Br

Y$'

!lE, /"

0t

IL

tT

I

b

oFsd

. sacu€JaJau

rfcuonba.:;: a3e"xeaY *11 xgPueddY

suoT+eqfil1.rad ageus * I xgPuaddy

sluame8PaTieomiov "9

rt1;suaP utroJFn*uoN "5

es€c"-{+Tsuop*ttr"ro$Tlm er{f 't

esuodsas "fcuanba"r;: aqtr '9.

srroT+€TTTcso aars "z

uoT+en?oJ+rrI 'k

OUVStrIEIH '',}T

TJ.IffiIINIT*NON "Ag SNTdYIV( fIVOIffit

,fq

696r f,rnf *raI r*5g 10/sdilr/t[u$3

'1"t',-rd-,

1 " Introduction

Transverse *ohenent ins*abilitf-esu driven by waka fi.eldsu nay

be stabiSised by fregueney spread", The ease Mere *his spread cores

from the longitudinal momentum spreaf, 0f *he beea is straightforward,

at leas* for e *oas*j"ng beamy beeause the xongttud:inar momentum is

a eonsts3"It, wb*"eh just effeets *he eoeffiei"er*s in the equations of

notion of the *ransverse ose*llatisns, ed henee tbeir frequency,

when the freqreeney spread esms frem a nonlinearity of the focrisirgu

*he si*ustion *s less elearu because the eohered motion is thea a

smaf,l ad&i*ion to the large i"neoherent mplS,tud.es *hat mab the frequeney

spread"u end it *s j":neensis*ent *o assure thet it san be .breated as a

Sinear superposi"tionu and. *t *s not obviouq that the large*anplitude

frequeney *s the epprtpriate effeetive resoaart fregueney for ealeuj.ating

the ad.di*ioaal motion &ue to a sma3l perturbetion*

Sinee the *"esu]* fs Euftc &Sfferen* f,mm the usuaL assumption,

we try *er nah it eomvinein6-by three &istinet attaeks; in sections

21 3u A" * But the essent*alsu in e rough eoneise fom, ane in Append.ix IT.

1 We eonsi&er on-ly the

the externel foeusing systemu

: spaee*eharge,

ease wlrere *hB non*3ffir Telm *ores fro*-

not tb.e ease where it esres from the beam

2. Free osei l lat ions

3n this seetion we stuQr e, "r:ather simple si'ema*ion to show tha"b

tir.e u-su.aJ- pi-cture of uon*-l-inearity eausing frequency tc d-epend on

amplitud.eu tllen amplitud.e range aaking a spnead" of resonant frequencies,

*an be mislead-ing" Tfe shall l-ook at "bb free*ringj-qg freq-uencar nf n bea^sr

r.ihi-ch aeeupies a. r"egion of phase*spaoe 'w-ith, ilni-for"rm d_ensity"

e onsi-de:: par-i.ieles that,move aceord-ing 'ho

f f + v $ x + F ( : * ) * 6 {z,r }

" s a-p n:1" tTdul€ l-LIaJaJJ-tp

a-tqe+Tns 'u {q ienba apg.u

;ro ;: d1s noau"€+-l-11u1'is

aq u€o .{aqfi '(vl'v:f oli ?.ilq

' a-gec s J0 ac "ror{c

ptl€ v €.r? saxEar'l8s s!--l :+

( z"z\

",i"{-siMir

s"6u

Y

- t"3T,{ uT Pelar+snlTT se d:trolcaCe'r1 V aq+ i'{-}Te lroJ?dtuoc

pa+JTqs at++TT e 1sn[ sT ilTT+rIo esot{ia lrot+nqTJ+srp 'f11suap ulroJTrm E

Je?Tsuos Jar{+€J sn +eT oS 'oJaz sT uoT+ou +ua;al{oa Jo apn+TTdue aqx 6 s1

+"q+ 'aIuT+ Jo luapuadepul 'fgalalduoc s 't uoT+nqT"r+sTp tr13suap aq+ +€tt+

.rfes e qans uT spolrad pu3 se?n+TTduE ur&o rTeI{+ q+Te pm-to* oF 11e

,taql geq4 sa+€+cT? uloJoar{+ sreTTT no;1 6so1c;pxed ;:o flgsuap ulroJTun

E i{+Te d"rolcefe'r+ "

qens 'fq pasolaua €are aI{+ sTT}J suo JI

"(v)aluaolan/aTuIoJJspoTJadTlF?unoJeq'TTT&Vs+oJazIIIoJJ

sapnlgdw g:o a9uex aq+ uI '(V)n1u7 Bol*ed aq+ ItT acuo +T punor

oF .(*o1aa$e;t1 s3q1 uo saToT+"red 119 '

aueld To* eq+ uT a1e19:ed € qcns Jo

ttTia srlJa+ cTtrouu€q *aqFlq aq+ Tt€us

ssa.:dxa uec

suoT+€TTTcso

*esdg11a lqSPrdn rre r{1q8no; s1

d:oqcefe-r+ eq+ Pu€ tosTe os aq

.f1"rge;: sg -rf1;*eauTT*uou ot{+ JT

scTriorrkreq *aq3;r4 + [dr + +(v)o] soc f = 16

a1o31.ted € Jo uoT+ou sq+

aro ?u€ V apnlg1dre rTaI{+ uodn puadap TTTa poprod aq4

.ra9;re1 ros .suor+errreso TT€us .1o pot'rad aq+ sT #

?u€

o=(o)ug e 6=(o)*1

asoddns u€c a& os 6acso;

Fu1"ro1sa: aq+ Jo q"-red -zeauTT-uou eq1 sluaso':da; (o)g aroq

*7,*

*3*

Ttre result of the shlft g is to make a thin lune-shaped" region

oecup iedbypa r t i e l es j us tou t s i de theA t ra j ee to r ya r r c l anempw

h:ne just i.nside it; eve:Xrwhe:"e else the density is as it was in

the unshifted case'

N o w e o n s i d . e r h o w t h e s e l u n e s n o v e w . i t h t h e p a s s a g e o f t i m e '

They nove ro'nd in tls phase pLane with the period' Zs/v(l)" Mo:re

p r e c i s e } y : t h e p o i n t s f f i w h i e h e o n s t i t l * e t h e i r b a s e l i e o n t h e

A trajeetory a-ad therefore go round with period exaetl-y zn/v(x)

f o r e v e r u w } r . i ] . e t h e p o i . a t s o * e l i e o n n e a r b y v a l u e s o f " n n p } i t u t l e

and. wilf go round' with perioas glqi.gF-gre.-Segq- to

gid"!L)"*=l:F. *PrL ql 'P4+ g ' rt is olear that the notion

of these li:aes determines the motion of'the eentroi{L sf the whole

beam, i.e* the eohererrt oseilfation" frus we have tbe soBewhat sur-

pristng result:

For e been having rmiforn phase-spaee densi\r fe^

a1.l aoplituites up to some Au *he free oseillation

period for oseillatioa of the ev€rage :ep 5-n the

linit that its amplitude is snall, is oqual to the

period. of individual palticl-es at a'uplittrde A' and

shovs no frequeireY sPreu'A**

f re queney spread or pqfll:r-g- llSl*-tl:ji*5?l19.Sifhus the

d.ensity is rt invisiblg,. r---****IbgJ notionuniform

discfose s the fre of the ertreme particles* We show i:r

Append:ix IJ how the effective

sueh E"s to give this resu-Lt"

coherent-oscillation frequencie s are

There

limit of snal'l-

are some more remarks about our valiility *in the

6 " i-n APPendix I"

is worth mentioning that this argument srith ths urliform

the lunes applies equally 'bo hi6her ortler modes like the

( tlrrobbrng-beam, bean*enrrelope , monopole) os cillatj-ons -

t +

d.ensity and

^ , , - - l . - ' - n l = nL t u a r A u p v + u l r

nu By 'rlri" we fitean 'i;here

no slow il i-ni:.uticn or

ihaL nonnail;tr resi-rl i !s

frequencf is inPrrre in

t ' ( l ) , cou , i r r 4 f r om the

is (in ihe J-imit of small coheren't ampJ-itud-e) -beating of 'che coherent arnplitude, of ,ln? SlUfrom siectnrn wielth- It remains true tha't the

the sense c'J' conta-''-ning higher imrltipl-e s of

non- l-inea.ri- tY ..

os

ecToq3

ox=]t 4;o+>+JoJ

:uot+nTos affes €q+ 11 an13

sra os uoSlenba au€s aI{+ sT (t-f ) o+ irBI{+ raTTreo + roJ I p*Y

'(V)a *o; a pa1d3:csqnsun ue 9u111'ri& ar€ eu 'd1;..re*e[ ro'{ '(e"Z) u'r

pauoT+uau .fpea"r1e sulJa+ cTuolIIJ€I{ ;raq91q fi{+ s+uosa*da; "q'q eJ'?r#e'

"T+{+ td+sdJsooY =ox

EoT+nTos aq+ a+Trla au (z"f ) Pat'rn+*adun aqq 'ro6

"rroT+ounJ €+Te? osrT0 eI{+ sT ( )S arat{&

(o+-+)gS=(x)*+x$rt+5

trq (l"f ) gtrlceltla-r deslndrnT uE Jo IIuoJ eI{+ uT uoT+€qrnp:ad

€. Jo +ce#Ja er{+ +SJTJ JepTsuoc o+ sT asn aa poq+aE er{f S

.g o+ tstroT+Jodo.rd oq TTTIa $s :C11eu1.r:

" tx Jo +r€d luepuadalug-g f,ne sploae ox aqeprdo':dde aq+ Jo

"+T uT JepJo +sJTJ uaql .raqpTq slera+ doap pue TTws g aumss€ €i&

o:r-x= r:f

rrTe+qo o+

* (o:c)'t + oxgd +o*

Jo uoT+n1os e+€Tddo'rdde aq1 +oar+qns pu€

+ogd:cs*=(x),*+x$n+5

aaTos TTPt{s a& oS "uoT+€qJn1'rad pu"ra}xa Tt?$s € o+

aplgred epnlgldue-egae1 € #o asuodsar aq1 aorni o+ peou ail

esuodse.: t[ouenbe.r,; eq6 "{

*5*

t6 the right

most general

hand. sid.e of

solution near

( l"t.)t o x o :

is again

The two quantities

and ean be found. bY

This requires

Differentiating

(in general K

For times later than

zero and we take the

fo r t> te s

*= (A+ 6A )cos [ v ( *+aa ) t + f +s r / ] +h "h ' ( l " e t )

5A and. 6tP are

satisffing (1"+)

constants which are treated as sma1I'

in the neighbourhood of t = to "

t - to+) t = t o * ) = O:c\x(

v l

(: 'z)+ + = t 6 - ) = 8 "

(1"5) and- introdueing

Aqu =P d A

d.epend.s on A)

the abbreviation

...-.:Ft t t

f o r * p t o

ana {1.?) gives us twof

eouations for 64 and 6itr lv-ith solutions

we

x

have

- X 0

+

ae u (t+r)

6ATrz tK

6pAv

sin( ut

eos(r"t

cos( ut

+ w)- , 1 \

+ tlt)

( : 'a)

l @ N d '

t^ . the time of aPPlication

i t" l " i tg i -ndePendent of the*

Th* quanti'bies EA

of Lhe imPulse, but

tine t, o

andu r l s J

6p depend onare ccnstan'Ls

!r't)+Pgdxa g

"q*q "l'

/\- ztt * z* { 4. * u\ = $x

+ffgdxas _*;*"E--

\)i '/

sprarr +rurr ;raddn eq+ +€ uor+€nr€^" t:="^:"""T,::1T#-Ti:ril--JJ:fiilT.:t;li, "il":; ";1;*oo" ,r*,s s+carra s+r ?u" 6 uo11eq'rnq"'xacl

,il-rat{ns J"orrTT 'fq uol1e1na F+"uITxoJdG? 1s'rr'} v+ "r'*-- --*--^-p 6"r..Tnrsod..radns J"orrTT 'fq uol1e1na

aJ{+ aunss€ a$ +€ri+ 1ce; a{* dq pag'rqsnf 6uo111so

o,r9aruT Ptls o1ofdxa {q

;::T I l_H u::*

;"""*- *,.."*o*, o*n a1e"r3a1u1 pue olordxa dq

r rr.r tla esec eq+ o&Tos -r€c s sT s 5tt''l' rad cTnou"req*aldurs r{+Tia asec eq+ o&Tos

itldrlgru +snq aD (u"6) uoT+€qrn+r-o "!*": -l-'-:ar.iln raorrartrau {

"I t;:; l"*-"**tad uor+cunJ*€+Top aq+ roJ sT srt{+ ;aquauau

"s uT -rapro +SJTJ aq+ s€

.I€Js€,t1uou1e19r3g€(6"f)uTs.Io+gEtT6ouePaI{+?a?u€dxaaaeqaAsJaI$|'

"r{'q +

a(ox -1)eoc( o+ a ** *

(ot"e )n("+ *+)rrr*(z/T - t) t

= sx r o+ € +

0= ryr o+>+

(6"( )

ty aog 9u1*15

g =Sg

(diffilffir*

-9*

g. =Yg

*7*

T h e l o w e r } i i t r i t i . s m o r e a w } m a r & * a s t h e r e a r e t e m s t h a t o s c i l l a t e

infiir-itely as t6 -+ - eoe but they 41tr eonverge to zero*if we assume

t h a t a l h a s a t } e a s t a l i t t l e u e g a t i v e i n a g i n a r y p a r t , l @

Tf one puts K * 0 (*oa h"h' = 0) : 'n (J- ' t l ) ' one reeovers

the f**lliar 1inear resonengs* The extra factor (t - VZ) in the first

line is possibly only an unrinportant eomeetion if K is snall, but the

seeond J-ine has a d.ifferent order sf (o * ') * elependenee' and is

something new"

&-* The unriforno-densitJr ease

frIo

phase*spaee t

would be to

by "d

; but we can easiS-y be a little more preeise' C

unperturbed. oseillation of (3.5) t

x s= A cos f r r t + rP ] +h *h *

@

& = ^A v sin[Yt + qf] + h"h"

The ellipse that fits the fundamental evid.ently has area n A"v

ou * plane, anf,

wan* to averege (3"tt) over a r:nifornly oeeupied region of

in order to eompare the result with that of section 2' Sinplest

multiply by srclAz *, integrate from zero to , anil iliviite

der the

Amonsi

( t * . t ;

in the

\

(:*"21/'d ( n A E u ) = 2 w ( 1 +

\.

So the response faetor averaged orrer

{ i . t r ) and ( t r . , z ) , to f i rs t o rd 'e r in

l \ \

2 I

the

T { i <

u AdA

u::-iform-lY fj-l1-ed area, from

* Th"u.t i'- s by Zsr Adi.

potTTJ ,{1uuo;:pm sq+ +€r{+ }Insar aq1 uiese a"=q nr-*"qg, ,anpr6#.r- o;raz"s€q-aila-*"q+

fG;t) +{*;; "r{+-"ni",i"*+b4"'saot

i"rqi ,,, uaql p.ro^,Bo1 pted .,treur8eus ar++TT e ,o anlg .:o uo11e;rFaluT Jo q*)&

"u* urroJap

uec pue *(f"t) ur sro+€ugtrouap o.rez lag aar d Jo a8ue;r oq+ uT salTpue T€a.r sT fD Jf * Er3

; Jo oa ;- o4 penba +ou sr m peplao,-rd

NI

,@*.dI

or r u Er| .* :*\ | :-;

u LzVn ) vI'Y

oq. lenba sp (t"t) oS

uuI4,VtL

b

wp

' ttv)a' roJ s?u€+s *n araqrr

. oo\,t,

/JqE

Yz (*"ro;* *- +-*)=

ffi - Jn$-ip. Er - ,sg-ikg =

,,P *z*

.Y''a

re?TsuoC 'pa1e.r9aquT oq uao sTqfr,

&' 8F4 *o*(

-8a

*?*

=gi:1 b9lgves as thou-6-h all the particles -hpd -q,.-IpF-.,a-narl-t..fg-€-qy_9-lgy

c*o.ye s po nd i n g,t_o - *the- p*e_rjp_-d at- the- . grea! g. g !- 9_ryp-l i t ud e "

l*::.1-.*"*"y.:-!"S*--hg:-PJ-'1-11kl--*'l-H1'e: However, real beams donot have a un-iforn phase-spaee density with a sharp cut*off, so rre

must extend. the theory to more general iiensity distributions before

seeing its practical irnplications*

5" Non*uniforn d.ensity

let the phase-space d.ensity be g:iven by

of partieles present is then, from (&"2)

p(A) - The number

p(a) d(rAeu)

-' $) u"o(e)

N= I' 0

@

fI f a

/ \ II

/ q , r \

/ c r )

and. the response

mo&tfication of

factor averaged.

( t - : ) :

over the part ieles is the appropriate

1N

t t vl (ry**t ""I l v - - u -U

j-nteresting toT'c i- s integrate this b"y parts" n ^ 1 av d f J

^ [

p (A) z.n

] =v u

u

"arraru.g puE rqgru"olrroJJ s+uauuros rnJasn srrros p€q osr' J 'rrrJoJl.rn Jaq+€J sr f+rsuapao€ds-es€qd aq+ JT uan.€ 6suo.rle11Tcso

TeuorsrTaurp_aarq+ Jo .-ojr+

Jo as€c eq+ uT FtrSdurap nepu€T Fu;*19 ur aAT+caJJe aq f,r:ur f11-reau11*uou ruorJ salcuanba;J Jo pEarde ari+ +eq+ +no pelulod aH "+xe+ aq+ JouorsJa.rl dreuufig1a.:d e FulpeeJ JoJ uo+Jol[6d quEi{+ o+ a^eq J

s4uaue$preTlorqcy "9

"+T pulJ o+ TT€J fllensn qceo"rdde +atrTp arom e asn oqla sraq+o +nq{poqlau uoT+€nba--d.os?Tl\ aq1 Fug.sn ,fq +t punoJi or{ra 6

tr.TE +a ++oTsETq13u saa.rge (t"E) 1e.r8a1ug uols.radsgp sit+ uT .\; rng

*e splap{ (r"5) pue .f,.ress"oo "*'ffi"iil:I;ffi=";i;-::; -€'LroJap rr€rus rsnsn eq+ o = f1 +8 ora'-uou sr ,d Jr "(1.5) o+rrt(5-V)S- =, a Fu11+nd lfq (5't) reaooar uec auo d4caqc e sy .Fugqlou

e+nqTJ+uoc ifllsuep EroJFrnl .1o suolg* 61 uopcas ur se : 1e.:ge1u1qgq+ opTsuT uoT+cun.f gu-r+q€Ta&, eq+ sT . d +ou ,W/op

+eq+ acT+og

st0n 0,-

/s I JVLa

"]

w 'o..-1d (v). d oF gi-

zY 4 ' '" _J;\y

q+Te &I8Toc*a pepgno.rds+TnTT

e.r€ oa (?"t) 9ugsg "qgnoua IT€usq+eq +€ seqsT[s^ a.ra+ +srTJ sr{f

trfP a sn

*O!*

o+ T€nba sT(z'6 ) oeql

4 4

Appendix Ts@

Snall perturbations" . - . - - , . *

* f f i

N o a p o } o r y i s n e e d ' e d f o r e a l c u l a t i r r g t h e c a s e o f s n a l l p e r -

turbations (srnal3" g on pagp 5 p srnall B' SA' 6{r' in section J)'

for we are nainly interested i-n eriteria for stabiJity of statS-onary

i n i t i a l & i s t r i b u t i o n s , a n d f o r t h i s t h e b e h a v i o u r i n t h e l i n i t o f

srnall pertuxtations is prinorcl-tal* One leaves until later the

b e h a v i o u r o f u n s t a b l e b s c i l l a t i o n s a s t h e y g r o w i n t o t h e r e g i o n o f

non-snall amFlitude, and the possibility of finite oscillations

grow'ing in a systeo where snalL oaes do notn 'Some care is however

n e c e s s a r y w i t h t h e t i n e v a r i a b l e . F o r i n s t a n c e , w h e n c o m p a r i n g ] c

with xo on page 5 we effectively made use of

cos [v(a + an)t + r/i

*o" [u(A)t * sr] *6ut sin[e(a)t + 4']f t r \

'T+L V

i s

d' cos[u(a)t + 4']is trrre that ffieorrect in the lirit of snall

is - #- sin[v(l)t + {l ,

fA , But clearlY it is in

t l

I t l -

"o ( : , t )

fact

lor" t l <. 1

-Lhat is- i ;he condit ion for (r . t ) to be a good approximation"

So we InaY saY:

f,or any ltl one can ehoose 6p small enough to

nake the aPProxination good"

- But, for any 16' l Sreater than zero the

approximation will fail at su'fficiently large

behaviour is typical of' a'pproxiroate solutions to

equaticns "

(r"z)

This so:r-L af

r1i-f f eren'rial

-3

aq+ +€ +urod ?axTJ aTqe+sun u€ pu" aTq"+s € errETd-as"qd 8uT+€q'o-r a{4+

uT sT aral& uoT+"iuTxo'rdde lensn aq+ o' " (,

'&roaqi suTl{c€u J€auTT*uou

uoJJ uisorr{ sT acueuosal € qons o+ uo Jo asoTo rlaql s8ul:q apn111due

uo acuapuadep-n asor{ir selcl4*red Jo JnoTr€qaq aq+ .'raaerog "sa1a11*ed

ffi .:o.g; poo8 aq o+ uoT+€ugxo'rdde Jno JoJ q9noua TT€lIIs g a{€w uec auo

+fl{+ snoT^qo +oTr sT +T os 6 r" o+ s?ua+ 4 s? ,t+pr'rul o+ s?ue+ +€q+

apnlgldrne alclped e saa18 (ul'f ) +eu+ s1 'f,11nc1JJTp rat{+ouv -;

"TTEIns aq. o+ peJepTsuoc aq u€c asTolr Jo

uoT+"TnuIT+s TEuJe+xa TaaaT +€q$ o+ dn elelncT€c u€o auo 3a1e': Sulduep

uTE+Jac E q+T& lfllgqep slcrPa"rd '&:oaq1 aq+ JT ''f1:e11u15

'+nE+suoca0T+ aq+ sT t a;aqla

! >> J"rtg

'r)

-aET+ lral € +s€e1 +€ roJ ift1e11ueuodxa gupm;r9

L >> +P"49 rf

sT sTq+ dsluelsuoc

ueaq s€r{ dg JT pu€

'l+l eerelpeuroJisP "f1sso:r9

l?,l ttne "ro; '+ng -

"uoT+stIJoJe?pug Jo.rJa

puno.r oB

.fut .rog * u.ec ar l+l

: {a9ed

s{J"uer 1e11e.red fllctxg

w/ (v)a +

QT

((Z"f) Jo uoT+€sge"rauag +qgTTE € .fq) 'C15p11ea roJ uoT+Tpuoc rno d'rf1rm1s

9rr18uqcs1tlg.}I*poogsTuoT+€ttrpoactcteuoT+eqJn+;ad11ems.zno

Jaq+at{& f,llecprauru {oarp o+ oTqTssod sg 11 6uo115u9oceJ t"cTlce'rd *o3;

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qryro.rF uaq q+TA c{+TT.rq€+cuT Jo as€o cg;:1oads 'rftre *og

tt1que1c1;:gns +€ ?a+Jgtts-eseqd pua

eq. TTTA lCaql o.rez urqX ;alear8

aseqd a1q1g113au q+Te dsaluT+

rlaql 1eq1 q8noua IT€ms t esooqo

1I

IIIf

sarmT €q+ Jo rnoTi€r{sq aq+ o+ fldde

-71 -

*13*

resonant amFlitude, with topologY like Fig"2'

Fig. z

l$o part ic les go

a patte:'n ean be

without eausing

to unlim.itecl aqflitude, e.ren at large tu and such'

immersetl in a region of urriform phase*space density

anything special j-n the behaviour of the average x"

Finally a remark on the systematic disregard of the

harxron-ic ter-ms and the fact that we have wo:"lmd onf.y to the

in the coeffici-ent K* In a nachine wrth a hiSh Q-value '

lnerl:-um a"nd- high fiel'd- range of a machine with Q of orily 5

higher

f i rst order

or i n t he

o r s o , i t

"lf Jo sJo&od laq8pl pue scguou.req

.raqggq aqq lcalFeu 1e-rt pruu sX.gqs-S resuTT-uou t$nqs o+ aTq€uos€or

.(11cag:.rad sT +T ses€c qcns uI 'uoTxeuuoc +€ti+ ug a1q18118au 1ou

.reg: ttq sT os- 'rt+TTTqe+suT aTqTssod e i{+Te pa+€rcoas€ s+JTrts xaldutoc

ro TEar JCIq+o at{+ s€ 91q se +noq€ aq f,w pea'rds-6 r€euTT-Irou aq+

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aq+ er€ os put TTws sT X +€q+ asues 6q+ uT rtsauTT {11ec11ce*d s1

arrTl{celtr oq+ uat{.1 "JTes+T b t{+Ts pe.reduoc TIelIs .re:an' aq o+ s+JTr{s

g xalduoc pu€ TEaJ aq+ TT€ pue pea;rds b ari+ roJ uoutroc alrnb s3

- iTt *

* 15 *

Lp:gndjx II

Average frequency

The behariour of the eentroid. of a reg'ion occupied. by particles

can be irrferrecl, when the ctensity is uniform, fron the bound.aries aLone "This is what we ditl. with fig""l, in section 2, But another anrL more

general method is to work out r*rat happens to the snaL] tlisplacernent

vector g for a general particle, and then average this (weighted

with the dtensity if it i.s. not uniforn) over the whol-e oocupierl area,

We do this for a notlel in which all partieles move in cireles

in an x* p plsse-plane, 1ib

* = y (a )p

$ = -u("t) x

with u rtepen&i"ng on the anplitude A" Tkis is

mod.elu but Hanrjltonia^ns ea3 be invented. to d.o

a little coherent oscillation in ad.dition will

( x+5x ) " = v (n+dA) , (p+

(p n sp)u = * u(e + m) " (x +

(rr , t )

a somevhat artificial

it* Now a particle w'ith

have

op)

s*) (='"z)

Subtraeting autt roz{d.ag to first ord.er in $*quantities we get

u( l )sp + pau

=-u(a )sx -x6v

u(l+a*)* v(a)cly .^Er o^

{S snA

5'x

s"p(m. t)

where

tTrom. the flefiniti-on of K^

6u

. . .

/ Y-r ' I

JoAo FuT9?JaAV "aTeJTc

e o+ BulPuodsa"r.roc d

11c safcTlred .;:o 1es

v ar{+ puno"I

puB x q+Ta

€ oi€1 ie6u s!!asald ug

+nq. 6v ptr€ {dg'

9r:o1e ,tg'xg

;:6 , ''_

aa*@

=uYo

Exga

r:r

.II)

sqroceq (f"ff) os Pu€

:...,.

6d6x uoT+ceJTP or{+

ro+coa aq1 Bnppear Sq. (t"€?g) dr4ma8 aq+ trord

g'6,t

-91 *

*17-

px -'

2x " +

pa '+

0

L'/2

L'/2

and (fg 6) e so averaged, gives

cEx

a

ip

J i

So u( r +

for tb

ft (ul") = av

* ZAtr

i.ntegral and

u(t + r/z)op

+(t + r/z)s*

d v as the effecti've frequencY

of, the alPl-itutle-A Particles"

density of partieles from anpl"itutle

eoryfletely clear what is the interpretation

, but Te can anJrwaY calculate it :

(rr"s)

1T K/21 =-hv

A" F I h

#- | , (r+r/z)znaem uo

"" cli l' l l - @{ ' s aa

(r " vz)

nd.

Ar --'l m

lus" lL J O

(n.T)

t. n/t)*l = fu*Av "n

= u(nr)

*

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Tqqy{"caTTo+sod"Tod

+prsH'&

dno.:g :koaq6 591rir€af s aT?n+s auTqa€m

(uado ):uffiffi

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(z

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