Wave Physics: Oscillations — Solitons — Chaos

296 Pages·2003·6.581 MB·English

by  Professor Stephen Nettel (auth.)

Wave Physics
Advanced Texts in Physics
This program of advanced texts covers a broad spectrum of topics which are of
current and emerging interest in physics. Each book provides a comprehensive and
yet accessible introduction to a field at the forefront of modern research. As such,
these texts are intended for senior undergraduate and graduate students at the MS
and PhD level; however, research scientists seeking an introduction to particular
areas of physics will also benefit from the titles in this collection.
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
Physics and Astronomy
http://www.springer.de/phys/
Stephen N ettel
Physics
Wave
Oscillations - Solitons - Chaos
Third Edition
With 67 Figures and more than 100 Problems
with Hints for Solution and Numerous Examples
, Springer
Professor Stephen Nette!
Rensselaer Polytechnic Institute, Department of Physics
Troy, NY 12180-3590, USA
Contributors to Chapter 8:
Professor Dr. Andrei V. Gaponov-Grekhov
Professor Dr. Mikhail!. Rabinovich
Institute of Applied Physics, Russian Academy of Sciences
ul. Uiyanova 46
603 600 Nizhnii Novgorod, Russia
Professor Dr. Martin Gutzwiller
IBM T.J. Watson Research Center
P.O. Box 218
Yorktown Heights, NY 10598, USA
The cover picture was composed from Figs. 5.1 and 2.6 of this book.
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ISSN 1439-2674
ISBN 978-3-662-05319-5 ISBN 978-3-662-05317-1 (eBook)
DOI 10.1007/978-3-662-05317-1
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Preface to the Third Edition
This is a text for the third semester of undergraduate physics for students
in accelerated programs, who typically are preparing for advanced degrees
in science or engineering. The third semester is often the only opportunity
for physics departments to present to students who are not physics majors
a coherent background in the physics of waves, required later for confident
handling of applied problems, especially applications based on quantum
mechanics.
Physics is a coherent subject. It is often found that the going gets easier
as one goes deeper, learning the mathematical connections tying together
the various phenomena. Even so, the steps that took us from classical wave
physics to Heisenberg's "Physical Principles of Quantum Theory" were, as
a matter of history, harder to take than later steps dealing with detailed
applications. With these considerations in mind, the classical physics of os
cillations and waves is developed here at a more advanced mathematical
level than is customary in second year courses. This is done to explain the
classical phenomena, but also to provide background for the introductory
wave mechanics, leading to a logical integration of the latter subject into
the presentation. Concurrently, detailed applications of quantum mechanics
are beyond the mission of the text. The concluding chapters on nonlinear
waves, solitons, and chaos broaden the previously established concepts of
wave behavior, while introducing the reader to important topics in current
wave physics.
Experience teaching the course has led in this third edition to the author's
fully presenting material dealing with Green's functions, rather than trying
to save course time by burying the material in problems and an appendix,
as heretofore. This extension has led, in turn, to a broadening of Chap. 6
on wave mechanics, with a treatment of the Green's function for harmonic
oscillators. In this way, along with additions to examples and problems, the
integration of wave mechanics with classical wave physics has been reinforced.
The text begins with a full chapter of mathematics that develops Fourier
analysis in the context of generalized functions. The idea is to expose students
to the new concepts while there is yet a minimum of "pressure" , and then al
low them to assimilate the concepts through the chapters on classical physics.
In this way, when it comes to the wave mechanics in Chap. 6, students should
\TI f>reface
be able to distinguish the new physics from mathematics. M. J. Lighthill, in
his monograph "Fourier Analysis and Generalized Functions" , says that "The
theory of generalized functions ... greatly curtails the labor of understanding
Fourier transforms". Application of generalized functions (Schwartz' "theorie
des distributions") avoids undefined integrals like J~ooeikxdk still prevalent in
texts on quantum mechanics. Chapter 2, on oscillations, may seem somewhat
in the nature of a review. The physics material thus starts at the point where
the first exposure in the freshman year has been made, but may have led to
only a partial grasp.
It is hoped that this book will be useful to various categories of readers,
honors math and physics students, graduate students studying for Ph.D.
qualifiers, as well as professors looking for examination problems. Problems
displace tedium by presenting a challenge. Here many are used to establish
significant results. This makes for meaningful activity, as well as for efficiency
in the learning process.
The prerequisite for the text is limited to a standard calculus based
freshman course in mechanics and electricity and magnetism. The text is
an outgrowth of the Honors Physics III course which the writer had been
frequently teaching at Rensselaer since 1973. He believes that some selection
according to the purpose of the course will need to be excersised. Only in this
way will the first six chapters of this extended edition fit comfortably within
a standard fourteen week semester. Selection is facilitated by the fact that
the more advanced topics occur towards the end of each chapter.
Alternatively, following a suggestion of Professor Lyle Roelofsl , physics
departments might consider offering Chap. 1, Mathematical Foundations, in
a mini semester, if there is one available.
Saarbriicken, Stephen N ettel
June 2002
1 L. Roelofs, Book Reviews, D.J. Griffiths Ed., Am. J. f>hys. 69, 922 (2001).
Preface to the Second Edition
A number of examples and problems to elucidate basic concepts have been
added, and typographic errors corrected.
The first edition has now been used a number of times at Rensselaer in
second year courses using the interactive method of teaching. This method
includes regular problem-solving sessions where students work together in
groups with aid from special work sheets. There is input from more senior
students, graduate and undergraduate, acting as tutors. It was discovered
that with this method Wave Physics can be used by a wider selection of
individuals to advantage than the honors students for whom the text was
originally intended. The main factor in a student's success appeared to be
the quality of his or her mathematical preparation. It is a pleasure to thank
the many students who participated as tutors. Special thanks go to Howard
Goldowsky, Byong Kim, and Richelle Thompson who carried much of the
responsibility over the various classes. Our teaching experience has influenced
the present revision.
Troy, Stephen N ettel
August 1994
Preface to the First Edition
This is a text for the third semester of undergraduate physics for students
in accelerated programs who typically are preparing for advanced degrees
in science or engineering. The third semester is often the only opportunity
for physics departments to present to students who are not physics majors
a coherent background in the physics of waves required later for confident
handling of applied problems, especially applications based on quantum
mechanics.
Physics is an integrated subject. It is often found that the going gets easier
as one goes deeper, learning the mathematical connections tying together
the various phenomena. Even so, the steps that took us from classical wave
physics to Heisenberg's "Physical Principles of Quantum Theory" were, as
a matter of history, harder to take than later steps dealing with detailed
applications. With these considerations in mind, the classical physics of os
cillations and waves is developed here at a more advanced mathematical
level than is customary in second year courses. This is done to explain the
classical phenomena, but also to provide background for the introductory
wave mechanics, leading to a logical integration of the latter subject into
the presentation. The concluding chapters on nonlinear waves, solitons, and
chaos broaden the previously established concepts of wave behavior, while
introducing the reader to important topics in current wave physics.
The text begins with a full chapter of mathematics that develops Fourier
analysis in the context of generalized functions. The idea is to expose students
to the new concepts while there is yet a minimum of "pressure", and then al
low them to assimilate the concepts through the chapters on classical physics.
In this way, when it comes to the wave mechanics in Chap. 6, students should
be able to distinguish the new physics from mathematics. M. J. Lighthill, in
his monograph "Fourier Analysis and Generalized Functions" , says that "The
theory of generalized functions ... greatly curtails the labor of understanding
Fourier transforms". Application of generalized functions (Schwartz' "theorie
des distributions") avoids undefined integrals like J~oo eikx dk still prevalent in
texts on quantum mechanics. Chapter 2, on oscillations, may seem somewhat
in the nature of a review. The physics material thus starts at the point where
the first exposure in the freshman year has been made, but may have led to
only a partial grasp.
X Preface
It is hoped that this book will be useful to various categories of readers,
honors math and physics students, graduate students studying for Ph.D.
qualifiers, as well as professors looking for examination problems. Problems
displace tedium by presenting a challenge. Here many are used to establish
significant results. This makes for meaningful activity, as well as for efficiency
in the learning process.
The prerequisite for the text is limited to a standard calculus based
freshman course in mechanics and electricity and magnetism. The text is
an outgrowth of the Honors Physics III course which the writer has been
frequently teaching at Rensselaer since 1973. He has found that the first
six chapters can be taught comfortably within a standard fourteen week
semester.
Troy, Stephen N ettel
February 1992
Acknowledgements
It is a pleasure to express my gratitude to Dr. Ernst F. Hefter, the original
editor at Springer-Verlag, whose interest has made this text a reality. I have
had the benefit of some hitherto unpublished work of Dr. Hefter's on solitons,
contributing to the presentation of nonlinear waves in Chap. 7. Thanks are
likewise due to Dr. Natasha Aristov of Springer-Verlag for her critical reading.
The writer is much indebted to Dr. Hans J. K6lsch, the present editor, for his
advice and encouragement during the preparation of this, the third, edition.
The contributions of Professor Martin C. Gutzwiller and of Professors
A. V. Gaponov-Grekhov and M.1. Rabinovich are deeply appreciated.
Professor G.B. Whitham of Caltech was kind enough to send me some un
published notes on inverse scattering in soliton theory. Professors P. Banderet
and H. Beck at the University of Neuchatel, in Switzerland, were generous
with various help. The "Further Reading" at the end of each chapter is an
indication of sources.
The honors students at Rensselaer Polytechnic Institute who have taken
the course which led to this text have contributed by their scientific curiosity
as by their tolerance. Later, it was discovered that Wave Physics can be
used to advantage by a wider selection of individuals than honors students
when using an interactive method of teaching. Special thanks go to Howard
Goldowsky, Byong Kim, and Richelle Thompson who carried much of the
responsibility for the various classes.
The idea of writing a text came at the time when I had been initiated into
'Ifanscendental Meditation, which imparted the desire to add to my profes
sional activity in some creative way. Lastly, I express gratitude to Irmgard
Blandfort for her moral support, and for taming our computer from time to
time.