Course on Waves

7/27/2019 Course on Waves

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Plancks constant h

Reduced Plancks constant = h / i n

Electron rest meProton rest mp

G r a v i t a t i o n a l c o n s t a i f ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ S G

Acceleration of gravity at sea level g

Bohr radius a

Avogadros No

B o l t z m a n n s c o n s t a n l ^^^^^^^^^^^^^^S k

Standard temperatur^^^^^^^^^^^^^^^BTb

Standard p r e s s u r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ H poMolar volume at Vo

Thermal energy k Tat kT0

Density of air at po

Speed of sound in air at v0

Sound impedan ce of air at Z0

Standard sound i n t e n s i t ) ^^ ^ ^^ ^^ ^^ ^^ ^| l o

Factor of ten in intensity

One fermi (F)One angstrom unit ( A )One micron (ju)

One hertz (Hz)

Wavelength of one-electron-volt photon

One electron volt (ev)

One watt (W)

One coulomb (coul)

One volt (V)One ohm (fi)

Thirty ohms

Impedance per square of vacuum for

electromagnetic waves

One farad (F)

One henry (H)

Useful Constants2.997925 X 1010 cm/sec = 3 X 1010 cm/sec

4.8 X 10~10 statcoulomb

1.6 X 10~19 coulomb

6 .6 X 10 ~27 erg-sec

1.0 X 10 27 erg-sec

0.9 X 10~27 gm1.7 X 10-2 4 gm

6.7 X lO- 8 CGS units

980 cm /sec2

0.5 X 10-8 cm

6.0 X 1023 mole-1

1.4 X 10~16 erg/deg Kelvin

273 deg Kelvin

1 atm = 1.01 X 106 dyne/cm 222 .4 x 103 cm3/mole

3.8 X 10-1 4 erg ^ ev

1.3 X 10-3 gm/cm 3

3.32 X 104 cm/sec

42.8 (dyne/cm 2) /(cm/sec)

1 jnwatt/cm2

1 bel = 10 db

10-13 cm 10-8 cm

10~4 cm

1 cycle per second (cps)

1.24 X 104 cm ^ 1 23 45 A1.6 X 10-1 2 erg

1 joule/sec = 107 erg/sec

3 X 109 statcoul = c/ 1 0 s tat coul0

s f ostatvolt =; 108/ c statvolt1/(9 X 10n ) statohm = 109/ c 2 statohm

1/ c statohm

At t/c statohm = 377 ohm

9 X 1011 statfarad = c2/1 0 9 statfarad

1/ (9 X 1011) stathenry = 109/ c 2 stathenry

In converting from practical units to electrostatic units we have approximated the velocity of light as 3.00 X 1010 cm/sec. Wherever a 3

appears, a more accurate conversion factor can be obtained by using the more accurate value of c. Similarly wherever 9 appears, it is moreaccurately (2.998)2.


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Recommended Unit Prefi xes

M u l t i p l es and

Submu l t i p l es Pref i xes Symbols

1012 tera T109 giga G106 mega M103 kilo k

102 hecto h10 deka dalO"1 deci dio-2 centi clO"3 milli m10-6 microIO"9 nano nio-*2 pico P

Useful I dent i t i es

cos x +cos y= [2 cos (x t/)] cos \ (x+ y )

cos x cos y [2 sin %(x y)] sin (x + y)

sin x+ sin y= [2 cos (x y)] sin (x + y)

sin x sin y =[2 sin (x t/)] cos (x + y)

cos (x y)= cos x cos yqp sin x sin t/sin ( x y ) =sin x cos y sin ycos x

cos 2 x = cos2 x sin2 x

sin 2x = 2 sin x cos x

cos2 x = { 1 + cos 2 x)

sin2x = (1 cos 2x)

sin x = x c 3 +

cos x = 1 |x2 +

(1 + x)n = 1 + nx + n(nl)x2 + cos Oi+ cos (#i + y) + cos (0i + 2 y) +

; x2 < 1 .+ cos [0i + (N l)y] = cos [01 + %(N l)y]

sin %Ny

sin i y


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waves


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N e w Yo r k St . Lo u is

Sa n Fr a n c i s co

To r o n t o L o n d o n

S y d n e y

mcgraw -h i l l book com pany


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wavesb e r k e l e y p h y s i c s c o u r s e vo l u m e 3

Th e prepara t io n of th i s cour se teas suppor ted, hi j a grant

f r om the Na t i onal Sci ence Founda t i on t o Educa t i on D e-

ve l opmen t C en t er

Fra nk S. Craw ford,Jr .Pro f esso r o f P hys icsUn i ver si t y o f Ca l i f o r n i a , Be r kel e y


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COVER DESIGN

Photograph ic adapta t ion by Fel i x Cooper

f ro m an or i g ina l by John Severson

WAVES

Copyr i gh t 1965, 1966, 1968 by Educat i on

D eve lop m ent Cent er , I nc . (successor by m erger

to Educat io nal Servi ces In corporat ed). A l l

Ri ghts Reserv ed. Pr i n ted in the U ni t ed Sta teso f Am er i ca . Th i s book , o r pa r t s thereo f , may

no t be r ep r odu ced i n an y f o rm w i t h ou t t h e

w r i t t en perm i s si o n o f Educa t i o n D evel o pmen t

Center , In c ., N ew ton, M assachuset t s .

L i b ra r y o f Cong ress Ca ta log Card Number

6466016

04860

7890 BPBP 7654


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Preface to the Berkeley Physics Course

This is a two-year elementary college physics course for students majoringin science and engineering. The intention of the writers has been to pre

sent elementary physics as far as possible in the way in which it is used by

physicists working on the forefront of their field. We have sought to make

a course which would vigorously emphasize the foundations of physics.

Our specific objectives were to introduce coherently into an elementary

curriculum the ideas of special relativity, of quantum physics, and of sta

tistical physics.

This course is intended for any student who has had a physics course inhigh school. A mathe matics course including the calculus should be taken

at the same time as this course.

There are several new college physics courses under development in the

United States at this time. The idea of making a new course has come to

many physicists, affected by the needs both of the advancement of science

and engineering and of the increasing emphasis on science in elementary

schools and in high schools. Our own course was conceived in a conversa

tion between Philip Morrison of Cornell University and C. Kittel late in

196 1. We wer e encourage d by John Mays and his colleagues of the

National Science Foundation and by Walter C. Michels, then the Chair

man of the Commission on College Physics. An informal comm ittee was

formed to guide the course through the initial stages. The comm ittee con

sisted originally of Luis Alvarez, William B. Fretter, Charles Kittel, Walter

D. Knight, Philip Morrison, Edward M. Purcell, Malvin A. Ruderman, and

Jerrold R. Zacharias. The committee met first in May 196 2, in Berkeley;

at that time it drew up a provisional outline of an entirely new physics

course. Bec ause of heav y obligations of several of the original members,

the committee was partially reconstituted in January 1964, and now con

sists of the undersigned. Contributions of others are acknowledged in the

prefaces to the individual volumes.

The provisional outline and its associated spirit were a powerful influence

on the course materia l finally produ ced. The outline covered in detail the

topics and attitudes which we believed should and could be taught to

beginning college students of scien ce and engineering. It was never our

intention to develop a course limited to honors students or to students with adva nce d standing. We have sought to present the principles of physics

from fresh and unified viewpoints, and parts of the course may therefore

seem almost as new to the instructor as to the students.

The five volumes of the course as planned will include:

I. Mechanics (Kittel, Knight, Ruderm an) III. Waves (Crawford)

II. Electricity and Magnetism (Purcell) IV. Quantum Physics (Wichm ann)

V. Statistical Physics (Reif)

The authors of each volume have been free to choose that style and method

of presentation which seemed to them appropriate to their subject.


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The initial course activity led Alan M. Portis to devise a new elementary

physics laboratory, now known as the Berkeley Physics Laboratory. Because the course emphasizes the principles of physics, some teachers

may feel that it does not deal sufficiently with experimental physics. The

laboratory is rich in important experiments, and is designed to balance the

course.

The financial support of the course development was provided by the

National Science Foundation, with considerable indirect support by the

University of California. The funds wer e admin istered by Educational

Services Incorporated, a nonprofit organization established to administer

curriculum improv ement programs. We are particularly indebted to

Gilbert Oakley, James Aldrich, and William Jones, all of ESI, for their

symp athetic and vigorous support. ES I established in Berkeley an office

under the very competent direction of Mrs. Mary B. Maloney to assist the

development of the course and the laboratory. The U niversity of Califor

nia has no official connection with our program, but it has aided us in im

portan t ways. Fo r this help we thank in particula r two successive chair

men of the Department of Physics, August C. Helmholz and Burton J.

Moyer; the faculty and nonacademic staff of the Department; Donald

Con ey, and many others in the University. Abraham Olshen gave much

help with the early organizational problems.

A Fu r t h er No t e Volumes I, II, and V were published in final form in the period from Janu

ary 19 65 to June 19 67. During the preparation of Volumes III and IV for

final publication some organizational changes occurred. Education Devel

opment Center succeeded Educational Services Incorporated as the

administering organization. The re were also some changes in the com

mittee itself and some redistribution of responsibilities. The com mittee isparticularly grateful to those of our colleagues who have tried this course

in the classroom and who, on the basis of their experience, have offered

criticism and suggestions for improvements.

As with the previously published volumes, your corrections and sugges

tions will always be welcome.

January, 1965

Eugene D. Commins

Frank S. Crawford, Jr.

Walter D. Knight

Philip Morrison

Alan M. Portis

Edward M. Purcell

Frederick Beif

Malvin A. Buderman

Eyvind H. Wichmann

Charles Kittel, Cha i rman

June, 1968 Berkeley, California

Frank S. Crawford, Jr.

Charles Kittel

Walter D. Knight Alan M. Portis

Frederick Beif

Malvin A. Buderman

Eyvind H. WichmannA. Carl Helmholz 1

Edward M. PurcellJCha i rmen


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Preface to Volume I I I

This volume is devo ted to the study of waves. Tha t is a broad subject.

Eve ryon e knows many natural phenome na that involve waves there are

water waves, sound waves, hght waves, radio waves, seismic waves,

de Broglie waves, as well as other waves. Fur therm ore, perusal of the

shelves of any physics library reveals that the study of a single facet of wave

phenomenafor example, superson ic sound w aves in w a termay occupy

whole books or periodicals and may even absorb the complete attention of individual scientists. Amazingly, a professional specialist in one of these

narrow fields of study can usually communicate fairly easily with other

supposedly narrow specialists in other supposedly unrelated fields. He has

first to learn their slang, their units (like what a parsec is), and what num

bers are important. Indeed, when he experiences a change of interest, he

ma y becom e a narrow specialist in a new field surprisingly quickly. This

is possible because scientists share a common language due to the remark

able fact that many entirely different and apparently unrelated physical

phenomena can be described in terms of a common set of concepts. Many

of these shared concepts are implicit in the word w ave.

The principal objective of this book is to develop an understanding of

basic wave concepts and of their relations with one another. To that end

the book is organized in terms of these concepts rather than in terms of

such observable natural phenomena as sound, light, and so on.

A complementary goal is to acquire familiarity with many interesting and

important examples of waves, and thus to arrive at a concrete realization

of the wide applicability and generality of the concep ts. After each new

concept is introduced, therefore, it is illustrated by immediate application

to many different physical systems: strings, slinkies, transmission lines, mail

ing tubes, light beams, and so forth. This may be contrasted with the dif

ferent approach of first developing the useful concepts using one simple

example (the stretched string) and then considering other interesting

physical systems.

By choosing illustrative examples having geometric similitude with one

another I hope to encourage the student to search for similarities and analogies betw een different wave phenomena. I also hope to stimulate him

to develop the courage to usesuch analogies in hazarding a guess when

confron ted with new phenomena. The use of analogy has well-known

dangers and pitfalls, but so does everything. (The guess that light waves

might be just like mechanical waves, in a sort of jelly-like ether was

very fruitful; it helped guide Maxwell in his attempts to guess his famous

equations. It yielded interesting predictions. Wh en experiments espe

cially those of Michelson and Morley indicated that this mechanical model


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could not be entirely correct, Einstein showed how to discard the model yet

keep Maxwells equations. Einstein preferred to guess the equationsdirectly what might be called pure guesswork. Nowadays, although

most physicists still use analogies and models to help them guess new equa

tions, they usually publish only the equations.)

The home experiments form an importan t part of this volume. They can

provide pleasure and insight of a kind not to be ac quired through the

ordinary lecture demonstrations and laboratory experiments, important as

these are. The home experiments are all of the kitchen physics type, re

quiring little or no special equipment. (An optics kit is provided. Tuningforks, slinkies, and mailing tubes are not provided, but they are cheap and

thus not special. ) These experiments really aremeant to be done at

home, not at the lab. Many would be better termed demonstra t ionsrather

than experiments.

Every major concept discussed in the text is demonstrated in at least one

home experiment. Besides illustrating con cepts, the home experiments give

the student a chance to experience close personal contact with phe

nomena. Because of the home aspect of the experiments, the contact is

intimate and leisurely. This is importan t. There is no lab partn er who

may pick up the ball and run with it while you are still reading the rules of

the game (or sit on it when you want to pick it up); no instructor, explain

ing the meaning of hi sdemonstration, when what you really need is to per

form you rdemonstration, with your own hands, at your own speed, and as

often as you wish.

A very valuable feature of the home experiment is that, upon discovering

at 10 p .m . that one has misunderstood an experiment done last week,

by 10:15 p .m . one can have set it up once again and rep eate d it. This is

important. Fo r one thing, in real experimental work no one ever gets it

right the first time. Afterthoughts are a secre t of success. (There are

others.) Nothing is mo re frustrating or more inhibiting to learning than

inability to pursue an experimental afterthought because the equipment is

torn down, or it is after 5 p .m . , or some other stupid reason.

Finally, through the home experiments I hope to nurture what I may call

an appreciation of phenom ena. I would like to beguile the student into

creating with his own hands a scene that simultaneously surprises and delights his eyes, his ears, and his brain . . .

Clear-colored stones

are vibrating in the brook-bed . . .

or the water is.

S O S E K l f

t Reprinted from The Four Seasons (tr. Peter Beilenson), copyright 1958, by The Peter

Pauper Press, Mount Vernon, N.Y., and used by permission of the publisher.


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Acknow ledgments

In its preliminary versions, Vol. Ill was used in several classes at Berkeley.

Valuable criticisms and comments on the preliminary editions came from

Berkeley students; from Berkeley professors L. Alvarez, S. Parker, A. Portis,

and especially from C. Kittel; from J. C. Thompson and his students at the

University of Texas; and from W . Walker and his students at the University

of California at Santa Barbara. Extre m ely useful specific criticism was pro

vided by S. Pasternacks attentive reading of the preliminary edition.

Of particular help and influence were the detailed criticisms of W. Walker,

who read the almost-final version.

Luis Alvarez also contributed his first published experiment, A Simpli

fied Method for Determination of the Wavelength of Light, School Sci ence

and M a t h ema t i c s 32, 89 (1932), which is the basis for Home Exp. 9.10.

I am especially grateful to Joseph Doyle, who read the entire final

manuscript. His considered criticisms and suggestions led to many impor

tant changes. He also introduced me to the Japanese haiku that ends the

preface. He and another graduate student, Robert Fisher, contributed

many fine ideas for hom e experimen ts. My daughter Sarah (age 4^) and

son Matthew (2) not only contributed their slinkies but also demon

strated that systems may have degrees of freedom nobody ever thought of.

My wife Bevalyn contributed her kitchen and very much more.

Publication of early preliminary versions was supervised by Mrs. Mary R.

Maloney. Mrs. Lila Lowell supervised the last preliminary edition and

typed most of the final manuscrip t. The illustrations owe their final form

to Felix Cooper.

I acknowledge gratefully the contributions others have made, but final

responsibility for the manuscrip t rests with me. I shall welcom e any fur

ther corrections, complaints, compliments, suggestions for revision, andideas for new home experiments, which may be sent to me at the Physics

De partm ent, University of California, Berkeley, California, 94 720 . Any

home experiment used in the next edition will show the contributors

name, even though it may first have been done by Lord Rayleigh or

somebody.

F. S. Crawford, Jr.


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Teachi ng Notes

Traveling waves have great aesthetic appeal, and it would be tempting to begin with

them. In spite of their aesthet ic and mathem atical beauty, however, waves are physi

cally rather complicated because they involve interactions between large numbers of

particles. Since I want to emphasize physical systems rather than mathem atics, I

begin with the simplest physical system, rather than with the simplest wave.

Chap t er 1 Free O sci l la t i ons o f Sim p le Sys tems : W e first review the free oscillations

of a one-dimensional harmonic oscillator, emphasizing the physical aspects of inertia

and return force, th e physical meaning of


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W ha t to om i t : Sec. 2. 3 is optional especially if the students already know some

Fourier analysis. Example 5 (Sec. 2.4 ) is a linear array of coupled pendulums, the

simplest system having a low-fre quen cy cutoff. They are used later to help explain

the behavior of other systems that have a low-frequency cutoff. A teach er who does

not intend to discuss at a later time systems driven below cutoff (waveguide, iono

sphere, total reflection of hght in glass, barrier penetration of de Broglie waves, high-

pass filters, etc.) need not consider Example 5.

Chap t er 3 Forced Osci l la t i ons: Chapters 1 and 2 started with free oscillations of a

harmonic oscillator and ended with free standing waves of closed systems. In Chaps.

3 and 4 we consider forced oscillations, first of closedsytems (Chap. 3) where we

find resonances, and then in opensystems (Chap. 4) where we find traveling waves.

In Sec. 3.2 we review the damped driven one-dimensional oscillator, considering its

transient behavior as well as its stead y-state behavior. Then w e go to two or more

degrees of freedom, and discover that there is a resonance corresponding to every

mode of free oscillation. We also consider closed systems driven below their lowest

(or above their highest) mode frequency and discover exponential waves and filtering

action.

W ha t t o om i t : Transients (in Sec. 3.2) can be omitted. Some teach ers may also

wish to omit everything about systems driven beyond cutoff.

H om e experi ments: Home Exps. 3.8 (Forced oscillations in a system of two coupled

cans of soup) and 3.16 (Mechanical bandpass filter) require phonograph turntables.

They make excellent class demonstrations, especially of exponential waves for systems

driven beyond cutoff.

Chap t er 4 Trave l ing W aves : Here we introduce t rave l ingwaves resulting from

forced oscillations of an opensystem (contrasted with the s t a n d i n g waves resulting

from forced oscillations of a closedsystem that we found in Chap. 3). The remainder

of Chap. 4 is devoted to studying phase velocity (including dispersion) and impedance in traveling waves. We con trast the two traveling wave concepts, pha se ve loc i ty

and impedance, with the standing wave concepts, i n e r t i a and retu r n fo rce, and also

contrast the fundamental difference in phase relationships for standing versus traveling

waves.

H ome exper i ments : We recommen d Home Exp. 4.12 (Wate r prism). This is the

first optics kit experiment; it uses the purple filter (which passes red and blue but cuts

out green). We strongly recomm end Home Exp . 4.18 (Measuring the solar constant

at the earths surface) with your face as detector.

Chap t er 5 Re f lec t io n : By the end of Chap. 4 we have at our disposal both stand

ing and traveling waves (in one dimension). In Chap. 5 we consider gen eral super

positions of standing and traveling waves. In deriving reflection coefficients we make

a very physical use of the superposition principle, rather than emphasizing bound

ary conditions. (Use of boundary conditions is emphasized in the problems.)

W ha t to om i t : There are many examples, involving sound, transmission lines, and

light. Don t do them all! Cha pter 5 is essentially the application of what we have

acquired in Chaps. 1 -4 . Any or all of it can be omitted.

H ome exper i ments : Everyone should do Home Exp. 5.3 (Transitory standing waveson a slinky). Home Exps. 5 .17 and 5.18 are especially interesting.


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Chap t er 6 M odu la t ions , Pu lses, and W ave Packets : In Chaps. 1-5 we work mainly

with a single frequency w (exc ept for Sec. 2.3 on Fourie r analysis). In Chap. 6 we

consider superpositions, involving different frequencies, to form pulses and wave

packets and to extend the concepts of Fourier analysis (developed in Chap. 2 for periodic functions) so as to include nonperiodic functions.

W ha t t o om i t : Most of the physics is in the first three sections. A teach er who

has omitted Fourier analysis in Sec. 2.3 will undoubtedly want to omit Secs. 6.4 and

6.5, where Fourier integrals are introduced and applied.

H om e experim ent s: No one believes in group velocity until they have watched water

wave packets (see Home Exp. 6. 11). Everyone should also do Home Exps. 6.1 2 and

6.13.

Problems: Frequency and phase modulation are discussed in problems rather than

in the text. So are such interesting recen t developments as Mode-locking of a laser(Prob. 6.23), Frequency multiplexing (Prob. 6.32), and Multiplex Interferometric

Fourier Spectroscopy (Prob. 6.33).

Chap te r7 W aves in Tw o and Three D im ens ions : In Chaps. 1-6 the waves are all

one-dimensional. In Chap. 7 we go to three dimensions. The propagation vecto r k

is introduced. Elec trom agn etic waves are studied using Maxwells equations as the

starting point. (In earlier chapters there are many examples of electrom agnet ic waves

in transmission lines, evolving from the LC -circu it example.) W ate r waves are alsostudied.

W ha t t o om i t : Sec. 7.3 (Water Waves) can be omitted, but we recommend the

home experiments on wat er waves wheth er or not Sec. 7.3 is studied. A teacher mainly

interested in optics could actually start his course at Sec. 7.4 (Electromagnetic Waves)

and continue on through Chaps. 7, 8, and 9.

Chap t er 8 Po la r i z a t i o n : This chapte r is devoted to study of polarization of electro

magnetic waves and of waves on slinkies, with emphasis on the physical relation between partial polarization and coherence.

H ome exper i ments : Everyone should do at least Home Exps. 8.12, 8.14, 8.16, and

8.18 (Exp. 8.14 requiring slinky; the others, the optics kit).

Chap t er 9 In t er f e rence and D i f f r a c t i o n : Here we consider superpositions of waves

that have traveled different paths from source to detec tor. We emphasize the physical

meaning of coherence. Geometrical optics is treated as a wave phenomen on the

behavior of a diffraction-limited beam impinging on various reflecting and refracting

surfaces.

H ome exper i men ts : Everyone should do at least one each of the many home ex

periments on interference, diffraction, coherence, and geometrical optics. We also

strongly recommend 9.50 (Quadrupole radiation from a tuning fork.)

Problems: Some topics are developed in the problems: Stellar interferometers, in

cluding the recently developed long-base-line interferometry (Prob. 9.57); the

analogy between the phase-contrast microscope and the conversion of AM radio

waves to FM is discussed in Prob. 9.59.


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Genera l r emar ks : At least one home experimen t should be assigned per week. For

your convenience we list here all experiments involving water waves, waves in

slinkies, and sound waves. We also late r describe the optics kit.

W ater wav es : Discussed in Chap. 7; in addition they form a recurring theme devel-

oped in the following series of easy home experiments:

1.24 Sloshing modes in pan of water

1.25 Seiches

2.31 Sawtooth shallow-water standing waves

2.33 Surface tension modes

3.33 Sawtooth shallow-water standing waves

3.34 Rectangular two-dimensional standing surface waves on water3.35 Standing waves in water

6.11 Water wave packets

6.12 Shallow-water wave packetstidal waves

6.19 Phase and group velocities for deep-water waves

6.25 Resonance in tidal waves

7.11 Dispersion law for water waves

9.29 Diffraction of water waves.

Sl ink ies: Ev ery student should have a Slinky (about $1 in any toy store). Fo ur of

the following experiments require a record-player turntable and are therefore outside

the kitchen physics cost range. However, many students already have record

players. (The experiments involving recor d players make good lectu re demonstrations.)

1.8 Coupled cans of soup

2.1 Slinkydependence of frequency on length

2.2 Slinky as a continuous system

2.4 Tone quality of a slinky

3.7 Resonance in a damped slinky (turntable required)

3.8 Forced oscillations in a system of two coupled cans of soup (turntable required)

3.16 Mechanical bandpass filter (turntable required)3.23 Exponential penetration into reactive region (turntable required)

4.4 Phase velocity for waves on a slinky

5.3 Transitory standing waves on a slinky

8.14 Slinky polarization

Sound : Many home experiments on sound involve use of two identical tuning forks,

preferably C52 3.3 or A440. The cheapest kind (about $1.2 5 each), which are

perfectly adequ ate, are available in music stores. Mailing tubes can be purchased for

about 25 0 ea ch in stationery or art-supply stores. The following home experimentsinvolve sound:

1.4 Measuring the frequency of vibrations

1.7 Coupled hacksaw blades

1.12 Beats from two tuning forks

1.13 Nonlinearities in your earcombination tones

1.18 Beats between weakly coupled nonidentical guitar strings

2.4 Tone quality of a slinky

2.5 Piano as Fourier-analyzing machineinsensitivity of ear to phase

2.6 Piano harmonicsequal-temperament scale3.27 Resonant frequency width of a mailing tube


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4.6 Measuring the velocity of sound with wave packets

4.15 Whiskey-bottle resonator (Helmholtz resonator)

4.16 Sound velocity in air, helium, and natural gas

4.26 Sound impedance5.15 Effective length of open-ended tube for standing waves

5.16 Resonance in cardboard tubes

5.17 Is your sound-detecting system (eardrum, nerves, brain) a phase-sensitive detector?

5.18 Measuring the relative phase at the two ends of an open tube

5.19 Overtones in tuning fork

5.31 Resonances in toy balloons

6.13 Musical trills and bandwidth

9.50 Radiation pattern of tuning forkquadrupole radiation

Componen ts : Four linear polarizers, a circular polarizer, a quarter-wave plate, a

half-wave plate, a diffraction grating, and four color filters (red, green, blue, and

purple). The comp onents are described in the text (linear polarizer on p. 41 1; c ircu

lar polarizer, p. 433; quarter- and half-wave retardation plates, p. 435; diffraction

grating, p. 49 6). Some experiments also require microscop e slides, a showcase-lamp

line source, or a flashlight-bulb point source as described in Home Exp. 4.12, p. 217.

Aside from Exp. 4.12, all experiments requiring the optics kit are in Chaps. 8 and 9.They are too numerous to list here.

The first experiment involving the complete optics kit should be identification of all

the components by the student. (Compon ents are listed on the envelope container

glued to the inside back cove r.) Label the components in some way for future refer

ence. Fo r example, use scissors to round off slightly the four corne rs of the circular

polarizer, and then scratch IN near one edge of the input face or stick a tiny piece

of tape on that face. Clip onecorner of the one-quarter-wave retarder; clip tw ocorners

of the two-q uarter- (half-) wave retarder. Scratch a line along the axis of easy transmission on the linear polarizers. (This axis is parallel to one of the edges of the polarizer.)

We should rem ark that the quarter-wave plate gives a spatial retardation of

1400 2 0 0 A, nearly independent of wavelength (for visible light). Thus the wave

length for which it is a quarter-wav e retarde r is 560 0 800 A. The 2 0 0 A is

the manufacturers tolerance. A manufactured batch that gives retardation 140 0 A

is a quarter-wave retarder for green (5600 A), but it retards by less than one quarter-

wave for longer wavelengths (red) and more for shorter (blue). Another batch that

happens to give retardation 1400 + 2 0 0 = 1600 A is a quarter-wave retarder only for

red (6400 A). One that retards by 1400 2 0 0 is a quarter-wave plate only for blue

(4800 A). Similar remarks apply to the circular polarizer, since it consists of a sand

wich of quarter-wave plate and linear polarizer at 45 deg, and the quarter-wave plate

is a retarder of 1400 200 A. Thus there may be slightly distracting color effects

when using white light. The student must be warned that in any experim ent where

he is supposed to get black, i.e., extinction, he will always have some non-extin-

guished light of the w rong color leaking through. Fo r example, I was naive when

I wrote Home Exp. 8 .12. You should perhaps strike out everything after the word

band in the sentences Do you see the dark band at green? That is the color of5600 A!

Op t i cs K i t

Home experiment


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Use o f Compl ex N um bers Complex numbers simplify algebra when sinusoidal oscillations or waves are to be

superposed. They may also obscure the physics. Fo r that reason I have avoidedtheir use, especially in the first part of the book. All the trigonom etric identities that

are needed will be found inside the front cover. In Chap. 6 1 do make use of the complex

representation exp ioot, so as to use the well-known graphical or phasor diagram

method of superposing vibrations. In Chap. 8 (Polarization) I use complex quantities

extensively. In Chap. 9 (Interfere nce and Diffraction) I do not make much use of com

plex quantities, even though it would sometimes simplify the algebra. Many teachers

may wish to make much more extensive use of complex numbers than I do, especially

in Chap. 9. In the sections on Fourier series (Sec. 2.3 ) and Fourie r integrals (Secs.

6.4 and 6.5), I make no use of complex quantities. (I especially wanted to avoid Fourier

integrals involving negative frequencies!)

A Not e on the M K S Sys tem o f E lect r i ca l U n i t sf

t Reprinted from Berkeley Physics Course,

Vol. II, Electricity and Magnetism, by Ed

ward M. Purcell, 1963, 1964, 1965 by

Education Development Center, Inc., suc

cessor by merger to Educational Services

Incorporated.

Most textbooks in electrical engineering, and many elementary physics

texts, use a system of electrical units called the ra t i ona l i zed M K S system.

This system employs the MKS mechanical units based on the meter, the

k i l og ram, and the second. The MKS unit of force is the new t on , defined

as the force which causes a 1-kilogram mass to accelerate at a rate of

1 m ete r/se c2. Thus a newton is equivalent to exactly 1 05 dynes. The

corresponding unit of energy, the newton-meter, orj oul e, is equivalent to

107 ergs.

The electrical units in the MKS system include our familiar practical

units coulomb, volt, ampere, and ohm along with some new ones.

Someone noticed that it was possible to assimilate the long-used practicalunits into a com plete system devised as follows. Write Coulom bs law as

we did in Eq. 1.1:

F = k3132

*21(1)

Instead of setting kequal to 1, give it a value such that F2 will be given in

newtons if q1 and 92 are expressed in coulombs and r21 in meters. Know

ing the relation between the newton and the dyne, between the coulomb

and the esu, and between the meter and the centimeter, you can easily

calculate that k must have the value 0.8 98 8 X 1010. (Two 1-coulomb

charges a meter apart produce quite a forc e around a million tons!) It

makes no difference if we write 1 /( 4 weo) instead of k , where the constant

c0 is a number such that l / ( 477c0) = k= 0 .89 88 X 1 010. Coulombs law

now reads:

1 qiq2

^ 477o r2

with the constant c0 specified as

Co = 8.8 54 X 10 -1 2 coulomb2/n ew to n-m 2

(2)


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Separating out a factor l/4 w was an arbitrary move, which will have the

effect of removing the 477 that would appear in many of the electrical

formulas, at the price of introducing it into some others, as here in

Cou lomb s law. That is all that rationalized means. The constant e0

is called the dielectric constant (or permittivity) of free space.

Electric potential is to be measured in volts, and electric field strength E

in volts/m eter. The force on a charge q, in field E, is:

F (newtons) = r/E (coulombs X volts/meter) (4)

An ampere is 1 cou lom b/se c, of course. The force per me ter of length

on each of two parallel wires, r meters apart, carrying current Imeasured

in amperes, is:

f (ne wton s/m eter) = ( - g ) M (5 )

Recalling our CGS formula for the same situation,

/ ( dy nes /cm ) = ^ ( esu^ )Z21 (6 )r c2 (cm:y se c2)

we compute that (jiio/477) must have the value 10~7. Thus the constant jUo,called the permeability of free space, must be:

ju,0 = 477 X 10~7 newtons/amp2 (exactly) (7)

The magn etic field B is defined by writing the L ore ntz force law as follows:

F (newtons) = c/E + qvX B (8 )

where v is the velocity of a particle in me ters/s ec, qits charge in coulombs.

This requires a new unit for B. The unit is called a tesla, or a w eber /m2.One tesla is equivalent to precisely 104 gauss. In this system the auxiliary

field H is expressed in different units, and is related to B, in free space, in

this way:

B = ju0H (in free space) (9)

The relation of H to the free current is

J H ds = /free (10)

/free being the free current, in amperes, enclosed by the looparound which

the line integral on the left is taken. Since dsis to be measured in meters,

the unit for H is called simply, ampere /me te r .

Maxwells equations for the fields in free space look like this, in the

rationalized MKS system:

div E = p curl E

div B = 0 curl B = Moo

8B

31

SE

01+ M0J


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If you will com pare this with our G aussian CGS version, in which c appears

out in the open, you will see that Eqs. 11 imply a wave velocity l/\/oMo

(in me ters/ sec , of course). That is:

oMo = \ (12)c

In our Gaussian CGS system the unit of charge, esu, was established by

Coulombs law, with k= 1. In this MKS system the coulom b is defined,

basically, not by Eq. 1 but by Eq. 5, that is, by the force between currents

rather than the force betw een charges. Fo r in Eq . 5 we have ju,o= 4 7 7 x 10~7.

In other words, if a new experimental measurement of the speed of lightwere to change the accepted value of c, we should have to revise the value

of the constant o, not that of ju


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Contents Pr eface to th e Berkeley Phy sics Course vA Fu r t h er No t e vi

Preface to Volu me I I I vii

Acknow ledgmen ts ixTeach ing N otes xi

A Not e on the M K S Sys tem o f E lec t r i ca l Un i ts xvi

Chap t er 1 Fr ee O sc i l l a t i onso f Sim p le Sys tems 1

1.1 Introduction 2

1.2 Free Oscillations of Systemswith One Degreeof Freedom 3

1.3 Linearity and the Superposition Principle 12

1.4 Fre e Oscillations of Systems with Two Degrees of Freedom 16

1.5 Beats 28

Problems and Hom e Experiments 36

Chap te r 2 Fr ee O sc i l l a t i ons o f Sys tems w i t h M any D egrees o f

Freedom 47

2.1 Introduction 48

2.2 Transverse Modes of Continuous String 50

2.3 General Motion of Continuous String and Fou rier Analysis 59

2.4 Modes of a Noncontinuous System with NDegrees of Freedom 72

Problems and Home Experiments 90

Chap t er 3 For ced O sc i l l a t i ons 101

3.1 Introduction 102

3.2 Dam ped Driven One-dimensional Harm onic Oscillator 102

3.3 Besonances in System with Two Degrees of Freedom 116

3.4 Filters 122

3.5 Forced Oscillations of Closed System with Many Degrees of

Freedom 130


Chap t er 4 Trav el i ng W aves 155


4.2 Harmonic Traveling Waves in One Dimension and Phase

Velocity 157

4.3 Index of Befraction and Dispersion 176

4.4 Impedance and Energy Flux 191

Problems and Home Experim ents 214


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Chap te r 5 Re f lec t i on 225


5.2 Perfect Termination 22 6

5.3 Reflection and Transmission 23 3

5.4 Impedan ce Matching between Two Transparent Media 24 5

5.5 Reflection in Thin Films 24 9

Problems and Home Experiments 25 2

Chap te r6 M odu la t i ons , Pu lses , and W ave Packets 267


6.2 Group Velocity 26 86.3 Pulses 279

6.4 Four ier Analysis of Pulses 29 5

6.5 Fourier Analysis of Traveling Wave Packet 308

Problems and Home Experimen ts 311

Chap te r7 W aves in Two and Thr ee D im ens ions 331

7.1 Introduction 3327.2 Harmonic Plane Waves and the Propagation Vector 332

7.3 Water Waves 346

7.4 Electromagnetic Waves 355

7.5 Radiation from a Point Charge 36 6

Problems and Home Experimen ts 381

Chapter8 Po la r i za t i on 393


8.2 Description of Polarization States 39 5

8.3 Production of Polarized Transverse Waves 40 7

8.4 Double Refraction 41 9

8.5 Randwidth, Cohe rence Time, and Polarization 42 7


Chap ter 9 I n t er f e r ence and D i f f r a c t i o n 451


9.2 Interference between Two Coherent Point Sources 454

9.3 Interference between Two Independent Sources 466

9.4 How Large Can a Point Light Source Re? 470

9.5 Angular Width of a Ream of Traveling Waves 473

9.6 Diffraction and Huygens Principle 47 8

9.7 Geometrical Optics 498



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Supp l em ent a ry Top i cs 545

1 M icroscopic Exam ples of Weakly Coupled Identical

Oscillators 546

2 Dispersion Relation for de Broglie Waves 54 8

3 Penetration of a Particle into a Classically Forbidden Region ofSpace 552

4 Phase and Group Velocities for de Broglie Wav es 55 5

5 Wave Equations for de Broglie Waves 556

6 Electrom agne tic Radiation from a One-dimensional Atom 557

7 Time Coherence and Optical Beats 558

8 W hy Is the Sky Bright? 55 9

9 Electro ma gne tic Wave s in Material Media 563

A ppend i c es 585

Supp l ementa r y Read i ng 59 1

I ndex 593

Opt i cs K i t , Tables o f U ni ts , Val ues, an d U sefu l Constants and

I dent i t ies I ns ide covers

O pt i ca l Spect ra fo l l ow ing page528


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waves


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Chapt er 1

Free Osci l lat ions o f Simple Systems

1.1 Introduction 2

1.2 Fre e Oscillations of Systems with One Degree of Freedom

Nomenclature 3

Return force and inertia

Oscillatory behavior 4

Physical meaning of u 2 4

Damped oscillations 5Example 1: Pendulum 5

Example 2: Mass and springslongitudinal oscillations

Example 3: Mass and springstransverse oscillations 7

Slinky approximation 9

Small-oscillations approximation 9

Example 4: LC circuit 11

1.3 Linearity and the Superposition Principle 12

Linear homogeneous equations 13

Superposition of initial conditions 14Linear inhomogeneous equations

Example 5: Spherical pendulum

14

15

1.4 Fre e Oscillations of Systems with Two Degrees of Freed om 16

Properties of a mode 16

Example 6: Simple spherica l pendulum 17Example 7: Two-dimensional harmonic oscillator

Normal coordinates 19

17

Systematic solution for modes 20

Example 8: Longitudinal oscillations o f two coupled m assesExample 9: Transverse oscillations o f two coupled masses

Example 10: Two coupled LC circuits 27

2125

1.5 Beats 28

Modulation 29

Almost harmonic oscillation 29

Example 11: Beats produced by two tuning forksSquare-law detector 30

30

Example 12: Beats between two sources o f visible light 31

Example 13: Beats between the two normal modes o f two weakly

coupled identical oscillators 32

Esoteric examples 36

Problems and Hom e Experiments 36


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Chapt er 1 Free Osci l l ati ons o f Simpl e Systems

1 .1 In t roduc t i on

The world is full of things that move. Their motions can be broadly cate

gorized into two classes, according to whether the thing that is moving

stays near one place or travels from one place to another. Examples of the

first class are an oscillating pendulum, a vibrating violin string, water slosh

ing back and forth in a cup, electrons vibrating (or whatever they do)

in atoms, hght bouncing back and forth between the mirrors of a laser.

Parallel examples of traveling motion are a sliding hockey puck, a pulse

traveling down a long stretched rope plucked at one end, ocean waves roll

ing toward the beach, the electron beam of a TV tube; a ray of hght

emitted at a star and detec ted at your eye. Sometimes the same phenom

enon exhibits one or the other class of motion (i.e., standing still on the

average, or traveling) depending on your point of view: the ocean waves

travel toward the beach, but the water (and the duck sitting on the surface)

goes up and down (and also forward and backward) without traveling.

The displacement pulse travels down the rope, but the material of the rope

vibrates without travehng.

We begin by studying things that stay in one vicinity and oscillate or vi

brate about an ave rage position. In Chaps. 1 and 2 we shall study many

examples of the motion of a closed system that has been given an initial

excitation (by some external disturbance) and is thereafter allowed to oscil

late freely without further influence. Such oscillations are called f r e e or

na tu ra l osci l l a t i ons. In Chap. 1 study of these simple systems having one or two moving parts will form the basis for our understanding of the free

oscillations of systems with many moving parts in Chap. 2. There we shall

find that the motion of a complicated system having many moving parts

may always be regarded as compounded from simpler motions, called

modes, all going on at once. No matter how comp licated the system, we

shall find that each one of its modes has properties very similar to those of

a simple harmon ic oscillator. Thus for motion of any system in a single

one of its modes, we shall find that each moving part experiences the samereturn force per unit mass per unit displacement and that all moving parts

oscillate with the same time dependence cos (u t +


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tion. The displacement is described by a vect or \ j/(x,y,x,t). Sometimes we

call this vector function of x, y , z, ta w ave f u n c t i o n . (It is only a contin

uous function of x, y , and zwhen we can use the continuous approxima

tion, i.e., when near neighbors have essentially the same motion.) In some

of the electrical examples, the physical quantity may be the current in a

coil or the charge on a capac itor. In others, it may be the electric field

E (x,y,z,t)or the magnetic field B(x,y,z,t) . In the latter cases, the waves are

called electromagnetic waves.

1 .2 Free O sc i l l a t i ons o f Sys tems w i t h O ne D egree o f Freedom

W e shall begin with things that stay in one vicinity , oscillating or vibratingabout an average position. Such simple systems as a pendulum oscillating

in a plane, a mass on a spring, and an LC circuit, whose configuration at

any time can be completely specified by giving a single quantity, are said

to have one degree of freedomloosely speaking, one moving part (see

Fig. 1.1). Fo r example, the swinging pendulum can be described by the

angle that the string makes with the vertical, and the L C circuit by the

charge on the capac itor. (A pendulum free to swing in any direction, like

a bob on a string, has not one but two degrees of freedom; it takes two coordinates to specify the position of the bob. The pendulum on a grand

father clock is constrained to swing in a plane, and thus has only one de

gree of freedom.)

For all these systems with one degree of freedom, we shall find that the

displacement of the moving part from its equilibrium value has the

same simple time dependence (called harm on ic osci l l a t i on ),

\ p(t)= A cos (a t +


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second, or hertz (abbreviated cps, or Hz). The inverse of vis called the

per i od T, which is given in seconds per cycle:

T = X . (2)

Th e pha se constant


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D am ped osci l l a t i o n s . If left undisturbed, an oscillating system will con

tinue to oscillate forever in accord anc e with Eq. (1). However, in any real

physical situation, there are frictional, or resistive, processes which damp the motion. Thus a more realistic description of an oscillating

system is given by a damped oscillation. If the system is excited into

oscillation at t = 0 (by giving it a bump or closing a switch or something),

we find (see Vol. I, Chap. 7, page 209)

xp(t) = A e~t /2rcos (ttf +


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F ig . 1 .3 L ong i t ud i na l osci l l a t ions.

(a) Spr ings re laxed and unattached.

(b ) Spr ings a t t ached , M a t equ i l i b r i um

posi t ion , (c ) Genera l conf igu ra t ion .

If we retain only the first term in Eq. (6 ), then Eq. (5) takes on the form

! r = (7)where

2 = f (8)

The general solution of Eq. (7) is the harmonic oscillation given by

\ p(f) A cos ( t +


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total force Fzin the + zdirection is the superposition (sum) of these two

forces:

F* K (z ao) + K (2a z ao)

= 2 K (z a).

Newtons second law then gives

^ = Fz - 2 K ( z a). (9)

The displacement from equilibrium is z a. We designate this by \ p(t):

44*) = z(t) ~ a.then

d 2xp _ d2z

d t2 dt2

Now we can write Eq. (9) in the form

= - < * . (10)

with

2 =2K

M(11)

The general solution of Eq. (10) is again the harmonic oscillation

= A co s (u t+ (p). Note that Eq. (11) has the form to2 = force per unit

displacement per unit mass, since the return force is 2 K\ pfor a displace

ment xp.

Example 3: Mass and springstransverse oscillations

The system is shown in Fig. 1.4. Mass M is suspended betw een rigid sup

ports by means of two identical springs. The springs each have zero mass,

spring constant K , and unstretched length ao They each have length aat

the equilibrium position of M. W e neg lect the effect of gravity. (Gravity

does not produce any return force in this problem. It does cause the sys

tem to sa g, but tha t does not affect the results in the order of approximation that we are interested in.) Mass M now has three degrees of freedom:

It can move in the zdirection (along the axis of the springs) to give longi

tudinal oscillation. Tha t is the motion we considered above, and we need

not repea t those considerations. It can also move in the xdirection or in

the ydirection to give transv erse oscillations. Fo r simplicity, let us con

sider only motion along x. We may imagine that there is some frictionless

constraint that allows complete freedom of motion in the transverse x di

rection but prevents motion along either yor z. (For example, we could


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Fig. 1.4 Transverse oscil lat i ons.

(a) Equil ibr ium configurati on, (b) Gen-

eral configurati on (for moti on along x).

drill a hole through M and arrange a frictionless rod passing through the

hole, rigidly attached to the walls, and oriented along x. However, you

can easily convin ce yourself that such a constraint is unnecessary. From

the symmetry of Fig. 1.4, you can see that if at a given time the system is

oscillating along x, there is no tendency for it to acquire any motion along

yor z. The same circumstance holds true for each of the other two de

grees of freedom: no unbalanced force along xor y is developed due tooscillation alongz, nor alongxor zdue to oscillation along y .)

At equilibrium (Fig. 1.4a), each of the springs has length aand exerts a

tension T0, given by

T0 = K (a a0). (12)

In the general configuration (Fig. 1.4b), each spring has length Iand tension

T = K ( l a0). (13)

This tension is exerte d along the axis of the spring. Taking the x compo

nent of this force, we see that each spring contributes a return force Tsin 6

in the x direction. Using Newtons second law and the fact that sin 6 is

x / l , we find

M d?x _ F x _ _ 2 Tsin 8a t z


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Equation (14) is exact, under our assumptions (including the assumption,

expressed by Eq. (13), that the spring is a linear or Hookes law spring).

Notice that the spring length Iwhich appears on the right side of Eq. (14)

is a function of x. Therefore Eq. (14) is not exactly of the form that gives

rise to harmonic oscillations, because the return force on M is not exactly

hnearly proportional to the displacement from equilibrium, x.

Sl i n k y a pp r o x im a t i o n . There are two interesting ways in which we can

obtain an approximate equation with a linear restoring force. The first way

we shall call the sl i nky approx ima t ion , in which we neglect ao /acompared

to unity. Henc e, since Iis always greater than a, we neglect ao / lin Eq. (14).

[A shnky is a helical spring with relaxed length ao about 3 inches. It canbe stretched to a length aof about 15 feet without exceeding its elastic

limit. That would give ao /a< 1 /6 0 in Eq. (14).] Using this approxima

tion, we can write Eq. (14) in the form

This has the solution x= A cos (cot + cp), i.e., harmonic oscillation. Notice

that there is no restriction on the amplitude A. We can have large oscil

lations and still have perfect linearity of the return force. Notice also that

the frequency for transverse oscillations, as given by Eq. (16), is the same

as tha t for longitudinal oscillations, as given by Eq. (11). Tha t is not true

in general. It holds only in the shnky approximation, where we effectively

take ao= 0 .

Sma l l osci l l a t i ons approx im a t i on . If aocannot be neglected with respect

to a(as is the case, for example, with a rubber rope under the conditions

ordinarily met in lecture demonstrations), the shnky approximation does

not apply. Then Fxin Eq. (14) is not hnear in x. However, we shall show

that if the displacements xare small compared with the length a, then I

differs from aonly by a quantity of order a(x /a )2 . In the smal l osc i l la t ions

app rox ima t i on , we neglect the terms in Fxwhich are nonlinear in x / a . Let

us now do the algebra: We want to express Iin Eq. (14) as I= a + some

thing, where something vanishes when x 0. Since Iis larger than a,

whether xis positive or negative, something must be an even function

of x. In fact we have from Fig. 1.4

(15)

with

(16)

I2 = a2 + x2

a2( l + c),


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Thus

1 = 1(1 +l a

= H 1 - ( r ) + ( H * + a( f ) 3 + - -

Discarding the cubic and higher-order terms, we obtain

d2x_ 2K 2 T0x . . . .

Therefore x(t)is given by the harmonic oscillation

x(t)= A cos (cot -)-


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'mass is M. Thus the retu rn force :per unit displacem ent pe r unit mass is

~2To(x/a)/xM .

Notice that the frequency for transverse oscillations is given by to2 =

2T o /M a for both the case of the shnky approximation (a 0 = 0) and the

small-oscillations approximation ( x / a


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Because of Eqs. (25) and (26), there is only one degree of freedom. We

can describe the instantaneous configuration of the system by giving @i, or

Q2, or 7. The cur ren t Iwill be most convenient in our later work (whenwe go to systems having more than one degree of freedom), and we shall

use it here. W e first use Eq . (25) to eliminate Qi from Eq . (24); the n we

differentiate with respect to tand use Eq. (26) to eliminate >2:

Lft=C_1


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nonlinear, as we can see from the expansion of sin t[/given by Eq. (6 ). Only

when we neglect the higher powers of xpdo we obtain a linear equation.

Nonlinear equations are generally difficult to solve. (The nonlinear pen

dulum equation is solved exactly in Volume I, pp. 22 5 ff.) Fortunately, there are many interesting physical situations for which linear equations

give a very good approximation. We shall deal almost entirely with linear

equations.

L i n ea r h om ogeneous equa t i ons . Linear homogeneous differential equa

tions have the following very interesting and important property: Th e sum

of any tw o so lu t i ons is i t sel f a so lu t i on . Nonlinear equations do not have

that prop erty. The sum of two solutions of a nonlinear equation is not it

self a solution of the equation.

We shall prove these statem ents for both cases (linear and nonlinear) at

once. Suppose that we have found the differential equation of motion of

a system with one degree of freedom to be of the form

= CxP + o p + W + y r + ( 2 8 )

as we found, for example, for the pendulum [Eqs. (5) and ( 6 )] or for the

transverse oscillations of a mass suspended by springs [Eq. (19)]. If the

constants a , (3, y , etc. are all zero or can be taken to be zero as a sufficiently

good approximation, then Eq. (28) is linear and homogeneous. Otherwise,

it is nonlinear. Now suppose that ipi(t) is a solution of Eq. (28) and that

\ ^2(t )is a different solution. For example, ipimay be the solution corre

sponding to a particular initial displacement and initial velocity of a pendu

lum bob, and xp2 may correspond to different initial displacement and velocity. By hypothesis \ piand xp2 each satisfy Eq. (28). Thus we have

CxPi + a\ pi2 + A f c3 + y 4 + -, (29)

C\ p2 + axp22 + P^23 + yxp24 + (30)

The question of interest to us is whether or not the supe rpos i t i onof and

ip2, defined as the sum \ p(t) = \ pi(t)+ xp2( t) , satisfies the same equation of

motion, Eq. (28). Do we have

^'^2~ = + 4*2)+ ('/'l + >/'2)2 + + ife)3 + ? (31)

The question (31) has the answer yes if and only if the constants a, j},

etc. are zero. That is easily shown as follows. Add Eqs. (29) and (30).

d^h_

d t2

and

d2\ p2 _


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The sum gives Eq. (31) if and only if all the following conditions are

satisfied:

Equations (32) and (33) are both true. Equations (34) and (35) are not true

unless aand f iare zero. Thus we see that the superposition of two solu

tions is itself a solution if and only if the equation is linear.

The property that a superposition of solutions is itself a solution is unique

to homogeneous linear equations. Oscillations that obey such equations are

said to obey the superpos i t ion pr inc i p l e. We shall not study any other kind.

Superposi t i o n o f i n i t i a l cond i t i o n s. As an example of the applications of

the concept of superposition, consider the motion of a simple pendulum

under small oscillations. Suppose that one has found a solution corre

sponding to a certain set of initial conditions (displacement and velocity)

and another solution \ p2 corresponding to a different set of initial conditions.

Now suppose we prescribe a third set of initial conditions as follows: We

superpose the in i t i a l cond i t i onscorresponding to \ piand ip2That means

that we give the bob an initial displacement that is the algebraic sum of the

initial displacement corresponding to the motion \ pi(t)and that correspond

ing to \p2(t), and we give the bob an initial velocity that is the algebraic sum

of the two initial velocities corresponding to and \ p2 Then there is no

need to do any m ore work to find the new motion, described by ip3(t). The solution \p3 is just the superposition ip1 + xp2 We let you finish the proof.

This result holds on l yif the pendulum oscillations are sufficiently small so

that we can neglect the nonlinear terms in the return force.

L i nea r inhom ogeneous equa t i ons . Linear inhomogeneous equations (i.e.,

equations containing terms independent of \ p)also give rise to a superposi

tion principle, though of a slightly different sort. There are many physical

situations analogous to a driven harmonic oscillator, which satisfies theequation

where F(t)is an external driving force that is independent of \ p(t). The

corresponding superposition principle is as follows: Suppose a driving force

F\ (t ) produces an oscillation \ pi( t )(when F\ is the only driving force), and

suppose another driving force F2(t) produces an oscillation xp2(t) [when F^t)

dt2 dt2 dt2

C\ p1 C\ p2 = C(\ p1 + \ p2),

a\ p12 + a\ p22 = a(\ pi+ \ f2)2,

/Sxpi3 + /i\ p 23 + M3, etc-

d 2xpi d2\p2 _ d2(xpi+ Xp2)(32)

(33)

(34)

(35)

(36)

i t b it lf] Th if b th d i i f t i lt l


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is present by itself]. Then, if both driving forces are present simultaneously

[so that the total driving force is the superposition F i ( t )+ F2(t)], the corre

sponding oscillation [i.e., corresponding solution of Eq. (36)] is given by the

superposition \ p(t) = 'pii.t)+

- - * * PS)

These two equations are uncoupled, by which we mean that the xcom

ponent of force depends only on x, not on y , and vice versa. Thus Eq . (37)

does not contain y , and similarly Eq. (38) does not contain x. Equations

(37) and (38) can be solved independently to give

x(t) =A i cos (cot +


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Fig. 1.6 Systems w i t h tw o degrees o f

f r eedo m . (T he masses are const ra in ed to

r ema i n i n t he p l ane o f t he f i gu r e .)

1 .4 Free O sc i l l a t ions o f Sys tems w i th Tw o D egrees o f Freedom

In nature there are many fascinating examples of systems having two

degrees of freedom . The mo st beautiful examples involve molecules and

elementary particles (the neutral Kmesons especially); to study them re

quires quantum m echanics. Some simpler examples are a double pendu

lum (one pendulum attached to the ceiling, the second attached to the bob

of the first); two pendulums coupled by a spring; a string with two beads;

and two coupled L Ccircuits. (See Fig. 1.6.) It takes two variables to de

scribe the configuration of such a system, say \paand \ pj,. For example, in

the case of a simple pendulum free to swing in any direction, the moving

parts tpaand \ pj, would be the positions of the pendulum in the two perpendicular horizontal directions; in the case of coupled pendulums, the

moving parts xpaand \pbwould be the positions of the pendulums; in the

case of two coupled L Ccircuits, the moving parts xpaand \ pbwould be

the charges on the two capacitors or the currents in the circuits.

The general motion of a system with two degrees of freedom can have a

very complicated appearance; no part moves with simple harmonic motion.

However, we will show that for two degrees of freedom and for linear

equations of motion the most general motion is a supe rpos i t ionof two independent simple harm onic motions, both going on simultaneously. These

two simple harmonic motions (described below) are called no rm a l m odes

or simply modes. By suitable starting conditions (suitable initial, values of

xpa, M dxpa/d t , and dipb/dt) , we can get the system to oscillate in only one

mode or the other. Thus the modes are uncou pled, even though the

moving parts are not.

Prope r t ies o f a mode . When only one mode is present, each moving part

undergoes simple harmonic motion. All parts oscillate with the same fre

quen cy. All parts pass through their equilibrium positions (where \pis zero)

simultaneously. Thus , for example, one never has in a single mode,

\pa(t) = A cosoof and M t ) = B sin cot (different phase constants) or

\pa(t) = A cos u\ t and xph(t ) = Bcos co2t (different frequ encies). Instead

one has, for one mode (which we call mode 1),

M t ) = M cos (i t+


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Each mode has its own characteristic frequency: toi for mode 1, t02 for

mode 2. In each mode the system also has a characteristic configuration

or shape, given by the ratio of the amplitudes of motion of the moving

parts: A 1/B .1 for mode 1 and A 2/ B2 for mode 2. Note that in a mode theratio is constan t, independent of time. It is given by the appro

priate ratio A 1/ B1 or A 2/B 2, which ca n be either positive or negative.

The most general motion of the system is (as we will show) simply a

superposition with both modes oscillating at once:

\pa(t) = A t cos ( u i t +


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F ig . 1 .7 Tw od imensiona l harm on ic

osci l la t or , (a) Equi l i br i um , (b) General

con f i gu r a t i on .

M ~ = - 2 K i X , and \ = - 2 K2y , (45)

which have the solutions

2Kx Ai cos (wit +


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general, is still not as general in app eara nce as Eqs. (43). Tha t is because

we were lucky! Our natural choic e for xand yalong the springs gave us

the uncoupled equations (45), each of which corresponds to one of the

modes. In terms of Eq . (43), we cam e out with \paluckily chosen so that

A 2 came out identically zero and with \ pbchosen so that Bi came out iden

tically zero. Our fortunate choic e of coordinates gave us what are called

no rma l coo rd i na t es ;in this example the normal coordinates are xand y.

Suppose we had not been so lucky or so wise. Suppose we had used a

coordinate system x' and y ' related to x and y by a rotation through

angle a , as shown in Fig. 1.8. By inspection of the figure we see that the

normal c oord inate X; is a linear co mb ination of the coordina tes x'and i f ,

as is the other normal coordinate, y. If we had used the dumb coordi

nates x!and y 'instead of the smart coordinates xand y , we would have

obtained two coupled differential equations, with both x'and y 'appear

ing in each equation, rather than the uncoupled equations (5).

In most problems involving two degrees of freedom it is not easy to find

the normal coordinates by inspection, as we did in the present example.

Thus the equations of motion of the different degrees of'freedom are

usually coupled equations. One method of solving these two coupled

differential equations is to search, for new variables th at are linear combi

nations of the original dumb coordinates such that the new variables

satisfy uncoupled equations of motion. The new variables are then called

normal coordin ates. In the present example we know how to find the F ig . 1 .8 Ro t a t i on o f coo rd ina t es .

normal coordinates, given the dumb coordinates x'and y ' . Simply rotate


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o a coo d ates, g e t e du b coo d ates a d y S p y otate

the coordinate system so as to obtain x and y , each of which is a linear

combination of x'and y ' . In a more general problem, we would have to

use a more general linear transformation of coordinates than can beobtained by a simple rotation. Th at would be the case if, for example, the

pairs of springs in Fig. 1.7 were not orthogonal.

Sys tema t ic so lu t i on fo r modes . Without considering any specific physical

system, we assume that we have found two coupled first-order linear

homogeneous equations in the dumb coordinates xand y:

^ = a u x - a 12y ( 4 7 )

- ^ jr = a 2i x - a 22 y . ( 4 8 )

Now we simply assumethat we have oscillation in a single normal mode.

That means weassume that both degrees of freedom, namely xand y , os

cillate with harm onic motion with the same f r equency andsam e phase

constant . Thus we assumewe have

x A cos (cot +


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of the linear homogeneous equations (51) and (52) must vanish:

an w2 012021 a22 w2 = { ( I I I 2) (f l2 2 CO2) 021012 = 0. (56)

Equation (55) or (56) is a quadratic equation in the variable co2. It has

two solutions, which we call coi2 and CO22. Thus we have found that if we

assume we have oscillation in a single mode, there are exactly two ways

that that assumption can be realized. Fre qu en cy coi is the frequency of

mode 1; C02 is tha t of mode 2. The shape or configuration of xand yin

mode 1 is obtained by substituting co2 = coi2 back into either one of

Eqs. (53) and (54). [They are equivalent, because of Eq. (56).] Thus

= ( A ) = - g L = co!2 - a n . (57fl)

\X/mode 1 VA /mode 1 A i #12

Similarly,

(M.) = ( A ) = J * L = c22 - B l L . (5 7b)\ X/ mode 2 V A /mode 2 A 2 d i2

Once we have found the mode frequencies coi and C02 and the amphtude

ratios B1/A 1 and B2/ A 2, we can write down the most general superpositionof the two modes as follows:

x(t) Xi(t) + x2(t) = Ai cos (coi* +


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F ig . 1 .9 Lo ng i t ud i na l osci l l a t i ons,

( a) Equ i l i b r i um , ( b ) Gener a l c on f i g u r a-

t ion .

modes, since there are-tw o degrees of freedom . In a mode, each moving

part (each mass) oscillates- with harm onic motion. This means that each moving part oscillates with the same frequency, and thus t he retu rn fo rce

pe r un i t d isp l acemen t pe r un i t m ass is t he same fo r bo t h m asses. (We

learned in Sec. 1.2 that oo2 is the return force per unit displacement per

unit mass. Th at holds for each moving part, w hether it is a single isolated

system with one degre e of freedom or is part of a larger system. The

only requirement is that the motion be harmonic motion with a single

frequency.)

In the present example the masses are equal. We need therefore only

search for configurations that have the same return force per unit displace

ment for both masses. Let us guess that the displacements may be the

same, and see if that works: Suppose we start at the equilibrium position

and then displace both masses by the same amount to the right. Is the re

turn force the same on each mass? Notice that the central spring has the

same length as it had at equilibrium, so that it' exerts no force on either

mass. The left-hand mass is pulled to the left because the left-hand spring

is extended. The right-hand mass is pushed to the left with the sameforce,

because the right-hand spring is comp ressed by the same amount. Wehave therefore discovered one mode!

Mode 1: \pa{t ) = \pb(t ), i 2 = (60)M

The frequency i 2 = K /M in Eq. (60) follows from the fact that each mass

oscillates just as it would if the central spring were removed.

Now let us try to guess the second mode. Fro m the symm etry, we guess

h if d 6 i l h d If di


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that if aand 6 move oppositely we may have a mode. If amoves a distance

\pato the right and 6 moves an equal distance to the left, each has the same

return force. Thus the second mode has ipb = \pa. The frequencycan be found by considering a single mass and finding its return force per

unit displacement per unit mass. Consider the left-hand mass a. It is

pulled to the left by the left-hand spring with a force Fz = K\ pa. It is

pushed to the left by the middle spring with a force Fz 2K\ pa. (The

factor of two occurs because the central spring is compressed by an amount

2 ipa.) Thus the net force for a displacement \pais 3K\ pa, and the return

force per unit displacement per unit mass is 3 K /M :

Mode 2: i//a = \pb, w22 (61)

The modes are shown in Fig. 1.10.

F ig . 1 .10 N o rma l modes o f l ong i t u d i na l

osci l l a t i o n , (a ) M ode w i t h l ow er f r e-

quency . (b) M ode w i t h h i gher f r equency .

We shall solve this problem on ce more, using the method of searching

for normal coordinates, i.e., sm art coordinates. The sm art coordinates

are always a linear combination of ordinary dumb coordinates, such that

instead of two coupled linear equations, one obtains two uncoupled equations. From Fig. 1.9b , we easily see that the equations of motion for a

general configuration are

By inspection of these equations of motion, we see that alternately adding


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y p q , y g

and subtracting these equations will produce the desired uncoupled equa

tions. AddingEqs. (62) and (63), we obtain

M ~ ( i a + i b) = K ( 4 a + U (64)

Subtracting Eq. (63) from Eq. (62), we obtain

M d2('Pad ~ ^ = - 3 K ( 4a - U (65)

Equations (64) and (65) are uncoupled equations in the variables 4a + 4b

and if/a 4b They have the solutions

4a+ 4b= 4 l ( t ) =Ai cos (toit + z), C022 = , (67)

M

where Ai and


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The system is shown in Fig. 1.11 . The oscillations are assumed to be con

fined to the plane of the paper. There fore there are just two degrees of

freedom. The three identical massless springs have a relaxed length aothatis less than the equihbrium spacing aof the masses. Thus they are all

stretch ed. When the system is at its equilibrium configuration (Fig. 1.11a ),

the springs have tension To.

Because of the symmetry of the system, the modes are easy to guess.

They are shown in Fig . 1.11. The lower mode (the one with the lower

frequency, i.e., the one with the smaller return force per unit displacement

per unit mass for each of the masses) has a shape (Fig. 1.11c) such that the

cen ter spring is never compressed or extended. The frequency is thus obtained by considering either one of the masses separately, with the return

force provided only by the spring that conn ects it to the wall. Fo r either

the slinky approximation (unstretched spring length of zero) or the small-

oscillations approximation (displacements very small compared with the

spacing a), we shall show presently that a displacement \paof the left-hand

Fig. 1.11 Transverse osci ll ati ons,

(a) Equi li bri um, (b) General configura-

ti on. (c) Mode w it h low er fr equency,

(d) M ode w it h hi gher fr equency.

mass causes the left-hand spring to ex ert a return force of To(\ p/a). Hence,


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p g ( p ) ,

in this mode the return force per unit displacement per unit mass, coi2, is

given by

M o d el: Wl2 = - 5 - , ^ = + 1 . (70)M a xpa

We see this as follows. Firs t consider the shnky app roximation (Sec. 1.2).

In this approximation, the tension T is larger than T0 by the factor l / a ,

where Iis the spring length and ais the length at equilibrium (Fig. 1.11a).

The spring exerts a transverse return force equal to the tension Ttimes the

sine of the angle between the spring and the equilibrium axis of the springs,i.e., the return force is T(\ pa/ l ). But T T0( l /a ) . Thus the return force is

Ti)(\ pa/a ), and this gives Eq . (70). Nex t consid er the small-oscillations ap

proximation (Sec. 1.2). In tha t approxim ation, the increase in length of

the spring is neglected, because it differs from the equilibrium length a only

by a quantity of order a(xpa/a ) 2, and therefore the increase in tension also

is neglected . The tension is thus T0 when the displacement is xpa. The

return force is equal to the tension To times the sine of the angle between

the spring and the equilibrium axis. This angle may be taken to be asmall angle, since the oscillations are small. Then the angle (in radians)

and its sine are equal, and both are equal to xpa/a . Thus the return force

is To(xpa/a ) . This gives Eq. (70).

Similarly, we can obtain the frequency for mode 2 (Fig. 1.11 d)as follows:

Consider the left-hand mass. The left-hand spring contributes a return

force per unit displacement per unit mass of To /M a , as we have just seen

in considering mode 1. In mode 2 the cent er spring is helping the left-

hand spring, and in fact it is providing twice as great a return force as is

the left-han d spring. This is easily seen in the small-oscillations approxi

mation: The spring tension is Tofor both springs, but the center spring

makes twice as large an angle with the axis as does the end spring, so that

it gives twice as large a transverse force com ponent. The total return

force per unit displacem ent per unit mass, CO22, is thus given by

Mode 2:


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y g , ; qy

motion motion of the charges in this case. Th e electromotive^ force

(emf) across the left-hand inductance is L d l a/ d t . A positive charge Qi onthe left-hand capacitor gives an emf C~XQ\ that tends to increase l a(with

ou r sign conventions). A positive charg e Q2 on the middle capacitor gives

am emf'" G-1 @2 that tends to d e c r e a s e ' T h u s we have for the complete

contribution to L d l a/ d t

Similarly,

Ldlg

dt

L d l b

dt

= c -'Q i -

= c - i< ? 2 - c - i< ? 3.

(72)

(73)

As in Sec. 1.2, we will:express the configuration of the system in terms of

curre nts rather than charge s. To do this, we differentiate Eqs. (72) and

(73) with respect to time and use conservation of charge. Differentiating

gives

t d2I a _ x dQ i j dQ'2

d t 2 d t d t

^ d2h _ dQz _ -i_i dQ z

d t2 d t d t

(74)

(75)

Charge conservation gives

dQ1 _ T dQ2 _ r ,

d t ~ d t ~ a ~

dQ z

dth. (76)

Substituting Eqs. ( 76) into Eqs. (74) and (75 ), ,We obtain the coupled equations of motion

d2I aL

L

d t2

d2h

d t2

= - c ~ n a + c~\ h - ia)

= - ~ G ~ H h ~ h ) - C U b.

(77)

(78)

F ig . 1 .12 Tw o coup l ed L C ci rcu i ts .

G enera l con f igu ra t ion o f cha rges and

currents . T he arrow s giv e s ign conven-

t i o n s fo r posi t i v e cu r ren t s .

h h

^ n n n n p r L L

Qi Qz Qs

~Q i - q2 ~Q s

Now that we have the two equations of motion we want to find the two

l d h b f d b h f l d


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normal modes. These can be found by searching for normal coordinates,

by guessing, or by the systematic method (see Prob. 1.21 ). One finds

c1Mode 1: l a h , wi2 r (79)q r ' - i ' '

Mode 2: I a = h ,


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frequency (0mod:

wav = + w2)> Wm0(j EE -((Oi (02). (82)

The sum and difference of these give

toi - (0av + wmod> *02 = Wav COmod. (83)

Then we may write Eq. (81) in terms of (oav and comod:

1p = A COS Wit + A COS (02f

= A COS ((0avf + tomodf) + A COS ((0avf tOrnodf)

= [2A cos wmodt] cos wavt,

i.e.,

Amod(t) COS C0av, (84)

where

Am od(^ ) - A CO S COm od t. (8 5 )

We can think of Eqs. (84) and (85) as representing an oscillation at angular

frequ enc y coav, with an am plitud e Amod that is not consta nt bu t rather varies

with time accordin g to Eq . (85). Equa tions (84) and (85) are exact. However, it is most useful to write the superposition, Eq. (81), in the form of

Eqs. (84) and (85) when coi and 2 are of com parable magnitude. Then

the modulation frequ ency is small in magnitude com pared with the average

frequency:

(Oi ^


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are not exactly constant, but only almost constan t. Their variation is

negligible during one cycle of oscillation at the average fast frequency

coav, provided that the frequency range or bandwidth of the component

harm onic oscillations is small compared with coav. (W e shall prove these

remarks in Chap. 6 .)

Some physical examples of beats follow:

Example 11: Beats produced by two tuning forks

Whe n a sound wave reaches your ear, it produces a varia tion in air pres

sure at the eardrum. Le t ipi and \ p2 represent the respective contributionsto the gauge pressure produced outside your eardrum by two tuning forks,

numbe red 1 and 2. (The gauge pressure is just the pressure on the outer

surface of your eardrum minus the pressure on the inner surface; the pres

sure on the inner surface is normal atmo spher ic pressure. This pressure

difference provides the driving force to drive the eardrum.)

If both forks are struck equally hard at the same time and are held at

the same distance from the eardrum, the amplitudes and phase constants

for the gauge pressuresj p iand xp2 are the same, and thus Eq. (80) correctly represents the two pressure contributions. The total pressure (which gives

the total force on the drum) is the superposition xp = \ pi+ \p2 of the con

tributions from the two forks. It is given either by Eq. (81) or by Eqs. (84)

and (85). If the frequencies of the two forks, i>iand v2, differ by more

than about 6 % of their average value, then your ear and brain ordinarily

prefer Eq. (81). Tha t is, you hear the total sound as two separate notes

with slightly different pitches. Fo r example, if v2 is f times v i , you hear

two notes with an interval of a major third. If v2 is 1.06i'i, you hear v2as a note one half -tone higher in-, pitch than v\ . However, if and v2

differ by less than about 10 cps, your ear (plus brain) no longer easily

recogn izes them as different notes. (A musicians trained ear may do

much better.) Then a superposition of the two is not heard as a chord

made up of the two notes v\ and v2, but rather as a single pitch of

frequency vav with a slowly varying amplitude A mod, just as given by Eqs.

(84) and (85).

Square law detector . The modulation amplitude Amod oscillates at the

>modulation angular frequen cy comod. Whenever thas increased by an

amoun t 2 w (radians of phase), the am plitude A modhas-gone through one

complete cycle of oscillation (i.e., the slow oscillation at the modulation

frequency) and has returned to its original value. At two times during one

cycle, Amod is zero. At those times, the ear doesnt hear anything there

is no sound. In between the silences, you hea r a sound at the average

pitch. Since; cos .comodtgoes from zero to + 1 , to zero, to 1, to zero, to

+ 1, etc ., we see that A mod has opposite signs at successive loud times.

Nevertheless, your ear does not recognize two kinds of loud times, as

ill di if f h i i h i f k Th


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you will discover if you perform the experim ent with tuning forks. Thus

your ear (plus brain) does not distinguish positive from negative values ofAmod- It only distinguishes w hether the magnitude of A mod is large

(loud) or small (soft), that is, whether the squa reof A mod is large or

small. Fo r that reason, you r ear (plus brain) is sometimes said to be a

squa re la w det ec to r . Since A mod2 has t w omaxima for every modulation

cycle (during which wmodtincreases by 2 t t), the repetition rate for the se

quence loud, soft, loud, soft, loud, soft, . . . is twice the modulation fre

quenc y. This repetition rate of large values of Amod2 is called the beat

f r e quency :

k beat 2(0mod COi 02. (86)

We can see thisalgebraically as follows:

A m o d ( f) 2 A CO S Wm od*.

[Amod(f)]2 = 4 A 2 COS2 COmodt;but

cos26 = ^[cos26 + sin26 + cos29 sin26} J [1 + cos 20],

Thus

[Amod(i)]2 = 2A2[1 + cos 2wm0dt ],

i.e.,

(Amod)2 = 2A2[1 + COS Wheat*]- (87)

ThusAmod2 oscillates about its average valueat twice the modulation fre

quency, i.e., at the beat frequency, coi w2-

The superposition of two harmonic oscillations with nearly equal fre

quencies to produce beats is illustrated in Fig. 1.13.

Example 12: Beats between two sources of visible light

In 1955, Forrester, Gudmundsen, and Johnson performed a beautiful ex

periment showing beats between two independent sources of visible light

with nearly the same frequency, f The light sources were gas discharge

tubes containing freely decaying mercury atoms with an average frequency

of j'av = 5.4 9 X 10 14 cps, corresponding to the bright green line of mer

cury. The atoms were placed in a mag netic field. This caused the green

radiation to split into two neighboring frequencies, with the frequency

difference proportional to the mag netic field. The beat frequency was

v\ v2 ~ 1010 cps. This is a typical rada r or microwav e frequency.

t A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Photoelectric mixing of incoherent

light, Phys. Rev. 99, 1691 (1955).


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0 5 10 15 20 25 30 35F>'! v> " . V . ' . iAvfliWliW1111liVft

Miimw0 5 10 15 20 25 30

1 beat

Fi g. 1.13 Beat s, xf/i a n d \ p 2 ar e the pres-sure va r i a t i ons a t yo u r ea r p rod uced by

t w o t u n i n g f o r k s w i t h f r e q u en c y r at i o

v \ /v i 10 /9 . The to ta l p r essu re is the

superpos i t ion ip i+ 2, w h ich is an

almost ha rm on i c osci l l a t i o n a t f r e-

quencyz'avw i t h sl ow l y v a r y i n g amp l i t u d

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