Author | Ingram Bloch | |

Isbn | 9781489900524 | |

File size | 20.9MB | |

Year | 2013 | |

Pages | 318 | |

Language | English | |

File format | ||

Category | physics |

The Physics of
OSCILLATIONS
and WAVES
Ingram Bloch
THE PHYSICS OF
OSC ILLA TION S
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
DOI 10.1007/978-1-4899-0050-0
ISBN 978-1-4899-0050-0 (eBook)
1. Electricity--Mathematics.
4. Waves.
I. Title.
OC522.B58 1997
530.4'16--DC21
2. Mechanics.
3. Oscillations.
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 describable 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 oscillations 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 arbitrary 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 oscillations 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 nonlinear waves (in Chapter 17).
Besides trying to keep the geometry simple, I have made no attempt 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 physics-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 numbers. 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 reformulate 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 adequately 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
2 The Effect of Damping
3 Sinusoidally Driven Oscillators and Circuit Loops
4 Sums of Sinusoidal Forces or EMF's-Fourier Analysis
5 Integration in the Complex Plane
6 Evaluation of Certain Fourier Integrals-Causality,
Green's Functions
1
21
33
51
73
88
7 Electrical Networks
104
8 Nonlinear Oscillations
9 Coupled Oscillators without Damping-
120
lO
11
l2
l3
Lagrange's Equations
Matrices-Rotations-Eigenvalues and EigenvectorsNormal Coordinates
Some Examples of Normal Coordinates
Finite One-Dimensional Periodic Systems, Difference
Equations
Infinite One-Dimensional Periodic SystemsCharacteristic Impedance
151
161
181
197
216
ix
Contents
X
14 Continuous Systems, Wave Equation, Lagrangian Density,
Hamilton's Principle
15 Continuous Systems, Applied Forces, Interacting Systems
16 Other Linear Problems
17 Nonlinear Waves
230
251
267
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 differential equations. The treatment proceeds from simpler to
more complicated systems and uses successively more sophisticated 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 xaxis and is subject to a force toward the origin that is proportional 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
2
Oscillations and Waves
(1.3)
X+ Ol~X = 0.
Such a system, with the equation of motion 1.2 or 1.3, is known as
an undamped, undriven harmonic oscillator.
A force like that in Equation 1.1 can be produced by any object
that is described by Hooke's law-for instance, a coil spring on
which the mass is hung-provided the origin is taken to be the
position of the mass at equilibrium. More generally, any system
in which a mass that moves in one dimension has a position of
stable equilibrium can be approximately described by the foregoing equations, if the mass has a potential energy that is an
analytic function of x in some neighborhood of the equilibrium
point. Letting x 0 be the point of stable equilibrium, we require
that the potential energy U(x) have vanishing derivative (condition for equilibrium) and positive second derivative (condition
for stability), at x0 :
U'(x 0 ) = 0;
U"(x 0 )
(1.4)
> 0.
The function U(x), assumed analytic in a neighborhood ofx0 ,
can be represented within that neighborhood by a Taylor series
aboutx0 :
U(x) = U(x 0 )
+ (x- x 0)U'(x 0) + Mx- Xo) 2U"(x 0) + . . . (1.5)
In studying small departures of the mass from its equilibrium
position, one is dealing with small values oflx - x 0 l and can approximate the force acting on the mass by dropping the cubic
and higher terms from the series 1.5. In that case, within the domain of analyticity of U(x), the potential energy can be approximated as
U(x) ~ U(x 0 )
+ !k(x
- x 0) 2,
(1.6)
where we have omitted the vanishing first derivative and have replaced the second derivative of U by the positive constant k.
Now, using the relationship between potential energy and force,
we have
Undamped and Ondriven Osclllators and LC Circuits
3
X
Figure 1.1. Simple pendulum swinging in a plane.
F
= -dU/dx = -k(x
- x 0 ).
(1.7)
In order to reproduce Equation 1.1, we have only to place the
point of equilibrium at the origin, x 0 = 0, or, equivalently, to
choose x - x 0 as a new variable, which choice does not affect the
second time derivative in Equations 1.2 and 1.3.
The most familiar example of a system for which Equation 1.2
or Equation 1.3 is an approximate equation of motion, valid for
small excursions from equilibrium, is the simple pendulum
swinging in a plane (see Figure 1.1). Here x is taken to be
measured along the arc on which the pendulum bob moves, with
the origin at the position of equilibrium. Thus, if R is the length
of the string and 9 is the angle of the string from the vertical,
x = R9, or 9 = x/R. The potential energy of the mass m is
U = mgh, whereg is the acceleration of gravity and his the height
of the mass above an arbitrary point at which U = 0. Taking that
point to be x = 0 (the bottom of the swing), we have U(x0) =
U(O) = 0, and
U(x)
or
= mgR(l
-cos 9) ;;:;; mgR9 2/2
= mgx 2/2R,
Oscillations and Waves
4
k
= mg/R,
(1.8)
and
F
~
-mgx/R = -mg9.
Thus wij = kim = g/R, and Equation 1.3 becomes
x + gx/R = 0.
(1.9)
It is easy to verify that this approximate equation of motion for
small swings of the pendulum is the same one that follows from
the familiar resolution of the vertical force mg into components
along the string and along the arc; the latter component is -mg
sin 9, which is, for small 9, the force in Equation 1.8.
A second example of the small-excursion approximation is illustrated in Figure 1.2. Here the mass m is constrained to move
(without friction) on a straight rod that lies along the x-axis. Attached to the mass is one end of a spring of unstretched length S0
and negligible mass; the other end of the spring is attached to a
ftxed point on the y-axis, (0, D), where D
>
8 0 (the spring is
always stretched).
A spring obeying Hooke's law, stretched from its unstretched
length S0 to a greater length S, exerts a restoring force proportional to S - S0 :
Fs(S) = -ko(S- So),
(1.10)
where k0 is known as the spring constant. Thus the energy stored
in the spring, equal to the work required to stretch it, is
U
= - f.s Fs(S')dS' = !ko(S- So) 2•
so
(1.11)
Here we are assuming that there is no dissipation in the stretching and relaxation of the spring, so all the work done in stretching it can be recovered when the spring relaxes. In this case, the
work U plays the role of potential energy of the mass m; it can be
expressed in terms of x by means of
Undamped and Undriven Oscillators and LC Circuits
5
ftxed support
l~x
Figure 1.2. Mass oscillating on a line with a restoring force produced by a
spring.
(1.12)
whence
(1.13)
Expanding the square root by the binomial theorem for the case
r < JJ2, and dropping all terms past that in x 2, we find that
(1.14)
6
Oscillations and Waves
where
k
= ko(l
(1.15)
- S 0 /D).
U0 is the energy stored in the spring at its minimum length D,
which is its length when x = 0. The effective spring constant k is
related to the actual spring constant k0 by the factor 1 - SofD,
and thus k is always less than k0, becoming equal to it only in the
limit of infinite D (before which any real spring would have been
stretched beyond its elastic limit). If D is only slightly greater
than S0, k is very small and there is danger that the Hooke's law
approximation, for the force accelerating the mass, may break
down. It will be valid only if the maximum value ofx 2 is small enough, and the largest allowable value of r depends on how
much larger D is than S0• The condition for the x 2 term in U to be
much larger than the next term (in .0) is
(1.16)
as one may verify by calculating the x 4 term in the binomial
series. IfD

Author Ingram Bloch Isbn 9781489900524 File size 20.9MB Year 2013 Pages 318 Language English File format PDF Category Physics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare 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). Download (20.9MB) Laser Beam Propagation In Nonlinear Optical Media Magnetic Anisotropies in Nanostructured Matter Dynamics for Engineers Space, Time and Matter Magnetization Oscillations and Waves Load more posts