The Physics of Oscillations and Waves by Ingram Bloch


645885cf4171c6e-261x361.jpg Author Ingram Bloch
Isbn 9781489900524
File size 20.9MB
Year 2013
Pages 318
Language English
File format PDF
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

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