The Physics of Oscillations and Waves: With Applications in Electricity and Mechanics

326 Pages·1997·20.91 MB·English

by  Ingram Bloch (auth.)

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