Vibrations and Waves in Physics

362 Pages·1993·22.588 MB·English

by  Iain G. Main

Preface to the first edition
It would be generally agreed that an undergraduate course on vibrations
and waves should lead the student to a thorough understanding of the
basic concepts, demonstrate how these concepts unify a wide variety of
familiar physics, and open doors to some more advanced topics on which
they shed light. The fundamental ideas can be introduced with reference
to mechanical systems which are easy to visualize and to illustrate; less
tangible phenomena can then be treated with the same mathematics,
leaving one free to concentrate on more interesting problems of formula-
tion and interpretation.
In such a course there is always a risk that springs and strings will take
over completely, submerging the real physics. The theory must be
developed with some care and thoroughness if the student is to com-
prehend it in sufficient depth to be able to apply it readily, and all too
often the physics has to be left half baked. This textbook is an attempt to
provide, in a volume of moderate size, an account which is systematic and
coherent, but which also treats the physical examples in some depth.
The diagram on p. viii shows the plan of the book, and indicates the
relative weights attached to fundamental and illustrative material. The
theoretical development (indicated by the downward flow through the
toned areas on the left) is continuous, but is punctuated by regular
excursions into physics. These excursions are made as soon as the
necessary theoretical ground has been cleared, but there is no reason why
the order should not be varied by any reader who wishes to press on with
the theory first.
The two kinds of material are treated in distinct ways. The prototype
systems of the expository chapters are true models, endowed with the
properties that we choose for them, and equations predominate over
xiii
Downloaded from https:/www.cambridge.org/core. UCL, Institute of Education, on 05 Feb 2017 at 01:56:02, subject to the Cambridge Core terms of
use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139170567.001
xiv Preface to the first edition
numbers. The physics sections, by contrast, give due emphasis to the sizes
of the quantities involved. One of the things we find here is that a great
deal of physics is 'extreme', showing much simpler behaviour than the
general equations would suggest. A few order-of-magnitude estimates
may reveal that we are dealing with a very heavily damped system, a mass
controlled vibration or a shallow-water wave, and in such a case we can
often achieve a fairly advanced understanding of the phenomenon with-
out detailed analysis. This approach is particularly valuable in the case of
those quantum phenomena, such as the scattering of light and microwave
dispersion in water, which can be illuminated by a semi-classical discus-
sion.
Although my hope is that the book can be read with profit at any time
during a first-degree course, the mathematics background of a typical
entrant to a physics-based course has been kept in mind. A working
knowledge of calculus and trigonometry is assumed, and the reader is
presumed to be learning, if he has not already learnt, how to tackle the
solution of the simpler differential equations and the most elementary
facts of complex number algebra. The notation of vector algebra (but not
of vector calculus) is called upon in a few places.
Central to any discussion of wave motion is the problem of what to call
the propagation vector k and its magnitude 2TT/A. The best solution
seems to be to use as few names as possible. I have therefore called k the
wavevector, using the same name for the signed scalar k which replaces it
in one-dimensional problems. I avoid terms such as wavenumber, which
in different hands can mean I/A (sometimes denoted by k) or 2TT/A, and
merely draw these pitfalls to the attention of any reader for whom this is
only one of several textbooks in use at once.
Any attempt to bring together different branches of physics must lead
to the multiple use of symbols. In this book R, for example, has to do duty
for resistances, reflection coefficients, the molar gas constant and the
bond length of a diatomic molecule, not to mention a response function
R (a)). In most cases there is no overlap and, to avoid introducing a host of
strange characters or cluttering the landscape with suffixes, I have usually
adopted the most familiar symbol. An exception occurs in chapter 17,
where k (the wavevector) and k (the Boltzmann constant) meet; at that
point the latter becomes k . I have not provided a table of symbols and
B
their various uses, which could only instil unnecessary anxiety.
Numerical answers are provided for all problems requiring them. They
are quoted with a precision (usually two significant figures) which is
consistent with that of the data. At first the student will repeatedly find
that his answers are 'more accurate' than the ones at the back of the book.
Downloaded from https:/www.cambridge.org/core. UCL, Institute of Education, on 05 Feb 2017 at 01:56:02, subject to the Cambridge Core terms of
use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139170567.001
Preface to the first edition xv
But considerations of precision should be in the back of a physicist's mind
at all times, not just in the laboratory. Numerical problems provide an
opportunity, which the pocket-calculator revolution has enhanced, for
the student to develop these reflexes.
The suggestion that I write this book came from Professor J. M.
Cassels. At one time it was intended that he should be a co-author, and we
worked very closely on most of the material which now forms the first half
of the book. That material has been re-worked twice since other calls
upon his time obliged him to withdraw from the enterprise, but many of
the ideas, and the more incisive turns of phrase, are his. He taught me
much, not least about the elimination of what Thomas Young's anony-
mous critic in the Edinburgh Review for 1803 called a 'vibratory and
undulatory mode of reasoning'.
I am deeply indebted to my colleague Dr M. F. Thomas, who prepared
detailed comments on two drafts of the book and worked all the prob-
lems. I also received valuable guidance from Professor M. M. Woolfson.
My family gave cheerful encouragement, and advice on how to draw
springs, and the professionals of Cambridge University Press displayed
generous open-mindedness in the face of my amateur ideas on book
design.
I am grateful to the University of Liverpool for an eight-month period
of Study Leave which enabled me to accelerate the completion of a
project which had been under way for the preceding six years.
Liverpool I.G.M.
April 1977
Downloaded from https:/www.cambridge.org/core. UCL, Institute of Education, on 05 Feb 2017 at 01:56:02, subject to the Cambridge Core terms of
use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139170567.001
Preface to the second edition
Encouraged by the friendly reception given to the first edition, I have
preserved its basic form and most of the details. The new and revised
material occurs mainly in the latter half, on waves. There is one com-
pletely new chapter, intended to provide an elementary introduction to
the so-called solitary wave, or soliton, which has become such a pervasive
feature of physical science. It seemed to me that it should be possible to
base an explanation of the solitary wave on the simplest notions of
non-linear waves, and chapter 16 is my modest attempt. Consequential
changes were necessary in chapter 12, but I believe the outcome is a
more helpful treatment of dispersion, even for those who have no
immediate need of the solitary wave material. Apart from these and
other, smaller, changes, I have added some 30 new problems, the majority
of which introduce new physical examples of vibrations and waves.
Most of the work was done during a further period of Study Leave
from the University of Liverpool. Of the many people who made valuable
comments on the first edition, I am particularly grateful to my Liverpool
colleagues Professor J. R. Holt and Dr A. N. James, and their students.
A number of new diagrams were drawn for this edition by Roderick
Main.
Liverpool I.G.M.
September 1983
xvi
Downloaded from https:/www.cambridge.org/core. UCL, Institute of Education, on 05 Feb 2017 at 01:54:44, subject to the Cambridge Core terms of
use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139170567.002
Preface to the third edition
For this edition I have added one new topic, made a number of
miscellaneous changes intended to improve the presentation and clarify
some of the arguments, and adjusted a few emphases to attune with the
changing times. (Microwave ovens are now commonplace, while ballistic
galvanometers have been consigned to the laboratory basement.) To keep
the length reasonable I have relegated to the problems a few peripheral
topics, such as Lorentzians.
The teaching of vibrational physics at undergraduate level cannot ignore
for ever the revolutionary ideas grouped under the title 'chaos'. I believe
that a straightforward description of the physical phenomena will provide
the student with a platform from which to approach the more detailed, but
also more abstract, consideration of these matters in terms of trajectories
and attractors in phase space. I have therefore expanded chapter 7 to
include a simple discussion of large-amplitude forced vibrations, intro-
ducing chaotic vibrations and related non-linear effects.
Parkgate, Wirral I.G.M.
June 1992
xvu
Downloaded from https:/www.cambridge.org/core. University of Liverpool Library, on 05 Feb 2017 at 01:53:05, subject to the Cambridge Core
terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139170567.003