C RIT ERIA FOR I D ENT IFYI NG AMP LIFYI NG WAVES AND A B S OLU T E INSTABILITIES

8/11/2019 C RIT ERIA FOR I D ENT IFYI NG AMP LIFYI NG WAVES AND A B S OLU T E INSTABILITIES

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haper 2

RITER FOR DENTIFYN ALIFYN AVESAND B SOLT E INSTABILITIES

This chapter deveops a genera method for distinguishing beteen ampifying and evanescent waves and for detecting the presence o f ab soute instabilitie s t i s emphas iz ed that athoug thesubje ct of primar c oncern in tis monograp is the be apas ma

intera ction, the subje ct matter of the present chapter is not restricted to this specia cas e Te aproach pr e sented her e is agenera one based ony on an anaysis of the dispersion equationand is terefore appicable to a ide class of uniform timeinvariant systems Section te proe m is defined and te instaiit terminoogy to be use d througout this monogr aph is expained Temathematica pro of of the method for identifying ampifying ave sand ab soute instabiitie s is given in Se ctions 2 2 and 2 Tepr opagation of a pus e disturbance is studied in Se ction 2 todemonstrate the equivaence beteen "amplifyig aves and"convective instabilities " The r eader inter e sted primari in theresult i find the criteria restated in Section 2 aong with somephysica interpretations and comments on the mathematca procedure for appying the criteri a The dis cus sion in Section 2 incudes a comparison of the present formuation t the other workson tis sube ct , some comments on te usefunes s of such criter ia ,and a brie f or d about te c once pt o f group veoc it o f pr opagatingaves in unstabe sy stem s Iustrative example s of the appicationof te cr iteria are g iven in Se ction 2 7 the se example s incude thesimpe quadratic equations obtained from te couping ofmodesformais

2 Statement of te robe

The genera type of system considered in this chapter is timeinvariant an d unifor m in (at eas t one spatia dimens ion (the z co or dinate B e cause o f this homogeneity in time and te spatiadimension z, inearized perturbations of the undriven sstem canbe take to be o f the for exp

(t kz] The re ation between

the frequency and te wave nume r k is g iven b the dispe r sionequation

, k ( 2 8


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Statement of he Problem 9

A poblem o inerest in plasma physics and esewhee is thaof determning the aue o "usable waves" or "instaiitis"

in such a system. A wave is said o be unsable f for some ealwave numer k a complex W + jW with negatve is ob-taned fro the dspersio equation signiying gowth n time ofa spatialy peiodic distubance o infinite extent. It was fis in-dicated y Twiss43,44 and Landau ad Liftshitz45 and vey cealypointed out by Sturrock46 that two ypes of instablitis can e distinguished physicaly: "convective" instabilites, and "asolute"or "nonconvectiv" instailities. n an infnite system a pulsedisturbance that is inialy of finite spatial xtent may grow time iout liit at v y point in space (a asolute instaility)

or may "popagate along the system so hat its amplitude even-ualy deceases wih time at ay fxed point in space (a convctive instabiliy)

It is perhaps at first surprisig hat a disurbance that is offinte spatia extent does not bow up in tme in v y case wherethe eal wave umers coresponding to unstabe solutions of thedispersio equation are "excitd." he epresentation of a spa tiay oundd distuanc, hov, qs he suppostiono many real wave umes in the fo45,46

f(k)ej [(k)tkz] dk2 ( 2 2 (see also Appendix A). The limiting value of his ntegra as t (and z is hed consant) is not necessarily infinite, even when someea k values yield soutions o the dispersion equaion [(k)] wihW snc a decaying function can be epresened as a supe-posiio of may rowing exponenias as in he usual theory o

Lapace ransfoms. he physica reason fo hs is that the pulsedsturbance my convect away fro its origi as gows amplude, as was pointed ou by Stuock.46

n a physical sysem the ampitude of the oscilation is ofcours limited by nonlinear ffects. For caity in the dscussions that follow, howeve, we will loosey refer to cases whethe linearized analysis idicates exponential growth i time as a"respons tendig o infniy"

Cearly the labeling o an nstability is always wih respeco a paticua eferece fame since a convective instabilitywoud appea as a asout instaiiy to an osev ovingalong with the "puse." Thus oe should realize that the tem"absolute" insability adopted hee does not imply gowth i timat every point i space in evey reeence frame


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Ampfyg aves ad Absolute Istablity

Absolute istabilty ovetve istablity

Figure 2 Evolutio of pulse disturbae i a ustabe system

To illustrate these deas sapsot vies of some hypotheti

al ovetive ad absolute istablties are sho i Figue 2 e see tat the stiguishig araterist of a absolute istability is that it "speads out" i both diretios at o so thathe the dstubae eahes a poit this distubae keeps ogroig i time at this poit The ovetive istabiity o theother had "propagates alog" the system as it grows i time sothat the distubae evetually disappeas if oe stads at a fixedoit othe phsial itrpetatio o the distitio betwethese to is that the pesee of a absolute istability impliesthat the system has a "iteral feedbak" mehaism so that os

illatios a gow i time ithout the eessity o efetiosfm some tiatio of the system heeas a ovetiv istabiity requies suh refletios (or a extera feedbak) foosillatios to gro expoetially i tme at a fixed poit ispae

I ma ases oe also may be iteested i the siusoidastadystate espose o a system at a patiua (a) fequey(It is uia howeve as the developmet i this hapte showsto asetai whethe suh a steadystate i tim a exist) poblem whih the aises is the itpetatio of solutios thatyield omplex wav ubers k k + ki fo this ea fequey. a "passive" system as fo example a empty waveguideoe ould state o puely physial gouds that this soutio epesets dea i spae away rom some soue; that is t epesets a "vaeset" (deayig) wave I a "ative" systemthat has a "poo" o eegy i its upetubed state hoeve (afor example a system otaiig a letro beam) suh a soutio oud epeset spatia

owth of a siusoidal time sigaThat is it ou epeset a "amplifyig ave" I ompliatdases it is ofte ot lea

phsical

hih situatio pevaisOe o e mai purposes of this aalysis is to detemie a mathematial poedue for distiguishig suh waves Note that ithe folowig e tems "amplifyig" ad "evaeset" wave ibe used oly i oetio ith eal vaues of the feque additio ote that w ae ot estitig the tem "vaesetwave" to the ase o ossess systems as is sometime doe


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rees Fuctio Formalism 1

I Sturrocks evelopet he coclues tat a covective i-stability is basically of te same type as an amplifyig wave. Tha

is e process of spatial amplificatio of a sinusoial time sigalis really a for of "spatal istability" of te system. Tis resultis also borne out by the aalysis i te follog secios a tecoectio betwee the two is cosiere i some eail i Sectio2.

Sice i complicate cses tere may be may solutios of teispersio equatio tat have complex k for real te crieriao amplifyig an evaescent waves eveope i the followig sec-tios are neee i orer to sort out ch imagiary part of coplex k for real represets te "spatial growt rate" of a co-

vective istability. In ost cases this growth rate i space at areal frequecy is a more useful measure of te stregth of a co-vective istability ta is te maximum egative imagiary parof for real k (temporal grot rate).

22 Green's Fnction Formalis fo the Response to a LocalizedSource

n orer to etermie te physica meaig of te roots of heispersio equatio as iscusse i the previous sectio we shall

cosier explicitly te excitatio of tese waves by a source. Thesimplet situatio to ivestigate is tat of a system ich is i-fiitel log i te zirectio a excite by a source cofie toa fiite regio of space (Figure 2.2). Te respose of te systeoutsie of the source regio is a linear combinatio ofsome of te "ormal moes or atural resposes" of te sys-tem. Tese nora moes are gve y te solutios to the is-persion equatio (2.1).*

U f o

sy se

I

S oe Ufo

s se

z

Figure .2. Driven system.

If we were cosierig only the questio of istiguishig be-twee amplifyig a evaescent waves we migt be tempte to

assume tat siusoial steaystate coitios prevail a pro

*The term "normal moes" will e tae to me both the solutios for k at some fxe fro Equatio 2. a the soutiosfor at some fxe k. Exactly wic situaio prevails i te fol-loig soul e clear fro te cotext.


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2 Amplifying Waves and Absolue sabiliy

cee o invesigae wheher or no any waves can be excied whichgr ow in space away from he sourc e regi on. Ther ar a lea s

wo objecions o hs procedure ) In order o negec he refe cions from erminaions of h sy sem, w mus le he lenghof he sysem approach infiniy before or faser han we le imeapproach infiniy o aain seady sa condiions . ( 2 ) W mayanicipae ha if ab solue insabiiie s are pr esen, he sys emil never aain such a seady sae.

Boh of hese difficulies ca be avoided by considering he exciaion of his infini sysem by a source ha is zero for OThis alws s o sudy he manner i which he sysem approacheshe s eady s ae , if indee d i doe s s o a a . f we ook a he as

ympoic ime respons of he sysem a some fixed posiion ouse of he s ource reg ion, we may find ha here is a dis urbanc eincreasing exponenially wih ime, in which case hre is an absolue insabiliy. On he oher hand, if hr e ar e no abs olue insabiliie s , his asympoic r e spons e , for he case of a sinusoidalexciaion, shod be sinusoida wih ime a he sourc e frequencyIf he asympoic ime response conains any norma modes haare spaiay increasing away from he sorce region, hese areclearly amplifying waves.

In his approach, we are suppressing he role of any ermina

ions of he s ysem in orde r o s ablish his basic "causaliy" ofhe wave s on he unifor m sys em; however , w should always keepin mnd ha hese erminaions may play an imporan role in hebehavior of a given phys ica sys em. This is dscs s e mor e fullyin Secion 2

e wil indicae he r e spons e o f e sys em i Figur 2 2 by hvariable , z , T ' which symbolizes any or all of he physicavariable s n he pr oblem He re T is he posiion vecor in heplane ransve r se o he z dir ec ion. Similary, he " sourc e fcion" wi b e wrien as s , z,

T

. Th r e spons e can be given in

erms of he source by a raion of he form

, z , T K , z - z , T , l s , z, d dz d

(2

where K i s e Gr eens funcion; ha is , i i s he re spon se a hepos iion , z , T ari sing from an impus e sourc e lo caed a , z, r"and he inegraion is over he "space-ime volume" occupied by he

sour ce s . er e , K is a funcion of and z - z , raher haneach of he s e variable s s eparaey, s ince he sysem is homogeneousin he se coordinaes . For noaiona convenienc e , we will ake hesource funcion o be of h for m

(2


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Geens Funcion Fomaism 3

whee f fo O. Thi s fom o f he souc e funcion is suffcenly gene a fo ou pupos e s e now pe fom aplace

ansfomaions wih espec o ime and a Fouie ansfomaion wih es pe c o he spaia coodinae z A Fouie ansfom in space can always be pefomed fo all finie imes becaus e of he finie sp ee d of p opagaion of any dis ubance Thefom of hese ansfomaions is illusaed now fo he soucefunions gz and f

and

g (z )

gk

f

[0-0to0

gke-kzdk2

g z ejkz dz

O-0-jo d

f)eJ

2

f 0 fe j d0

2 . 5

2 . 6

2 . 7

2 . 8

The inegaion i n Equaion 2 . 3 is ca i ed ou along he e alk axis and he inegaion in Equaion 27 is caied ou aong

he line i The inegaion in Equaion 2.7 mus be caid ou beow al singulaiies of f in ode ha f be zeofo O. Simia ans fo ms apply o a ohe quaniie s; he s eansfoms wil always be wien wih he same symbo as hephysica vaiable excep ha he funcional dependence is eplacedby and/o k

Fo he pupose of disinguishing beween amplifyng and evanescen waves a some ea fequency we will usualy considean exciaion of he fom

2.9

and heefoe


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4 Amplifying Waves an Absolue Insabiliy

f 2 .0

Also noe ha s ince we are assuming a localize source, gz for z ; from his fac an from an inspecion of quaion 2 6i follows ha gk is an enire funcion o f k has n o pole s in hefinie kpane as long as gz is a reasonable" funcion of z Asan exampl of g z , cons ier h spaial puls e sho in Figure 2 . 3 .The ransform in his case is jus

z

I

NFigure 2.3. Example of gz.

-

k sin kgkd

2 .

By applying he ransforms o quaion 2.3, e ransform of

he response can be wrien as

2 .2

were

2 .3

The funcion G , k, T is ju s he r ans form o f he Gre en sfuncion "weighe" by he ransverse epenence of he sourcefuncion. s imple cas e s , e sourc e funcion T rT can be chosen so as o selec only one of he ransverse eigenmoes for cons ieraion a a me , alhough i is no nec es sary ha his be one .The acual response in space and ime is recovere by applyinginverse ransforms; i can be wrien in he form

z

=

1+0/+o-jG(w k)f(w)gk)e

(w

t-kz) d

dk

21 o 2 .4where he epenence on T is suppresse from here on for simpliciy in noaion.


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Proof o f rieria

Equaion 2 4 summarize s he de sir ed formais m; in he nexs e cion he gene ral charac er of he aspoic imi of he r e

sponse in ime wi be deermined by invesigaing his inegraexpression.

23 Proof of Crtera on mplifyng Waves an Absoute Ita-biites

The general formaism expressing he response of an infinieyong sysem o a localized source a is "urned on" a 0 wasdeveloped in he las se cion. In his s e cion we shal sp ec ializeo he ca se o f a sinus oida sour c e in ord er o bring ou he appear anc e of ampifying wave s mo s cleary . The re spons e given byEquaion 2 .4 can be wrie in he form

, z

wher e w e define

dF zfeJ ,2

j dkF, z G kgke-kz 0and where f is given by Equaion 2.0

2 .

2 .6

The inegra in Equion 2 is carried ou along a line belowhe real- axi s , as sho in Figur e 2. 4. The causali condiion

w

r

aplace coo

'/ /F z aatc

sae eon

Figure 2 .4 . Anayic r egion of F, z .

demands ha F z be analyic below he line i in order hahe r espons e be z ero for O A quesion which immediaelyarses is how large mus be . The answer o his que sion wilbecome cearer during he discussion of he anayic coninuaionof F z; however , one can pr edic in advance on purely phys ica


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6 Ampliyig ave s ad Ab so lute Is tabil ity

gouds a it sould suciet fo to be age tan eas te s g ot ate i time o ay us table mode at i s

sould be lage tha te maimum egaive imagiay pat of o eal k2 F(, z) a s a Sum o omal ode s e uion F( z )

otais e z depedee o e e spose ( t , z ) Pysially eko tat te espose i a soueee gio z > d soud beepessibe as a sum ove e nomal modes o te udive sysem I simple a s e s , e Ge e s utio G(, k) as pole s ite ompex kpae (o some fed comple on e Laplaeotou) a just te oma mode wave umbe s es e ae the oot s of e disp e s io equatio ( 2 ) fo that paticula Fom

Equati 2 6 , ad e a a g( z) 0 o z > d it ollows hatg(k)e kz 0 o k j ad z > d As a eample oside thepaiula g (k) futio give by Euatio 2 e itegal iEquatio 26 a eeoe be losed i e loweaf kpae oz d as log as G(, k) is suiietly ell behaved at k j (Figue 2 ) his as sumpio is a e as oable oe si e this losu eo e itegal allows F( z) to be expessed as a sum ove te app opiae omal mode s by e heoy o es idues oe at g (k)is a etie utio ad teeoe does ot otibute ay temsto e esidue evaluatio

x

=kr

x

Foe cotoclose as so

f o

z

Figue 2 F(, z ) as a sum o omal mode s

I mo e compliated a s e s , G( k) o a ied a have b ahlie s i the kplae his a be ite peted pys icaly as a tiuum o omal mode s as o eample he Va Kampe modesfo logitudial osilaios i a ho ollisioess plasma o inas e s ivolvig adiaio fom ope sucue s Fo s impliitye sha ot oside e s e as e s i e followig dis us s io is

appoah an be eeded o ove hese ases but eah ase ivolvig suh ba lies mus be hadled idividually As a example , e bac lie s ha o c cu in te ase of a ho coi si onle s s plasma ae ons ideed in Appedix B

Havig es ited ou se lve s o he a se ee e oly siguaities of G( ) i he pane fo some o he Lape otou


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Poo o Cieia 7

a e pole s a he o os of (, k 0 we ca wie he ucioF(, z as a sum of omal modes i the om

(2

o z d he sum i Equaio 2 7 is ove all o os + o, k 0 ha have wave umbe s k i he lower -hal kplaead wher e is some qu cy o he Laplace coou (i

Fo z < d, e iega i Equaio 26 is closed i the uppehalf kplae , ad a s imila ep e s sio fo F( , z valid fo z

- d

i s obtaied , ec ep ha he sum is over al poe s o f G(, k i theuppe-half k-plae [k(] oe hat te depedece o e ucio F(, z o z is hat of (a sum o epoetial e ms ha ald ca

away fom the souce egio o ay o he Laplace coou 2

Aaic Coiuaio of F( , z he detaied e spo seo ay give physical situatio coud i picple be compuedby caryig ou the pescibed itegaio alog the Laplace coou i Equaio 2 Sice ou ai is oly o discove somegeea haacisics o the asymptoic espose, howeve, iis coveie o deom he Laplace itegatio as far as possibleito the uppehal -plae . I this p oc edu e , it i s clea hathe espose i he liit as t is goveed by he lowest sigulaiy of F(, z) f(} i the -plae I paicula , i F(, z} i saalyic i he eie lower-ha plae ad alog the eal ais,he doma em aises om he pole o } at ' sice the cotibutio fom he e st o f he itega be come s epoeially salas (Figue 2.6

'

Figue 2.6. Deomed Laplace coou o as ympotic e spos e o asysem wih o absoute isabilit ie s .

Fom the p ec edig discus sio, he sigulaiies o f F(, z} aecealy of pime impoace i deemiig the asymptoic ime

e spo se of he syste m o eplo e he aalyticiy o F(, z}, iis coveie o hik of hodig the eal pat o ied whievayig the iagia y pa of Figue 2 7a aualy, tisp oc e s s mus be epeaed o al ea pa o Fom te o igial deiiio of F(, z, as gi;by Equatio 26, (ad fome epr e s eaio i Figue s 2 ad 2 7, it is clea tha F(, z


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8

w

Ampliying avs and Absolute Instability

Fourier contour in Eqaon 6x

w-

.

u -Jx As e f eque vesas so, te poles ofG ) te ple

ove s so (b).

Pole cosig

e axFigue 27 Eploing te anaytiity o F(, z

s a el-behaved untion o unles s on e o te pole s o G(, k

ro s s e s the ea-k ais , as i s iustrated in Figue 2 7b entis o ur s , F(, z as deined by Euation 2 6 , umps in value(as e oss e equeny by a amount equa tothe e sidue at te pole that os sed the ea-k as at s , ithe -plane , te line s o omple o eal k obtained by so lving(, k = 0 o al eal k a e ban ine s o te untio F( , zas deined

b E quation 26 (Figue 28

}otous of

o ple

f o ea

wFigue 2 8 B an line s o F(, z

For a pole o G( , k to os s he ea-k ais o some i teloea pane (su as - i Figue 2 7 it mustollow tat e dspersio equatio (2 yields omple sotionsith negative imaginay pa ts o some e al k at i s , it mustbe tue tat te iniite, homogeneous system suppots unstableaves It i s no lea tat shoud b e ho sen larg r tha temaimum

roth

ate in time o an unstable ave to satis theausalit r ui m nt, as as stated beoe e assume atis aimm grot rate i time is bounded, that is, that tereae no nstable aves it an ininitey ast goth ate in time

e untio F(, z an be analytiay ontinued through teseban ines tis anayti otinuatio is eeed by edeig

F, z i its intega orm as

' F( , z G( , kg(ke-kz (2 8


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20 Ampyng aves and Absoute Instabty

2 . Ab so ute Ins tabtes he e s a oe undamenta duty th the anayti ontnuaton o F( z) hen two poes of

G( k) e ge though the ontou C n the k-pane to o a doubeode pole (Fgue 2 0a ) . Sne the nteg aton n Equaton 2 8

cnu ckr

:W Ws-us ;_ _ .u - -l Ws Wrs JUs

(blFgue 2 0 egng of poe s though ontou C leadng to anabs oute ns tabty

us t be aed out beteen the to e gng poe s ths e sultsn a sngulaty of the unton ( z ) at that value of In Fgue 2 l 0b a s e le t the paa ete s tend to zeo the topoe s of k ege n the kpane as sho One oud expe t ntutvey thout pefo ng any detaled ag eb a that the fun

ton ( z) shoud tend to nnty n the t as - s tends toze o sn e the ntegaton path C beo e s s tuk and annot bedeo ed aound the to e gng pole s *

he appeaane o ths sngaty s pe haps best se en othe expesson fo ( z ) as a s um o e sdue s at te poe s ofG( k) as gven by Equaton 2 7 At a doube oot o f k fo e(s ' ks ) e ave (/k G

-) ' ks 0 and the dspeson equaton

nea the doube oot s appox atey

( 2 2 0 )

ote aso that the ndton fo a doube oot o k fo the dspe son equaton s the sa e a s the ondton fo a sadde pontof the unton (k) that s /k = O.

Oe an easy hek by ee enta eans that the nteg aaong th ea -x axs o [/ (x - j)] o [ /( x - j )Z] s finte n thet as tends to zeo heeas the ntega of [ / (x - j ) (x + j)tends to infnty ike as tends to zeo.


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Proof of Crtra

If w us Equaton 220 n Equaton 2 w fnd that

z 2 (8G-I 8ZG 8k ks' s

2

( 2 2)

ar s for tr z > or z < (Equato 2 2 corrct wt a sg wc can b dtrmd oy from a ta corato of t po oc Trfor ths mggof t pos of G k troug t cotour C as to a brac

po of z at = s ot aso tat ts branc po obta for bot z > ad z a tat t xprsso for F zs t sam bot rgos for s Ts brac po of , zos ot ars f two pos tat ar bot bow or bot abov thcotour C mrg to a oub po c two trms or on)t tr t sum of rsus Equato 2 7 a t can

b sown to cac ach otr n th mt s Ts s to bxpctd, sc a oubordr po tat s s of a codcotour maks a ft cotrbuto to th cotour tgra wnt tgra s vauatd by rsu cacuus at th mt

of as s ts to zro s now ft bcaus th ntgraton pth C dos ot btw th mrgg poThs brac po of { z must b takn to account th n

tgraton t pa a t owt sguarty t pabcoms t domat trm as t (Fgur 2 ) ot agan

gur 2 . Intgraton pa wt about stabty

tat ten ireowraf -pamust bxpordto dtrm

c s t owst suc guarty I ts cas w av a absout stabty bcaus th sturbac bowg up tm at

every pot spac In th mt of t th asmptotc r-spos ca b vauat as

Ts formuato of t conto for a about stabtyas b gv by Drfr6 for th cas of t oubstram stabty a pasma


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22 Amplfyg aves ad Absolue stablty

(t, z )

(2 22

where the double root of k ours for k ks ad rs


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Convetive Instabiliies 2

and o this ea son it an appear on one side o he sou e ( s ayz > d) and not on the other (z < d) (e eall that te banhpole type o singulaiy arisig rom megg poles, hih asdsu s se d beo e, must ne e s sarily appea on both sides o thesour e ) I e is in the loe hal plae , it ollos ta in ourmodel o the system e ae alloing o unstable modes ith anininit l shot av l n

h e an teeoe ignoe hese essential sngularities o z) i the ormulation o the stabilityiteia i e ae analyzing a model in ih thee ae no unstable modes ih a inte goth rate in time o an inntelyshot avelengh k hese essenial sngulariies o (, z)oen do our at eal equenies hoeve , hen one us es l os s

le s s model s o the system , as , o example , a old , ollisionle s s model o a be amplas ma sys em A simple eample o thisype i s briely dis us se d in Se tion 2

2 Response t o a Pulse in ime he spaetime dependeneo the as ymptoti r e spons e a s given by Equaio 2 2 2 i s e s se ntiallyindependent o the soue ime untion hen thee is an absoluteinstability presen* In patiula, i (t) oespods to a pulsein time , then () i s an entie untion o and the asptotir e spons e is alays dete mined by he loes sngulaity o w, z )in the plane , even i this s ingulaiy is i n te uppehal plane

he onsideration o a pulse exitation also veies ta heiterion deived is onsistent with he physial desription oab solute instabilities gven in Setion 2 Fo a puls e s oure ,te esponse at the instan the soure ampliude etuns o zeroill be o nite spatial exten, sine e spee d o popagaion oall signals i s inite he behavior o this iniely exended aveom on he undiven system as t an hen be deemiedthis response ill reman inite o deay i time at every ixedpoint in spa e unle s s z) has a branh pole in the loerhalplane , hat i s , unle s s the s yste m suppot s an ab solue insability as detemined by te iteion given n Setion 2

In the nex s etion, a mo e detailed analysi s o the popagationo pulse disturbanes is given i ode to bing out some additional aspets o te spatial and tempoal goh o signals in unstable systems

2 Popaation o Pulse Distubanes and Relatios Beeenempoal and Spatial Groth Rates o Convetive Instabilites

he us ual statement on the stability o a ave i s hethe o r

not the dispeson equaion ields omplex o eal k (ithi < 0 I as s tated in Se tion 2 that a puls e distubane on

*his statemet is u e s o long as te sour e doe s not have anexponentially inreasin amplitude at a ate that is geate thanthe goh ate o the absoute instabilty


17/39

2 Amplyng ave s and Abso lute Instabty

a system s unstabe by ts denton l aays bop n ampltude , eve toug t may appea to deeae n tme

a a ixed pont beause oud onve along te system as tbos up e vadty o ts statement s not obvous om teanayss n Seton 2. sne e aays nvestgated te aspot e spons e a a xed pont n spae e statement s an mpo tant one , oee , and s e sabsed n te p es ent se tonIt s son, by aong z and t o tend to ninty at a xed rao(veoty) tat a veo ty aays be ound o a pusedsturbane appeas to nrease exponentay t tme at temamum gro rate o any unstable ave [mamum ( ) oea k] s veo ty s popo sed as a denton o e popagaon veoty o e pulse on an unstabe sysemSturok, n s poneeng ork, oted a vey ose oneto beteen e oepts o ampyg aves and onvetve nstablte s s oneto s epoe d n te s e ond pat o sse ton, e e t s son at te deee te se oneptses oly n te extaton beg onsde e d (puls e o snusoda tme ) ad no n te pope tes o te medum It s s on, oexampe , at n a sys em suppos ovetve s tabesad as no absoute nstabltes, tee must exst one o moeo ot s o te dspe son equato t ompex k o ea

ae ampyng aves (tat s , tey ente e e spos e on esde er e tey appear spatay gong and not deayg) Inaddton, upper and loe bounds o e maxmum ampatonate tems o te temporal got ate o onvetve nstabtes are gven

2 Popaaton o a Puse Dstubane te anayss oSeton 2 te esponse as vestgated n te mt o t o ed (nte values o z o demonstate tat a puse dstubane does bo p n ampltude (eve t onvets alogte sys tem) n evey a se ee te syst em suppor ts usable

aves , e sa nve stgate te mus e r e spos e o e sys temn te mt o t ad z t

z = V + ( 2 . 2 )

ee V s a eta ied veloty ad emans te eoud, o ou s e , ande ts p obem by ans omg to aeeene rame movg t respe to te aboatoy amee oose no to do ts beause o te ompaons ntoduedby a eatvstaly oet tansomaton e asymptote sponse e aluate te eoe , s e oe measued n te ab oao rame t te abo aoy tme t, en te mea sugsrumet move s t te veoty V. )

e plo o omplex o rea k n te laboatoy rame sas son n Fgue 2.2 ee ( s te maxmum negatve m


18/39

Covecve sbes 2

mum

." -Slo

.. v ( ).

Fgue 2 . 2 . Skech of coe fo el k

gy o f fo e l k he ll be oved h obseve ovg h velocy

)V - k=k ( 2 . 2 6 )ll se e he dsubce c e s e s e ) s he veloc iy gve b y Equo 2 . 26 s se s be defo o f he o gon velocy of he puse o unsble syse he k she el ve umbe fo hch he u egve mgyp of fo e l k o c cu s s he co sod el

ofo ove s esul e e he ulse esonse l z ) s

fuco of d he poso z ' h z) beg gve byEquo 22. Fo Equo 2. he uls esons cbe e s[+j 0 . .1 ) _ G k) J

(w-kV)t -Jkzo dkdw' e e

( 2)J -0

(2 27 )

Fo muls e ec o e d s ce k f)g) f e defe e f equecy vble

= kV ( 2 . 2 8 )

hen we c e he espose fo whch s coleely logous o h gve n Equos 2.1 nd 2.16

( 2 . 2 9 )

hee


19/39

2

ad

Amifig Wave ad Absolue Inabili

J

= G{ k)e -jkz dk2

G{ k) G{ kV k)

( 2 0

( 2

The compuaio of he resose a ow ro cee d i exactly the ame mae r a hat give in Se cio 2 , excep tha i r epac ed by

We now oe ha i V i e equa o V (Equaio 22, he

poit k = ad = r - j i a sadde oi o (k) a doub eroo of k a hat becaue

ddk

d dk - V at k = k

(2 2

I additio i i clear from Equaion 228 tha = m or re k and therefore he maximum egaive imaginar part

o {k) fo r re a k i o equa o For h re a o we kowha o poes of G{ k) can have crosed he rea-k axi for anym < - that i he C c oour deined in Se cio 2 i thereal-k axi for m - ( Figure 2 The wo coractig

Pols gg og contour: Fn va ato:

Figure 2 Locu of pole i k-ane

pole s that for h sadd e poi of (k) on he r ea -k axi (ak = k) merge a ho in Figure 21, ad thi addle oi of{k) i a inguariy of { z) because he pole mus be mergig hrough he C conour I then olow fro he of Sec ion 2 hat he r e on e a ay iiial po iio z icr ea e in

tie a exp ) for he choi ce V = Vne ight worry a bi about the s ec ial ca e where he er gingpole o in Figue 2 ju t "gaze " he r eal-k axi Ti iuaio would leave oe doub about wheher or ot he poe acualy cae from opoite halve of he k-pae or < The aroach o o he sadde poi owever ca be made a


20/39

Convecive Insabiiies 27

ay ange beween 0 and 18 0 and no a 9 0 a sho inhe igure

We ao noe in passig ha i is quie obvious from he preceding formaion ha no vaue o V can ead o a aympoicre sponse which incr eas e s aer han exp ()

2. 4 2 Conne cion Be wee Ampifing Waves and ConveciveIs abiie s The anaysi in Se cion 2 indicae s ha her e ia very co s e connecion be we en ampificaion and isabiiy; anampifyig wave ms as o in a ens e be an uns abe wave sinc ehe condiion for he ro o o c r o he re ak axi for some inhe ower-haf pane is precisey he same as he condiion forcompex wih a negaive imaginary par or rea k I is cear

her efore ha a nec e sary condiion for a sy em o ppor ampifying wave i ha compex wih a negaive imaginary parbe obained from he dip er ion eqaion In he ca se o f a y em free from aboe inabiiie we migh expec iniiveyha his condion hod be ficien a we ha is i shoudenre he exisence of ampifying wave for some rea freqencyThis suficiency i proved in he foowing

We now speciaie o he case of a sysem ha has no absoeinsabiiies and i driven by a ource of he form

( } (z}f(} (2

If we choo e a sins oida or c e for f(} as wa done in Sec ion 2 he asympoic response is gven by ( z} wih he(rea) freency of he sorc e (Reaion 2 9 ) We can her eor einerpre F( } as he "seadyae response of he infiniesysem o a insoida drive

If on he oher hand we excie he y sem wh an imp esorc e f(} ( hen he imps e re sponse t ( z) can be givenin erms of hi "seadysae response a

f+ dt ( ) F( z}e 2 ( 2 4 The inegraion ovr can be carried ou aong a ine paced

an infiniesima amon beow he rea axi becase we are resricing or aenion o sysem s re e from abs oue inabiiie Equaion 24 provide he desired connecion beween sinusoida

and ps e r e spons e s and hence a o beween ampying wave and convecive inabiiieIn he firs par of hi secion i wa sho ha ( may

bow p in ime even for convecive insabiiie if e ake heimi and z wih z and r eaed by Eqaion 2 2 Inhe pr e sen for mais m he r e sponse in hi ame imi can be


21/39

28 Amplifying Wave and Abolue Intability

written in the following form where we expre F z a a umover te normal mode by Equation 2.1:

2 .35

We cider explicily i te followng oly the cae V 0 adamplificaio in te +z drecio imilar re mark apply to hecae V < 0 ad z < O

I he pr evou aaly i it wa ow tat I z icreaea exp t a 0 for V V wi V given b y Equaio 2 26 .I i clear by inpe cion of Euatio 2. 3 5 sice i real in thetegraio tat oe f te noral mode mu be an amplifyingwave over ome band of real frequency for te iegral i Equatio 2.35 o diverge a In fact for V = V e itegralin Equation 2 3 will icrea e m ore slowly tha exp MVt where i the maximum amplifcaio rae maxmum m k for real

-

(ikr)-5Iope= "

-:

I

w

( (bFigure 2 .14 . Sketch of complex k for re al

Figure 2 . l 4a We ave te r efre e abl e d te following lowerbound on e maxium ampfication rate of a ytem free fromabolute instabilitie

M V max i for real kr at max

k 1

2 . 3 6

The exact evaluation of the aympoic limi of I t z from Equatio 2 3 5 could be ac complied i priciple by a addlepoin technique. I thi metod te domian conribuio o te inegralcome s from iteg raig through the po it of atioary phae wher e


22/39

Appication of the Criteria 29

(237

Note that chooing the veocit V eqa to VM (Figre 24b)wi make te point k = krM M W = M a point of tationarha e and hence for thi veocit he r e pon e wi incr ea e aexp (VMt) a t 0 Since we howed in the fir t par t of thiection that for an veocit V the repone wi increae moreow than exp (t we can ao tate te foowng uer bondon the maximm ampification rate

(2 38

In proving that thi i an per bond however we have madeu e of the fact that ki/ = 0 at the maximum of ki ( for rea herefore we have actua aumed in thi proof that the ampification r ate i e than infinit ince the er o der ivative ofki at the maximum woud not e true for a cae in whch ki haa poe at M Such cae are ometime obtained when ideaiedmode of the tem are ue d

2 5 Comment on the Aication of the Criteria and Some Ph-ica Interpretation

2 5 Amifin Wave The cr iter ia on ampifing and vane cent wave deveoped in Section 2 2 and 2 3 c an be tated inthe foowing manner o decide whether a given wave with acompex k = kr ki for ome rea i ampifing or evanecent determine whether or not ki ha a different ign when thefr equenc take on a arg e negative imaginar part If it doe then the wave i ampifing other wi e it i an evane cent wave

i tatement ha a impe ph ica interpr etation i f we thinkof driving the tem with a ource that i ocaied in pace andha an exponentia increaing inuoida time dependence* Fora ufficient ar ge exponentia growth in time of the or ce theprincipe of

cauait

woud imp that a wave houd deca awafrom the orc e The re fore an ampifing wav hod have theropert that it growth contant chan in a the frequencacquire a ufficient arge negative iaginar part correpondingto exponentia growth in time

Norma ince one woud ike to know wich wave are apifing over the entire range o f r ea frequenc the ocu of the

*Thi interr etation wa et ed b Proe o r A Be r priorto the athematica deveopent given in Sec tion 2 3 and wa aubtantia timuu toward it deveopment


23/39

30 Amplifig Wave and Abolue nabii

roo of k from ( k 0 mu be aced n he complex k-paefor man uch real frequencie I i conveien although no

e ce ar re cor d he lcu o f he e roo hodig he rea parof fied (Figur e 2 5 . Mahemaica hi oper aion i ju

2 2

Figure 25 Mappig of (k) for a ampifig wave

a mappig of he lie of coan rea par of io he compexkpane ha i , our r e ulig locu lo i r eal a conour ap

rer e enaion of he fcio (k i he comlex k-lane For ade r ion equaion higher ha fir or der in and k of cour ehere wod be ma hee (or brache) of he funcio (k) beingrace d ad u one as depiced i Figure 2 . 5 .

2 5 . 2 Aboue Iabiiie Thi mapping ope raion al o idicae wheher or no a ab oue i abiliie are r e Anabolue inabili i obaied wheever here i a double roo ofk for om e comlex in he lower -half plane for which he womerging roo come from differe halve of he comex kplane(upe r and lower ) when ha a large ngaive imagiary par ( Fig

re 2 . 6 he codion for a double roo of k a k k for ome i he ame a he codiio ha he fcion (k) have aaddle poin a k = ince ) ( - k ) near hi poiThe general form of uch a addle poi houd be a ho inFigure 2 6 I apearance hould be obviou when he lie ofco an real par of are co rced in he klane even if hedni" of he e line i fair rogh a will be eviden whenhe crieria are applied o variou example i he laer chaper

There i an inereing feaure abou he addle poi of (k) iluraed i Figure 2.6 which alo occur i everal phical ex

ample di cu ed in Chaper 4 ad 5 n he figure he roo haha complex k fo real ener in he re oe for d when r > r over he range of reafre quenc be ing con ide re d Since ki 0 for real , hi mode iin a ene, an ampifyig" wave for r r and a "evanecenwave for r < r Sice he a mpoic ime re pone ari e argel from he coribuio ear he adde oi i he pae


24/39

Applicatio of the Criteria

E n

pon f oZ Root ofcompx f o l En t npo f oZ >

3

2 Figure 2 6 . A addle poit of (k idicatig a ab olute itability Cir cle idicate r eal quare idicate addlepoit

ad ot fro the itegral alog the real -axi (ee Figure 2.11,thi fact i of little co equec e i the cotext of the r e po e ofa ifiitely log y te to a localize d our c e . It could pe rhapbe of igificace i a fiite y te however a i br iey di

cuedi Sectio 2.6.It i importat to bear i id that oly the addle poit of

(k that correpod to a erger of root fro dferet halveof the coplex k-plae idicate a ab olute i tability. The fol lowig dicuio preet a phyical iterpretatio for thi retrictio.

Imagie a ifiite yte excited by a ource that i a patialimul at z = . If the ourc e ha a coplex frequecy with theiagiary part of thi frequecy larger tha the groh rate of ayutable wave i the yte the wave ut all de cay away frohe our ce a poited out befor e Figure 2 . la Thi idetifie wc wave appear for z > (k ad which wave appear i ther e po e for z < (k If we viualiz e the growth rae of the

Repone at a fed tme

t

1-jk

+Z,z

Impule source Jk 1-e z(a = r - j k * k_

Rpon t f d tm

/( c

.

Z

(b) w = Ws = wrs - jcs 'k = k i uer-half k-lae

Figure 2.1 Reonace coditio i a ifiite yte


25/39

32 Amifig Wave ad Ab o ue tabiit

ource a decreaig the for ome comex = r - theiuatio might are where k+ ad k be come equa. Clearl

he pr e ece of a atia imu e te of our c e mu t caue adicotiuit i the reoe or oe of it aial derivative atz = Whe k+ = k_ however we ca form a reoe thatdoe ot have a dicotiuitie ad ca be moothl joiedacro z = 0 Figure 2.b). Thi ituatio i therefore a tpeof atial "reoace" of thi ifiite em at that particularfrequec ( ) becau e i re ec e doe ot r equire a ourc e.Thi i re ci e what he aali i Sec tio 2. 3 idicated . fwe excite the em wh a ue i time raher tha a iuoid,f() i a etire fucio of i Equaio 2.1 5 ad the re po e

of the te i deermied b he gulariie of F( z ) in the-ae . Al of the iguarie of F(, z ) (excet he e eialigulariie ariig from the ifiie hor waveegth "reoace " di cu ed i Secion 2 . 3 .4) oc cur at pre ciel the fre quecie for which w e have uch a "oiig" of k+- ad k-tpewave ice thi oiig" i precie the ame a he tatemet tat two root of he dierio equatio merge throuh theC cotou, a wa di cu e d i Se cto 2 . 3 4 I fact it waho i that e ctio ha the " ead- ate r e o e" F(, z ) i the fuctio of z for z 0 ad z 0 for i agreemewi thi hica icture of a "reoace" of the ifiite em.

. 5 . 3 Aicatio of the Cr iteria i Sime Ca e . A waetioe i the reviou dcuio, the aicatio of theecrieria requre i geera hat a raher comete coformalmaig of the fuctio (k be ca rr ied out. Ufor tuael thiuuall require exteve umerical comuatio however iome cae a few hortcu ca be aplied which eae the labor.For itace if i i kow hat o compex in he ower-haf-ae are obtaied for rea k the there i clearl o poibii of either amlifig wave or abolue iabiiie .

Aother technique which i omeime ueful i that of aalzig the dierio equatio for comex = r + i with 0 thi imit ofen allow a explicit oluio for all oible k evefor higher -o rde r di e io equatio . A a ime illutraioof thi t echique coid er te die r io equatio for ogitudialo ciatio i a oe-dime ioal beam-la ma tem . For acod be a ad a cold ama thi i ( e e Sectio 3 .1)

Aw, k

For ther are pex rt fr for k are

v

( 2 3 9

If / / 0 the roo(2 .40


26/39


27/39

34 Amlifig Wav ad Abolut Itabilit

a . Acc or dig to i cu io i Sctio 2 . owvr bot root of k ar wav tat tr i t dowtram id

w t tm i citd b a localizd our c . A a mattrof irt locu of t root i k-pla a t gativimagiar art of W i varid i lottd i Figur 2.18 for vr al vlu of ral W that t r oagatig wav witk do idd mov ito upr-alf kpla for mallgativ imagiar part of Wi owvr it vtuall cro tral-k axi for ufficitl larg gativ imagiar part of .

vo p b

3

3

Complex wave fo / e f r e q u e n y

\

4

75 95 1 1 5 3 4

0. 1o

4

Progatig av

wth < H

W

P

O

I

r

.

-

0 Figur 2 18 . Locu of k for odim ioal bam-plama

m ( Figu 3.2) .Ti brakdo o f t coct of grou vlocit i a tm

tat uot utabl wav i alo raoabl from th moru ual ictur of t r opagatio of a pul o t tm Thpoit i tat t atial Fourir ctrum of t pul if uli fiit i t , d ovr al ral wav umbr o tat tdomiat cotributio a t 0 com from rgio wr coml for ral k ar obtaid ad ot c ar il from ak i tquaimoocromtic ctrum of k at t = T logical t

io of t coct of grou vlocit to tm that uort utabl wav a rgard t roagatio of a atial ul wagiv Scio 2.4.

2 6 2 C omarison wih Previous Work. The criteria der ived

in tis capter diffe in one way or another fromte criteria tat

have been previously ublished (see References 4, 4, 5 1 - 5 3 ) .


28/39

Dicuion 3 5

A o f he e work incuding he pr e en formuaion have re ied upon an inve igaion of on he di pe r ion equaion o e

abih he crieria on amifing wave and aboue inabiiieFor hi re a on i i of iner e o copare br ief he e cr ieria wih he reen formuaion

In Surrock pioneering work he recognied he fac ha hedi pe r ion equain hod c ontain he nec e ar informaion andhe eabihed he mehod o ooking a pue in pace or imewhich wa foowed aer auhor (A br ie f indicaion of higenera approach in he cae of diinguihing beween aboueand convecive inabiiie wa given b andau and ifhi Even in he conex of hi fr muaion however Surro ck did no

car efu con ider he effec of he branch poin of (k) and k()when deforming conour in he compex k and pane a wapoined ou in Refer enc e 5 and 5 2 . The e inguariie ar e oimporance in an d per ion equaion which i of higher or derhan fir in and k for hi re a on her e i ie cor re pondence of Surrock reu wih he reu f he preen formaion.

Fainberg Kuriko and Shapiro conider e apoic behavior o a diurbance ha i iniia in he for of a paiapu e , in or de r o diinguih beween ab oue and conve cive in

abiiie . The cr ieria he obain ae ha an ab o ue in abii reu whenever here i a adde poin of (k) in e owerhaf pane and beween he conour of compex (k) for rea kand he re a axi Figure 2.9 Her e i one of he brance hee ) of he func ion (k and i i imporan in heir cr ier ia

S le o o

r

C oo of _.

colex f o ea

Figur e 2 9 Cr ieria of Fainber g Kuriko, and Shapiro .

ha he adde poin being conidered ie on he ame hee of

(k) a he conour. Thi crierion i imiar bu no idenicao he one derived in he pre en work he requireen ha headde poin be inide he conour of comex for rea k andhe rea k axi and on he ame hee i amo e ame ahe re quiremen ha one o he roo of ( k ) = 0 in he kpanecro he r eak axi and merge ino a doube roo of k for ome


29/39

6 Amlifig Wave ad Abolute Itabilit

a e brig te frequec u from te Lalace co tour If teroo t ro e te r eak axi twice oever ter e i a o ibl

it tat the to cr iter ia are ot i agre emet (Actuall if itcroe a umber of time the to criteria mgt diagree.Ti i di cu e d i more detal Aedix A wher e it i owtat the dfferece te two formulatio arie becaue te brachoit of ( were ot carefull codered i Referece 1 I taedix a rater artificial umer cal examle di cu ed to illutrate the e dea (Note al o tat the r oblem of ditiguihgamlifig ad evaecet ave i ot coidered i Referece 1

Polovi ha co ider ed te que tio of di tiguhig amlifig ad evaecet wave from a baic formulatio wich ver

imilar to te oe rooed b Sturrock.*

A erturbato i coidered tat i aumed to be a ule i time at ome fxed oit ace. If thi perturbato vaie for z te te wavei defied a amlifig othe rwie te ave i defied a evaecet . Te r e ultig crite rio tate that f tere a addle potof k( i ide of te cotour of comlex k for r eal ad the r eakaxi ad o te ame hee t the wave i evae cet otheri e iti a amlifg wave Th mathematcal cr iter o oe ot beara re emblace to te oe erived the re et work. It believed tat te differec e betwee te e two c ritera ari e from

t differ ece i the bac hical defiitio of amlfig adevae cet ave . A a coute rexamle to the Polovi cr iter iocoider te dierio equato derived for forardwave iteracti i a eakcoulig aproximatio ( e e Sectio 2 Equatio 2.44):

(2 .41)

t eal ow that all addle point of k( i ti cae are ote reak axi ad ther efor e Poovi c rter io would tate thatbot comlex roo t of k for real are amlfig ave . Thi at variace wit te wellko re ult of both theor ad expe ri met o traveligwavetube amlfier.

It hould alo be metoed tat Buema ha develoed a criterio amlfig ad evaecet wave that volve determigthe admittace of a pr obe e rted to te tem I the aali however, he coder imultaeoul a fite tem adurel real frequecie a rocedure tat i ubect to te crtici m give i Sectio 2 . 2 He al o clearl doe ot coider teroblem of ditigihig abolute ad covective tabilite.

*Mo t o f Poovi work i coce red with terpr etig the two ave eakcoulig diagram like toe dcued i Sectio 2l hi ba c formulatio of te pr oblem i of itere t thi cr itique .


30/39

Discussio 3

2 6 3 Us efule s of Crieria The cr ieria ha we have de r ived on aplificaio ad is abiliy were ba se d o oel o f

ifiie sys e drive by a localie d sour ce Th e reao fordoig his wa ha we wished o obai iforaio abou he waveson he uifor syse wihou reference o y paricular se ofboudary codiio i Havig doe his however i is ipora o clarify exacly wha iforaio has bee obaied byhis procedure i regard o he behavior of yes of fiie leghI his respec i is perhaps worh whie firs o give soe warigs abou higs ha our crieria do say abou yse of fiielegh

1 e should o be ed io he incor rec coclusio ha asyse which suppors an absolue isabliy is alwas usableSuch a yse uually u be loge r ha so e cr iical leghbefor e he sye be coe uable a for exape i he cseof he backward-wave o s cillaor . The saee ha a sysewhich uppor s ab solue isabiliie i usabe oly a o s cil laor is als o no rue A wellkow couerexape is ha ofhe backward-wave aplifier he asolue isabiliy is uppr eed i hi ca se by kig he ube hor e ough ad ye ane gai ca be realied by operaig close o he oscillaion co

diion2 I is als o o rue ha a s yse of fiie legh which suppors oly convecive isabiliies (apifyig waves is necessar ily sable ha i s ha pe rurbaios cao as o grow i iea every poi i space All his aee rely ay i ha anaplifier ca becoe a oscillaor if here is a sufficien reflecio of a aplifyig wave ari sig fro eriaios of he ys e as for exape i he case of isaches a he oupuad ipu of a ravelig-wave-ube aplifier

I i believed ha hese crieria provide very useful infor

io however a log as oe exercies due cauio whe applyig he r e sul o phys ical iuaios To lis a few of he po si ive saeens ha can be ade:

1 The cr ieria on aplifyig ad evae s c e wave s el oeif i is ever possibe o have expoenial paial aplificaion ofa igal wihou coide rig nuber of boudary vaue prob1es in ha i wihou cosider g he c ouplig of he s igal i and ou of he ys e

2 If he uifor sys supo r s ab o lue isabilies hen

eporally growig oscillaios ca occur wihou he ecessiyof ay r efe cions fro eriaios or exernal feedback Iany phy ical iuaios i i obvious ha no e chani for uchre fe cion (or r ever s e propagaio exis in which ca se he"sabiliy of he finie syse cn be deerined aos direclyfro hese crieria excep for he "saring engh" quesion already discussed


31/39

38 Amifying Wave and Abolute Intability

3 . The "roagation velocity defined in Section 24 i uefulfor etimating whether a convective intabiity would grow to large

amlitude befor e r eaching the end o f the y tem For many efoldng time t o have evolved the ength of the ytem

Lhould be

much larg er than V where V ad are defined in Section 24It i inter e ing to note in thi r egard that the lower bound thatwa derived on the maximum amlification rate CM (Inequality 2.36how tha the above -mentioned condition requir e al o that L 1 .Thi i another examle of how the aalyi in Section 2.4 allowone to connec the concet of a growing ule diturbance with heamification in ace of a iuoidal ignal

In br ie f the c rite ria de rived her e uy addiional infomatioabout "otentially untabe ytem beyod the imle requirement tha comlex with egative imaginary art for real k beobtaned from he di e r ion e quation o f the infiite (unifor m) y em . addition thi develoment ha cearly demon trated thatthe roblem of ditinguihing amifing and evanecent wave canever be earated from the que tio of tability Thi would al obe true whenever y tem of finite ength ar e con idere d.

264 Amifig Wave in the Peence of an Abolute Intabil

. It i of inter e t to review bri efy the exame of an ab o luteintability reeted in Section 2.4 in the light of the reviou dicuion on ytem of finite ength. In he ca e illu trated in Figure 2 16 , a wa mentioed befor e there i an abr ut ranitionof the root that ha comlex k for real from amlifying o evane cent for ome rea in he vicinity of (the real ar f ) hefrequency of the ab o lute intability. Seve ral hyi ca examle exhibiting thi behavior will be r e ented in Chater 4 ad 5 Inthe context of an infnite (uiform) y te m of cour e thi i ofo ignificace be cau e ther e i an abo lute intability r e ent.Difficutie in interretatio do arie however in regard to the

behavior of yt em of finite length In a y tem that i oo hor tfor the abolute itabiliy to be excited a for exame abackwardwave amlifier it i meaningful to eak of real frequencie . Sic e the fr equency at which hi "tran ition occur i not unique* the criteria baed o he infiite ytem are otable o give any definitive informatio concering the meanig ofthee root "ampifying" or evanecent in the finite ytemin thi ca e . The only

g n ral

concluion oibe in the cotextof the r e ent work i hat to anwe r thi que tion the y temof finite lengh mu t be con ide ed exlicitly in uch c a e

*Note tha he br anch line o f F( z can b e oriented at an arbirary angle i he lane and no ju a 9 0 to he rea xisa ho i Figur e 2 . . It can be ho hat thi will make a correonding alteration i the real frequency at which thi traniion from "amlification" to evanecence occur.


32/39

Exampl 39

2. Examl

In hi cion om impl xampl of h applicaion of h

criria ar rad in ordr o illura h chniqu involvd.In h fir par h quadraic quaion obaind from a coupldmod d cr ipion of wakl c oupld m ar anald and di cu d whil h cond of xampl i dra from h fildof plama phic.

2 . 7 WeakC oupling Di er ion Euaio .

Th opraion ofmo bam-p micr owav ub can b undr ood b a wakcoupling formalim. In hi cion w al h cririaon amplifig wav and abolu inabilii o h dprioquaion ha rul from uch a wakcoupling formulaion. Whall conidr onl h coupling of wo propagaig wav in lol m; h four po ibl p of dipr ion diagr am hacan rul ar hown in Figur 2.20. (Trivial variaion in h

A

/

t / V

B IV

I D I

VFigur 2 . 20 . Wak coupling dipr ion diagram .

diagram ar o con idrd a diffrn p Th quaiondcribing h diagram in vicini of h ircion of hwo uncoupld wav ar h following

A

B

k (k - ;) =

k -;(k+ J= -

2 .42

2 . 43


33/39

0

(C

(D

Amplifyng ave and Abolute Intabliy

(k -

k

-;2

) -k

(k - ;)G J k2

2

In the e equaton , for algebraic mplicity the pont of inter ecton o f the two uncoupled wave ha be en hifted to the or gnof the ( - k) plane . e conder each of thee cae eparatey:

(A). In th cae k i real for al rea and converely ir eal for all r eal k There for e there ar e no ntabte , and

the reutng coupled wave are ordinary propagatng wave.(B. Now there are compex root of k for real but i realfor all r eal k. Ther e are no un table wave and hence we canmmediatey conclude that the copex wave for r eal ar e bothevanecent wave.

C) In th ca e there are coplex roo t for k for real ancompex root of for re al k A - the two root become

26

and

27

and therefore both root are in the lower-half k-plane for i - It follow that the root with k > 0 for real i an amplifying wavein the z direcion and the roo wth ki 0 i an evanecent wave.Ther e i al o no po biity of an ab olute ntability becau e bothwave coe fro the lowerhalf kplane and therefore they cannotmerge through the contour C the deformed Fourer contour. Moreover , it e a ily hown that all double ro ot o f k oc cur for r eal .

A a mater o f nter e t the r eade r can check for hm elf thathe lower and upper bound on the maximum ampificaion rate ofa convective ntablty derived in Section 2 are atifed for thimple exaple

(D) Now k i real for al real however, there are complexroot of for real k. By olvng the quadratic for k we findthat a double root of k occur when

1

1 - 2

j

+ V2 2 8


34/39

Examples 41

As 1 w I . 0 the roo s be come

(2 .49

nd

(2 .50

nd therefore for wi. -0 one root s n the upperhlf kpane

and on root s n te owerhafkplane We now hav n a

soute instbiity at the freqency given by Equion 2.48, and atte wve number

( 2 . 5

It is intee sting to note al so tht the s ptil pter o f he a syptotic esponse in this case which s given by e Jksz wth ksgiven by Eqtion 2 . 5 is exponentiay incr ea sing in the dirction o f the " wv" (se e Eqation 2 . 22. Th s whenV > Vz' ths sptal patten has n exponenta enveope ncresng in the directon ken by t

g r ou p v l oc i ty

of wve 1 when tewves are ncoupled

As a mtter of ne e st , the o ci of te roo ts in the kpane fothese four caes are sketched n Figue 2.21 for complex wth = O. Note, however, hat it ws not n c s s a r in ese simpe

cases to perform a detied mapping of (k) in the comex kplane as wold be he ca e in mor e complicated sato A f the s e r e suts are in agr eement wth the pr edictons bs ed

on the concepts of smalls gn energ y and power.49,59 Digms and B r o f the tpe that rest frm a wek couplng of two passive wves o r fom a copling o f two ctive wve s B y "ps sivewave, " we mean a wave that hs postive smalgna energy, andthe ter "active wave" means tht t wv carrie ngtivsmal- signl ener gy Diagrms C and D ar e the type that r esut fom a couin f an ac tve wv wit a pas s ive wve

hen t group veocitie s o f the uncouped wves a re n thsame dire ction as n C we hve amplifiction as for ntance in a traveng-wve ampifer 47 hen te grop velocitie s o f the ncouped wave s ar e in opposite dr e ctons , as nD an absolte nstbity r e ts , a for exampe n a bckwdwve osciato , 5 8


35/39

42 Amplifyi Wves d Ab so lue sbiy

j j( A ) ( 8 )

+ k kFre uec v o

) )

ko, o

k j

k

1k ko ko

Fiure 2 . 21. Loci of roos i kple for wekcoupli exmple s r 0 d V > V)

2 . 2 Doble Sr em erc os We ow cos ider dis pe r so equio h de s cr bes oe e of he wellkno ele cro sic sr emig) isbiliies i plsm . This exmpe i s e s srivil h he previous oes d llsres some ddoechiques for pplyi he crieri whou resori o deledmercl cculio

The dispersio equio we cosder is

1 = ( 2 . 2 )Ths equo descrbes he lonidnl wves in sysem ofchred prcles wh wo eqldensy "cold") srems echwh plsm freqecy p s ' whch flow s ech oher heql nd oppose veloces V nd re immersed in bckgroundof siory cold pricles wh he plsm frequency p se e Appendx C Ths is gener liio of he ds pe r s ion equio ( 2 . 39 )which described sle srem flown hroh plsm

If we defne

K = ( 2 3 )

he Equon 2.2 cn be solved for k s


36/39

Exames 4 3

(2 4

We consder fst the secal case of a dolestream nteracto n the asece of the acrond lasma p 0 or K : ) th cas e a dole root o f saddle on of ) , doe s oc cr te lowerhalf ane or

W : s ps- J ( 2 To how that th saddle ont corresonds o an aolte n talty we s etch n Fre 2 2 2 the roo t lo c n te klane for

pre mana . A sow n he fe, t s convenent in hiseamle to setch te loc n the lane frst and then to devethe lane oc from the kZ loc As evdent from the fre, e have a s olte tlty n ths ca e a t te pre manaryfreency s

req e c v r t on

w i

W

r

S d d e t ( l

. ps -

;

Fe 222 Dolesream nstalty wtot a statonarylasma 0 ) .

n the case that ncldes a statonary plasma, we fnd fromEato 24 that ole rot of for an n the lower halflae occrs only when

(2 6


37/39

44 Amplifying Wae and Abolue Inabiliy

Wen Inequaliy 25 i aified, e addle poin occur a hefrequency

( 2 5 7 )

t can be own, in a imilar manner a for e pecial aep = 0 a wen Inequaliy 26 i aifed e addle poingen by Equaion 257 doe correpond o an abolue inabiliya On e oer and, wen Inequaliy 2 5 i reer ed , wekno a her e canno be an ab olue in ability becaue er e i no addle poin in te lower plane I i readily o fromEuation 24 a ere are alway complex rot of for rea k,a reult indcaing e preence of conecie iabiliy

To deermine te poible amplificaion rae n e cae pp >p 2 , e compex k alue for real are keced in Figur e2 2 3 In or der o idenify wich complex roo co rr e pond am-

0\\I-"

- re T m k " " > ompex 0- Re J

I-pp

0

-:

+

t_

r

-

,07

' _

,/

Figure 22 Complex k for real (pp p2) plfying wave and which ne correpond o enecen wave (realizng a he analyi in Secion 24 a poved ha amplifyingwe mu exi in i ca e ) we mh pe form numeri cal compuaon n a pariular ca e I i perap more nrucive,however , o ue e omea inri cae bu more gener aly u e ful argumen now o be oulined


38/39

Exames 5

e shall sho hat he o roos of k ha hae ki - 0 as p are amplifying aes ha is, here is an infnie ami

ficaton rae predicted by his model in boh the and direcions hen > s/2 To demnsrats , i is coneniento race he klane loci for a rather deous roue for Route ain Figure 22b raher than the more standard roue obtained by

t

2root

k

Q

? 0

2

ah

{

I

-#

-

k

r

O

2

a l Pa

t k l rooti

Roueb

Roue aa Loci of o of the rs of k b Freey ariaions

Fre 2 2 Loci o f k roos p > s/2 ) The roos of kfollo he rajecories indicaed in a a he freuecy ar

ies aong Roue a in b

holdng he r eal par o f fixed on the de sire d re al r euecy andbringing i from -0 o ero Route b The alidiy of this aroachis dis cus s ed in he folloing paragrah The adantage of raeingalong Roe a is hat e can bring up o a poin jus belo he realaxis = r j , and a he s ame ime keep 1 I ery large in hepr oce s s , s o that we no hich root of he fourhord er eaion2 e are dealing wih Pah a in Figure 2 2a In he figure ,he o roos indicated in he klane are hose ha become

2 5 8

kv c: s 2 5 9

as - 0 e hen brin r down from large posie alues ohe de sir ed r eal freuency slightly bel in he presen case,using he re al diagram o follow the ro o o f iner es Figure 2 2 In his process, e reain he infnitesimal negaie imaginary arof in order o aoid go ing dir e cly rough any "ndeerminatepoins lie Fr he re as ons dis us se d in e folloing ara gr ah, he use of his s mal imaginary art of ells us how oj oin correcly he k ros on he opsie sides of such singulaons hen his pr oc e s s is carri ed out, e find ha he wo ros


39/39

4 Apifyin Wves nd Absolue Insbiliy

of ine es hve pe foed he j e oie s so in Fie 2 . 2nd his veifies he esul sed elie

I sill ens o deeine he condions fo wi is devie woks nely unde wha ondions is e hese nswe is obained by goin alon P a as would be obined by oin alon Ph b? A lile ou will convince usa e ony wy we ould eh diffeen poin in e kplnefo he sae eal vlue of whe aveli dfeen pas whihbo s wi e s e o o of k and he se poin nely i = ) is if bnh poin of k is enlosed wihin ese wophs n e -plane If we do no enlo se suh b n po in en wen we s on pacul see of k piul oo

of k we sl be on he se see when we ive e endpoin even if dffe en pahs e os en Al is e ly sys is i f we we e o go hou poin h is double oo of k bnh poin of k we would lo se ak of wih oowe e followin If hee e no suh doube oos of k in helowe hlf plne s i s he s e he e en e po edue isvlid Te e e bnh poins of k on e eal xis in hisexple ho suh s pp nd his is e eson fo einin sll n iv iinay pa of wen veln onPah We epasize in his ik s useful in son

o whh wves e plifyin onl wen i s known h no doubleoos of k occu in he lowe-hlf plne

The pysil esuls we hve obned e ineesin nd woha bief dis uss ion We showed in Secion 25 snle sein pas esls in n infine plifiaion e in he diec-ion of e se and no bsoue insabiliy Two ollidin sein e bsene of pls on e ohe hnd wee shown in hisse cion o e sul in n b solue insbiliy whih s uie esonble physi lly sine e sys e hs builin feedbk Whenwo se ollide in he p es ene of plsma oweve e be

hvio obined depends on e elve densy of e se ohe plsm Inequiy 2 5 . Fo sfficienly enuous plsmhe bsolue insbiiy beween he se is obined whees dense plasm esuls in infinie mpificion in boh semdiecons nd no bsolue insbiliy

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C RIT ERIA FOR I D ENT IFYI NG AMP LIFYI NG WAVES AND A B S OLU T E INSTABILITIES

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