The Physics of OSCILLATIONS and WAVES Ingram Bloch THE PHYSICS OF OSCILLATIONS and WAVES With Applications in ELECTRICITY and MECHANICS Springer Science+ Business Media, LLC Library of Congress Cataloging-in-Publication Data Bloch, Ingram. The physics of oscillations and waves with applications in electricity and mechanics I Ingram Bloch. p. em. Includes bibliographical references and index. ISBN 978-1-4899-0052-4 ISBN 978-1-4899-0050-0 (eBook) DOI 10.1007/978-1-4899-0050-0 1. Electricity--Mathematics. 2. Mechanics. 3. Oscillations. 4. Waves. I. Title. OC522.B58 1997 530.4'16--DC21 97-15933 CIP ISBN 978-1-4899-0052-4 © 1997 Springer Science+ Business Media New York Originally published by Plenum Press, New York in 1997 Softcover reprint of the hardcover 1st edition 1997 http:// www.plenum.com All rights reserved 10 9 8 7 6 5 4 3 2 1 No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher Book Designed by Reuven Solomon This book is dedicated to MARY BLOCH whose encouragement, advice, and labor have been essential to its completion Preface Except for digressions in Chapters 8 and 17, this book is a highly unified treatment of simple oscillations and waves. The phenomena treated are "simple" in that they are de scribable by linear equations, almost all occur in one dimension, and the dependent variables are scalars instead of vectors or something else (such as electromagnetic waves) with geometric complications. The book omits such complicated cases in order to deal thoroughly with properties shared by all linear os cillations and waves. The first seven chapters are a sequential treatment of electrical and mechanical oscillating systems, starting with the simplest and proceeding to systems of coupled oscillators subjected to ar bitrary driving forces. Then, after a brief discussion of nonlinear oscillations in Chapter 8, the concept of normal modes of motion is introduced and used to show the relationship between os cillations and waves. After Chapter 12, properties of waves are explored by whatever mathematical techniques are applicable. The book ends with a short discussion of three-dimensional vii viii Preface problems (in Chapter 16), and a study of a few aspects of non linear waves (in Chapter 17). Besides trying to keep the geometry simple, I have made no at tempt to discuss the use of an electronic computer on any of the equations in the book; the only cases in which such use would be justified are a few of those in Chapters 8 and 17. Also omitted are topics whose understanding would require a knowledge of either relativity theory or quantum theory. One of the purposes of the book is to prepare students for the study of quantum mechanics. My experience has been that, in trying to learn new mathematics, students of quantum theory are likely to slight the new and difficult physics which is the main point of the course. Students have found it helpful to learn much of the mathematics in working on problems in classical phys ics-especially so when the classical problems have their own interest. Students planning to use this book should have at least a little knowledge oflinear differential equations and of complex num bers. Students with more extensive backgrounds can use some parts of the book (e.g., Chapters 1-5, 10) as review. For students who have not studied vector analysis, instructors may wish to re formulate in one dimension the three-dimensional topics in Chapter 16. Helped by such flexibility, physics majors who are juniors, seniors, and beginning graduate students should be ade quately prepared to use the book. It is important for physics students to work on problems. There are problems at the ends of chapters, and brief answers to a few of them in the back of the book. More problems can be found in some of the books cited in the notes at the ends of chapters. I greatly appreciate the patient help that I have received from Sidney and Raymond Solomon. Ingram Bloch Contents 1 Undamped and Undriven Oscillators and LC Circuits 1 2 The Effect of Damping 21 3 Sinusoidally Driven Oscillators and Circuit Loops 33 4 Sums of Sinusoidal Forces or EMF's-Fourier Analysis 51 5 Integration in the Complex Plane 73 6 Evaluation of Certain Fourier Integrals-Causality, Green's Functions 88 7 Electrical Networks 104 8 Nonlinear Oscillations 120 9 Coupled Oscillators without Damping- Lagrange's Equations 151 lO Matrices-Rotations-Eigenvalues and Eigenvectors- Normal Coordinates 161 11 Some Examples of Normal Coordinates 181 l2 Finite One-Dimensional Periodic Systems, Difference Equations 197 l3 Infinite One-Dimensional Periodic Systems- Characteristic Impedance 216 ix X Contents 14 Continuous Systems, Wave Equation, Lagrangian Density, Hamilton's Principle 230 15 Continuous Systems, Applied Forces, Interacting Systems 251 16 Other Linear Problems 267 17 Nonlinear Waves 290 Answers to Selected Problems 309 Index 316 1 Undamped and Ondriven Oscillators and LC Circuits AI but two chapters of this book deal with the behavior of physical systems that are describable by linear dif ferential equations. The treatment proceeds from simpler to more complicated systems and uses successively more sophis ticated techniques of analysis. In this chapter we deal with the simplest oscillating system. Let us consider a mass m that moves without friction on the x axis and is subject to a force toward the origin that is propor tional to the distance of the mass from the origin. Thus the force is F = -kx, (1.1) and Newton's second law implies that (1.2) or, if kim=~ 1
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