Table Of ContentStephen Nettei· Wave Physics
Stephen Nettel
Wave Physics
Oscillations - Solitons - Chaos
With 67 Figures
Springer-Verlag Berlin Heidelberg GmbH
Professor Stephen NetteI, Ph. D.
Rensselaer Polytechnic Institute, Department of Physics
Troy, NY 12180-3590, USA
Contributors to Chapter 8:
Professor Dr. Andrei V. Gaponov-Grekhov
Professor Dr. Mikhaill. Rabinovich
Institute of Applied Physics, Russian Academy of Sciences
ul. Ulyanova 46
603 600 Nizhnii Novgorod, Russia, CIS
Professor Dr. Martin C. Gutzwiller
IBM T. J. Watson Research Center
P. O. Box 218
Yorktown Heights, NY 10598, USA
ISBN 978-3-540-53295-8 ISBN 978-3-662-02825-4 (eBook)
DOI 10.1007/978-3-662-02825-4
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© Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg New York in 1992
Softcover reprint of the hardcover 1s t edition 1992
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Preface
This is a text for the third semester of undergraduate physics for students in accel
erated programs who typicaHy are preparing for advanced degrees in science or
engineering. The third semester is often the only opportunity for physics depart
ments to present to those of these 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 vari
ous 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 his
tory, harder to take than later steps dealing with detailed applications. With these
considerations in mind, the classical physics of oscillations and waves is devel
oped 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 fuH chapter of mathematics that develops Fourier anal
ysis 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 allow
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 general
ized functions ... greatly curtails the labor of understanding Fourier transform".
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 weH
as professors looking for examination problems. Problems displace tedium by
VI Preface
presenting achalienge. Here many are used to establish significant results. This
makes for meaningful activity, as weIl 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 com
fortably within a standard fourteen week semester.
Troy, February 1992 Stephen Nettel
Acknowledgements
It is a pleasure to express my gratitude to Dr. Ernst F. Hefter, my editor at Springer
Verlag, whose interest has made this text a reaIity. I have had the benefit of some hitherto
unpublished work of Dr. Hefter's on solitons, contributing to the presentation of nonIinear
waves in Chap.7. Thanks are likewise due to Dr. Natasha Aristov of Springer-Verlag for
her criticaI reading of the text. The technical typist is Charmaine Darmetko. I am grateful to
Mrs. Darmetko, especially for her confidence since the outset when the task ahead loomed
so large.
The contributions of Professor Martin C. GutzwiIIer and of Professors A. V. Gaponov
Grekhov and M. I. Rabinovich are deeply appreciated.
Professor G.B. Whitham of CaI Tech was kind enough to send me some unpublished
notes on inverse scattering in soliton theory. Professors P. Banderet and H. Beck at the
University of Neuchätel, 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.
The idea of writing a text came at the time when I had been initiated into TranscendentaI
Meditation, which imparted the desire to add to my professional activity in some creative
way. Lastly, I express gratitude for encouragement by my late parents and by my older
brother.
Contents
1. Mathematical Foundation for Wave Physics ................... 1
1.1 Introduetion to the Coneepts ............................. 1
1.2 Important Integrals ..................................... 5
1.3 Complex Numbers ..................................... 7
1.4 The Dirae Delta Funetion ............................... 9
1.5 Fourier Analysis ....................................... 11
Problems .................................................. 22
Further Reading ............................................ 27
2. Oscillations of Mechanical and Electrical Systems .............. 29
2.1 The Systems and Their Equations ......................... 29
2.2 Natural Motion of Systems .............................. 32
2.3 Forced Motion ........................................ 36
2.4 Oseillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Summary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Problems .................................................. 44
Further Reading ............................................ 47
3. Waves on Stretched Strings ................................. 49
3.1 Equation of Motion of a String ........................... 49
3.2 Natural Motion of a String .............................. 52
3.3 The Normal Modes .................................... 54
3.4 Forced Motion of a Stretehed String ...................... 61
Problems .................................................. 65
Further Reading ............................................ 70
4. Electromagnetic Waves ..................................... 71
4.1 Maxwell's Equations in Integral Form ..................... 71
4.2 Maxwell's Equations in Differential Form .................. 72
4.3 Plane Eleetromagnetie Waves in Free Spaee ................ 76
4.4 Distributed Eleetromagnetie Systems - Cavities ............. 82
4.5 The Veetor Potential and Related Solutions
of Maxwell's Equations ................................. 86
4.6 Dipole Radiation ...................................... 91
X Contents
Problems .................................................. 95
4.A Appendix ............................................ 103
Further Reading ............................................. 105
5. Light - Physical Optics, Refraction ........................... 107
5.1 The Nature and Generation of Light ....................... 107
5.2 Diffraction ........................................... 111
5.3 X-ray Diffraction ...................................... 117
5.4 EM Waves in Dielectrics. Refraction ...................... 119
Problems .................................................. 125
5.A Appendix ............................................ 130
Further Reading ............................................ 134
6. Wave Mechanics ........................................... 135
6.1 Origin of Schrödinger's Wave Equation .................... 135
6.2 Postulates of Wave Mechanics ........................... 139
6.3 Motion of a Free Particle.
The Heisenberg Uncertainty Principle ..................... 149
6.4 Wave-Particle Duality and Loss of Determinism ...... . . . . .. 157
6.5 Scope of Quantum Mechanics ............................ 159
Problems .................................................. 165
Further Reading ............................................ 173
7. Nonlinear Waves on Water - Solitons ......................... 175
7.1 Linear Surface Waves on Water .......................... 176
7.2 Dispersion. Group Velocity .............................. 179
7.3 Nonlinear Waves ...................................... 181
7.4 Solitons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 187
7.5 Inverse Scattering ...................................... 191
Problems .................................................. 199
Further Reading ............................................ 205
8. Nonlinear Phenomena - Chaos ............................ " 207
8.1 Nonlinear Physics - Chaos and Order
By Andrei V. Gaponov-Grekhov and Mikhail 1. Rabinovich ... 208
Further Reading ....................................... 221
8.2 Quantum Chaos
By Martin C. Gutzwiller ................................ 222
Further Reading ....................................... 232
Hints for Solution ............................................. 233
Subject Index 243
1. Mathematical Foundation for Wave Physics
Abstract. The chapter begins with an introduction to the main concepts in gener
alized functions theory. The development of the mathematics of wave physies itself
begins by evaluating certain essential integrals involving gaussians. Abrief section
on complex variables yields additional integrals. After its intuitive introduction,
the Dirac delta function is shown to be an example of a generalized function,
that is, it is given by a certain family of equivalent sequences. The members of
such sequences must be good functions, for example, gaussians. The designations
"equivalent", "good" are carefully defined. The Fourier representation as a decom
position of functions of spatial variables into harmonie functions of all possible
wavelengths is introduced. Using the analogy of basis vectors in three dimensions,
the idea is developed that harmonie waves form a complete basis in an infinite
dimensional function space (Hilbert space). The formula for the Fourier transform
is obtained in analogy to projecting out vector components, the Dirac delta func
tion serving as the analogue of the Kronecker delta. Worked examples include the
Fourier transform of a square wave, the computer synthesis of the square wave
from its transform, and orthonormality of the Dirac delta functions. Problem 1.17,
which develops the Green's function for diffusion, is partieularly recommended
as background for Chap. 6, "Wave Mechanics".
1.1 Introduction to the Concepts
Waves play a pervasive role in nature. There are the mechanieal waves, includ
ing seismic waves, sound waves in air, water waves. There are electromagnetie
waves, and underlying all matter, quantum mechanieal waves. These diverse wave
phenomena are understood on the basis of a few unifying mathematical concepts.
In the study of particle mechanies we use vectors to describe displacements,
velocities, and other quantities of motion. In three dimensions any vector F can
be constructed from three building blocks, the standard unit vectors i,j, k:
Al, A2, A3 being the component amplitudes whieh determine F. The correspond
ing building blocks for waves are, in one dimension, sin kx, cos kx, if we are
