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Eindhoven University of Technology
MASTER
De tweede-orde dispersierelatie van een enkelvoudig homogeen, isotroop plasma inéén dimensie
van Someren Gréve, S.C.
Award date:1979
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De- twtt~att -o~<:le olisper~icrelahe VONt ~ ekke-l vol,(.di 9 W.o~o9rcd-1 1 i~troop pla..~~a.. ik i.é-... di ~e-MSie
V erslaB vo.~ \t0t a.tsh .. ()~1.00rk Va.~,~t 3a.~ valt SoV~otere>lil. ~riN~
dat ~e-~Cl.O.kt i~l~~t k va.kBroe-p trl,qoretiScke t-.la.hu.c.rkl,(.~d.e
o~der bi di l-13 Va.l't
F.W. S\~kr-a:w r. La.K-tbert rr~ck~ÎScktZ Hose-Scb\ Eit-tdkore\1
I
bl'Z.
0. 3a.Me~;~ valt i ~j 1.
1. Drt tl:ei"Sht·- ordlè bere~ki!.tj Gta!,.l(~(l;l.o so.~~trd4 2
Vla.~ov_ Poisso~-tpla.~VIota. ~~-t de disp~lerela.he.
i.o I1.1\eiài h~ 2
1. L D<z- eers~<e-orde bere~i~ 2
1.2 De d.is p12n~a erela.h'<Z 3
1.3 Be~a. Jeri "'-~prorY~M.t.I<Zs voor cill' dï~persi0relc:die <2k
de ~t2:f'S~<Z _orO.e di èleldrlsc~ ko\t st-o.~.tre 8
2. De +~ede _orde bct-eke-lili 1-\3 aa11t ~ V\a.ç.ov _Poi ~à-1 11
Fla.s""a é:lil cl~ t~d(t'" _or-drt di ëlek~n·5clteko!.tc:,huàe 1.o IV~IeiO.i 1.1.~ 12..
2.1 be ll.1terakti e Va.lil drie- solvel-t ik C<At Vla.-:.ov- Poi:5Sow l'l
pla.Shltct
2."2. De tCAJee-dtr _orde bereke-t-ti ~ 11 2.'3 be +c.ue.ccl~-orcAe di ël<.ëk+risclw.- kot.t~Ja~-the E(z) 23
2-~ Nor ~ere:t.t Va."t dt2 tt..U<tede-- ocdl2 di Q lekl-ri~k~ kolit.'ta~otttl' 2it 2.5' Be lilader-i ~5 f-or ~le5 voor~ t~e--or-de- 16"
die lektri~ ka.t~~a.wttl' 2.-5'a. l3e~ acleri lilf1S tOriM.U.le- I(Oof ecz) a,\~ d<2:1 dr\ e- t3o\Vew 25'
O.ri q; i ol-1 ~"' gd 'fet-t Zj ~ 2.S"b \3elila..d.eri k[jS tor~ltl' voor fCl) ai<E>d.e porkp3ot~ eet.~ 2QJ
0\ektro~ 5olr is <l.IA. d(è tU)<Ze> Wi -ssd t.eer"~~.-toe <jolve~.c
3. UIA <2ÁektrolileU f1d/- (li\A ~ lOke\t gclt~IA be i1.0ee:de- orde dispe~ierdcttle va"' ~e-IA ekk.e-1 volAd'3 3o
plasVI.ta. IIA.e~~;t di Vl-t.IZLot~e
3.o !V~ I~ i O.i 1.1~ 3o
"3.i be: alge: \'\té-11~ o pi o~si t-tg 3o
6.1 a. E t~tl ~e op ~rki 1.-tlje-L.t .b~ d.o a.l ~ Vl4.e-ke ~o~i u3 '30' 3.2. Dt:. si+lAa.he cAa.td.e:, po!Mp~o\f- <ZI.-t Je ke, c.vr~~eL -41
tuzr6Ja- ~o\ Ve:41t drie- fo~ eolvCM 'S"'
_Jr_
3. 2o. De- reso ~l.(tie VOOYWdat'"tie- k'= k tk11; ~ 1 .:1f t~" ~2
1. 2 b be- r~~o~~ta.~he-VooY"t.va.ark k" k't-1<' i~= 1( 1~Ht 11 4s-3.3 Des,i+u.a:he Ja.\- &<ë y>o~p9o\bzt.t ttf;k tvisc;,diOcr-~ 41
0olH·wu ela:ktrol1evBO' ~~~~ ?§~t~ e~.t O.tt a.wckre.
Wi ~sa:l (.Uev-~&e 0ol~a:~ lo~aolfis 3.'3a Ek":>a.~t~ ra:~l-1a.~A.htt ~0
'3. "3b Nifl+ q;k::>o.kte ~kcu.thc 5g /.i Nu.~ er ie ku bttrek~i 1-t~ Val-1 d~ kok4 p\exe-IU<l"tkcl6 kit 63
t r ll' b.occ Lot h <lS.
4.o !~leid.il-t-9 63 ~.i. Hef l'ro ~raWt M-t'l 63 4.2 De po~ p~olt a:l.f de tl.c)(l:(l: IUiS5el LOeY~tie- 9o\vöt "ZS\.f b"f
cl.rie iohett Bolv~
4.3 De poVKptJdt e~ ll<Zit IUi~sel I.UeY~"'~9o~2~1.t t~ b1 &:\e.:k~ro~~ eo\ve~ eiA de a.~tJ.crf!1 (i)IS'i>el ~de ljc:Áf
i ~~L-t ioltte>Lt ~olt
L~erah.wx 12
]k ~~t ar-s~deeYVerslaB {.c)orcle~.o ew kele as~ktea.. \raL-~ de- ·h.)ce&e-oró.t
cli'5per~ie rclcd-itZ vo.w a.c~c ek \cc\ Vo1.u:li3 plets~ kebd~J. Ër- (,.)orcÜ LÛ ~ ~e9a.al-1 Vo.t-1 ete~ \.to~ot3ee~.e <21.t i so+roop plaS Ma. da.r kScl.trevet.t (i)(KcU d.oor <k <lq.'L-tt:Ái~ek'f.io~altZ"Ifer-~Sk;k.ef"* Vo.~.t V\ctSb\"(lM. PeisSok
Net ~ku.lp va."' eet. Fou.rier- trc:ut5\-or*oiie ~oar ~\ctats e4;t. +-sd wordt <lel.\ e€;tste_ <ll.t t~GleorO.e storiH.9sberektkik.5foe~epo."btaa.~-t ~+stelSel di He-rewti a.o.l ver~~kikf3<ZM Vo.IA Vla::, cv cw Po·,s~dc. Bij c).rt ~r-<f>t~- orde s\ori kS~~~k~ t.vortÁe:M i~~t k.d kort dleop\o'SSi ~e1. beka.~.t.deld 0.1e lopekde- \ok~•tuc:J.l~ta.le Boltop\ossi KCJ~ 'Zf)'-'. B~ cl<è t{.c)a:~cle_orcl~ ~ri~s~~ l.üotd.t Q.~i\ttet"ak+i0 v~ d.r~e lopek.ol.e
lok~ikclikale lo~~J.e>eolvfll..t, J.ie ~d-dk~lk reso~+ie~1.1 'beko.."'àdd.
~ierb~ i~ lferOHc\er'5telJ dat ket ~Skis c;loor l1,Ü4c!el VttL-r+acvoeY" Yal-t
eker~ie \fc;u, bl.t.ite~a.~ {tet,1 po11Apeolttckreët'Ot t.U~rva."' r).e av....pli+u.d~r tbtls Okatkatt kelskis. De.u po1Mp'3olf (.Oissdtc::lau et.ter~i0 t.Uhwzt·hutt CHtclcr-e- ~olv-e~A d.ic a::u v-~1 kle,kere C.H\tp\if~.tc:AE> lwbbe~~t. -kierhj iCZ>Voord.l
o:f Ke~ ~a.aY de lcoKdÎtLeS Wa.a(b~ at y:>o~~CJO~ tc_ue~ Gla.k~roe\<ZMde-r;o\v~ Vfi:K oorzaak -1-.
J~ olt v-ersla.~ I.VOrcÁek VooriA.)aàrdet.t coe'fdeVr&U Vcx:>ra.w.ttc>I•'J.ude fZIA 9=>~~
<jc:tal Vaw. de f->OIMf' ~o[f-, "Zoda.+ de:74 po!M.IO~olt huee C((:lk5roe:~eu.de 5olveLA \ref'oor-za.akt-. t:>e:ü Vconua.o.rde\1\ '5"" bcreke-&.tol voor het ~va.l da.ter ~ol~- ~oltkop~~~~~ i~h.t.~Ci>fZL-1 drie i~t.t.~dv~ <l.w voer~+ ec-val ~ter 0o~- ~ o~ k.o ppdi ~ 1 ~ +u.~~et.t t<.U<U c.ldctroLclt14.cadve~..c <Z:k ~~ iou.e:Lt _
g c4. B~ ok f IC(a.r sf e> se ral L s ~ el ddro~ lJO\t de f:'OH-tp~o\t . BS e.L-t ~e <U1It vow.d.i~~ ~itc.t.aJ\es bJt.Ltek a~ttaly}iScke~ka.cA.eri~~torHuJ~ C(f?r~ l.ûord~ o\le> ~t verba .. d t(A.~'5cz;l.td(tl ~oltCj~~a.llek 1 Gl~p\;tude m~ de pon.tpcoolt (2;1.4 d.e Je~pi~::,- qaa~;t.7jrOe:Üt~~lcOI.tc;ta.ubz <(JeVeH..
Ee~ ~nekt bent:k:e-ki~ fdee!fvoor la-clere tnzku.:~aàie r~ultakw. t:>c~ ~IA ~ra.}-i se~ ~<jt~a>\1~. klcub~ -zs~~r be~.tac::lerl~s.F~es <Qe.bru.lkt Vct~ot c:l0 PlasVka. _ Jispe-r-siefu~.tktie. J.te ~~ beperkt ~ldi9keid&eJëbi(Z'-d ~ebbe:ovt
1. . .pe- <re-Yst<e or~ krcbi ~ a:tlll e01 -:o.tKelil~ Vlasov-PoisSClApla~tkc:t e.vt
de ch-s!Oet'5icr~l~
1-t dit kc4F.A W< worelf <UCTsf eek a:Y'bw _orde btrekdti ~ aa~.< ca. ç,a ~~teld V1 a.SoV _
Pois~ pbs~ U.:t~a::rcA e.Jtt.Mrl1a. dz dispe:rsicr~aticbe:-kuu:Jdd. Dctar&i+ i~-t de lifer~r{.o.a.Fr-iedt:k g:>ttld; How~C3oi!AéY~) u.ituocn~ wardtbebkoldd, UX>rcltdif ~ia- ÏH U brt liYtLY?f~.
u ll ~at-t~spu.~-tt is de rfi ~.tel i Kt/Ck'5Î okdb botsi ~~loze- ver~Ski~ VaLt vlascv ek de
é~di ~e~-<~oWàle va~Ski ~ \fa~A Poissel;f
(1..1)
é)è: = 2 ~'5 tZs ft=-:. dv (1.2) ox. ::. ~o
Hieril-! i:'> r-;, de: vadeliL('3'5fuwktie I ks ~t aaHtal d<re~~ peYVOI~I'l'l(l<lek~icl. Wis
Je: hta~~o 1 e<; de la.dil-'t~ Vak cle de_elf)csScod(elektrok, ÎOk)S. E6:,t) \~ket e\ektrisc~ ve\cf ew Eo de di ëlekl·nSc\e kot.<·sb~.«te lra.k vakutt,.".
be l[erde:liVll)Stu~-tk+ic- toordt ~reVC"Vt dlsdesD~ Val;tw,t fufic~-ta.ire \fetdeli~StUVI'Jill F~cv) e\.1 /UI-l kleil-le vastori~ dadrop: J:~,c),f~ v).
~-;. = fsCv) t ~(x,+;v)::: r5[V)t JJ f[k,c.v;V)expi(h_wt)dkdw (1.3)
V cx;lr'" t="scv) wotclt vack:r i VI kd ~kde vcrsiClij ole haxwelll.rerde lif,(~ ~brui kL ~ckM \'}toet JP(v)clV=1· Voor kt vdö Sc-kV'~ VelA~
t [)(,t) = JJ E(k,w) expLékx _w+)&k cl(A) {i.~)
..-'\. "-
b{x,l-) evr t()(J;v)~\ll e:ersk_orcb 3roo+~. N~ beku.lr 'fQllr (1.1-2)~ (1.3-0
krS0eJA c.ueals <Ue dc+~.Utecleew ~~re-orde ter~ tce3lab:
_3_
Wro kul-t~~~~ cokdefwri~tr-~tra~.tsfor-~dte vcr~djki~o,osckrjve'-1. We krS~IA da.~tt:
i.-ékv_ w) f(k,w;v):: _ <l~ E(k,w) ~f'i.cv) nes oV
ik \:(k,w) = ~"' ~ Jf::.ck. w;v) dv l?o
(j..5)
{.1.. 6)
0.1)
(1.8)
Rls \01l: (_1.7) I i ~-tb~ reckts deb docr t:(kv_()})et-t c.lcwr\1o. i k+e8rere~-~ ov~ Ir ka.~-t ~f bekul p Vo\lt (i.ö) j ~ s( k, w; v) ~~di VKI kurd 1.\X)rde-w. ~ kr~~ u..::c·.
ft_~ Vl." el ((}Fscv)fov clv)Ë(k.w)= ec•)c~.w)E(k,I.V)=o (1..~) \. ~ Eo~o;. k2. ) (V- W/k)
Hiev- is dus é'1Ck,w) cle- eas-1-e.ctà<Z Jiekktri.sc~e ko~'5ta.1.1tc. &ka.lve de~riviale oploS5i\ltj ECk,tu)=-o isercdcde-oplo5::>11.{3:
eü\k,w): t- .l hst~s"' fof;cv)/oV dv::o s ~oh\:.k'l (V-W/k)
qUe,e~tt di c lope~~tde oplo.5SiK3et-t volol001 waar60 k e1.1 ~ oa~-t (1. to) vold~. (i. to) ku+ daL< ook de- dispcrsicz- relatie.
1.2 be di~persiercla+ie
(1..1o)
V oor reë Ie Lu doe-n 2ic~ kovwpli ka.1ie.s Voc:K. De 1~-t.te~ta:lle~ '3ctQfdw-ee1-1
"5; ~ulierpuL-tt. Vcor 0e~pte ~aa~.t~yooeltlde-edvet.t istU kot~.trkx. De-it.ite~foaJ is do~~o +e l:ercke~.te-~-t l.ue vtc~eL-t:
"f t :: u.H i. 0 {1..11)
We kv-s~ cb~t~:
E \ k. V ) = 1. -:4 hs ~-z J t1Fscv)fov clv S to f\t!l k1 {_V - "f"/k)
(i.12)
- Lj -
h f?o cla~~t 5o.aL:k- iL-~te-3ratie~mtcler de 'Si~-tBularifeit dcar.f:tlcs j/o da[,. ka~-1 de
{ulllkf-itt (1.12) at.lal~tîsc~ worde1.1 voor+~w olcxxde. i t-t~r<:die~ te~~ '20a.ls
i 1.1 fi~ C i..1) is aa~;~5e~o..
Voor O.e 1-1.ulde- orde vao\eliKB<;;fuL11dilZ word~deHo-xv..:dLva-deliw~f>fu."'ktie~o~etr. De b0Wtl bepaald deelties'SCOrt s bekore~~tJ~ drifk~ keid toordf us 9fl'toe:»~d.
Hier-i 1-1. (s k8 ck BolbVItaL-~k ko~sb~.t t~ ek 'Ps de t61.tpera+<.w..r bebn&de by ke-fdeel~ es <::.oorl s. HeL-t ka~,o actkfokew Jat:
tv.>
{t.n)
- [otsév)/dV dV= l f1 t f~ r tZXp-'Xld'Xl=l\ltf~C{f~;)f (1..1~) J(_ v- rt.p+/k) 'vst '(rT Jcx- ?:.) J 'vs" J
-c.c>
I~-t de:-ze <.Mttln.Akki~ ~oat Zé)) vro-depla'S~a._di~persiefu.kk+ie ot FrieJ_Cowte-fu~.tkiie cÁle als vol3+ is ~d~ti~i~rcl:
2{~0== n- 1~ rt~p- ?.2
v\'X (~d 'Xrüel) (1.1b) _J (-x -1t)
LL.t0e6reide- to.bellelA ~IA Voorde~ }w .. hJie bkikbaar. q1<i>Lw. ))o ~at de i ktefyct+ie:ll.)e3 lak~S de- P,e_as .te fu~ktieZ(~j) l·":>a~a.l~ti~~ voort~7et vcaket~al M llfk f..{ odoorde iL-~~ra.+lewes OH.derhcl'i>ik5u.l1'ere
pu~t x= r +e~VI.tevt -zoals i~-t fij (1.1.).
dl())= -2(1 +')2(?)) d)
be- VeY~eljl<i~(i.t2)(A)Qrdt ck:u., ~~kuif=' Va~.t(1.11')€t.t (1.q)
EC•)+(k. 'lfJ ) = 1- ~ h~es-z. Zl()s) ~ 2~ohlf>k2 Vs.2
be o~a}b~.t~\S~ variab€-le- is
)s = LW+L~-)Ik - Lls
v~ Vl.'
We. deti 1-liert:VI ~u. de- vd~el!ldc Brod~d.e-111.
b~ (?las~afrck~t.t+ie;
OWI toi·ee111 CjotZde besdo··svi~ te-ko1Me1.1 c.oorole~~o de-vol~de ~r~eerck cl; ~eksielo~ 3rool-~eoe~.-t i~tt~vcerd. \"}'e~pe-r-atuu.rs Verkou.d.i~3:
(t.q)
(t .1.8)
(t.t~
(t.1.1)
(1.1.2)
{1.23)
Dri~:
De volge~d.z 5roc:J-l-ledoo evr rela+i e5 -zs 11 voor vuc:kre u.itwerki "'8 zetr h?."' d.ij
\.v e Spl i h:.e111 )' 5 i \11 lle.H re ie Ie~;~ ~ZC1il i l'lltaj i Lila. i r clee-1 :
(Vf!. = 4Jpe re Ks = K~ lfês
k ke 5:--
Vel'
, ... = l'e V e) b~ I
(1..25)
(1.26)
(1.27)
(1.28)
(1.30
(1.31)
{1.33)
(1.31-t)
_,._
r·=G.~ (t. 35Cl)
(s = ie~ (i.3S.t)
r>" <:;, .I<.(,' (1. 3bo.)
i<:; c Y-:. · t<~ '11 (1..3b.t)
\f~:: ~~ + i-fs ('1. 37)
Me+ bdtu.lrvc.u~ ( 1.2o- 26) e~ 0 .28_ z~) ka111 de lfer-'.:11jk;"'9 (i. t~) als \ro~i ~I<Yevet. !.vore!~~~~:
(t.~)
De iwJex j ke:~ bd-rekkiVt51:f>lk ctee~B~SDorl,oie~et-t 5ebrwkt 0\4 oe 1re-ktoatHe te nor-~r~~·1. 8\s n1e~ ~e-t ruktuliel va"' (i. 38)'5e1Sk stelt Gtalll I'Lu I kr-5lj+ l'l1en de
c\j 5pero;,i er el ah e.
He .... kalft Jct~~t v-ersckillllVlJe relo.tles·tt<'5'5el-l de variab~le~.t Vi~-tJet.t.
be 6e\a1115 rskste- Zjt-c
1: ~c.:, alsiu.wktie vo."' ke; be r~ik dispers·tek-roV~-t~~'~~. 2: r~ als1u.~otktie Vcu .. te(' ))t; á'vtaB' 11a.irc Oi.sperslekro~l'ltc 3: (::. o.l~1uwktk2 VaL~~ Ws i PlAAts Valil de polt!t.. ·,lil l-t~t ko~~exevlo.k
Voor be pao.l de l.Uo.ardev. vow. f~ ka.v. de piGtsl'ka _di s.rer~i~t!AI'Ik+ie l(r:;-.) clocree~;~ voL<dié)l
tu.~~tkhes be~clerd lUordev..
Voor I i"' I ~t.l ~si OVtt kop~<Zi~ de r<Zile eL-ti ~~i ~;ui re dclör zie~ btt rukso~~ttwikkeli~-t~ is absdu.uJ ko~-tve-ree~+
{1. 3j)
(Ho)
De asy~éo+i-scke Ol·dwikkeli~ot3 i~ ik kt al~e~Jive.vBewt. No.arl-\4ot~ fs zJcJerwo~cH ko.l,.1 ck fu~~tk+ie i()c,) k·,er~c'le- beter bet-tatierel IUorde ....
(1. 4~)
(1.'15)
Wc be~ ke"' Vt.u de d.i;:,per-sierela.tie- e-11 u-rste _oro~ Jit:lektrisc~e kow'::>ta1.1.+c
Vak <Ze~.o~ llM.kelvow:.L;:; plasV~-~a. Ele-k+rolile1.t el/I io~ote~.o~ worde~o~ o.al!l~~se-v~ cloord<Z"
i Vl.dicc~ (l rest=' i.. <.05 r.uord~ e <je~-toetwJ e.t ~s worc:H ó ~e~~d.
~s eiA ~':> "Zjl1 i ':""e;<;;p. _1. Voor- de oisper-sie-rda.tie -z~·~ er+w~ 3roepz:~ Va.l-1
Op\oS'SÏVL<JCW.
Dzl.-1 <i:~{oep Vo.IA. oplo§i~t~f.., bebort b~Je elektro~ Bolv~ (Ook wel /..a~m.tt.ïr_ '?J?\vet..t ~e-LtDevwd). D;+zjt-t ~tao31rek~+e5dvet.. (<.o~ UJfe)
bc ctt..tdere- ~yo~r Va~tt orloS.S\I.{~e~.ot bzl..toor\- b'6~e l00CIA50lvev~ {Ook tUil\ aKCX?'J-is.cke-_
~IV~ZIA of akocsti<5ck~Z_ioketABdlf~lf1 ~e'-'~V~otd).Di+ ~l'!la.a.~trek~t<Z 5olv<ät ( o~w~UJpi.)
Bs de loa.~ fre k~ te io\ltc~A~olv~ ko.t-t owder tepaa.ld~ ol'k~to.IA.d.i3 hcd~~.-< ) c: ~ 4 2011. Voor )e '?flcHda.~~t cta.f ~eL~1.Neel'k tULVI d<tt<\\_?lj· Voorr}.e eewor~e>role
tl'e;-kt.VDttie lileiM.eJA. we i0i
He+ beblp va11 (1.4o -~•) e~ ( 1. 4~ _lj-4) (J.)orctf de ecn.te .orde d.iilek+riSc~c- ko~":>Ta~«te (1)4. /t- - I . )
f LKe 1 "fi.) c"2te 1.:.8
r~ é')+ c ~. <\f.:) ';( 2. r:; 'h + ~~...:. 85 ~! Q cö~3
(1..48)
He+ de dispersi~relatie é')\k:e .t\Vi)=o lto~t ucJ(1.tq)e1-1 (1.lf6) ~d·verba~-td +-tcsseM ke e~ 0;_ e~tt h.tS5e\il Re e~~r fi. U\4- (tAf-) kr~~~ c.ue ck rela-l-ie +v.-s~ ~e e~A Wi.; de reitie disperosiehoVMnu:
~- "-' -ke lfrT' 0 l = Cke'2. t1)2 ru (1.óo)
Ook k~lA u_ii (t. ~0') e~ ( 1. .1-t'j) de re la he tc.t'=><:.el-1 ~Le-t iOL ~e~adlch)Ofcb; cÁe
pla.ah:. Vctll\ d.e po\e;w i~ hd ko\1.tplexe vlak:
_to_
(151)
De rela+i€"S {1.5"0-51) '251-1 clo.ar (A)e: beko.~eri~si-orVl.u-1 kslje:brwk+ he-bbe~tt trtaar- voor
ee~ beper-kt ~bid ~daî3. 2cker Vlo\O(llf <Jel~ do..l- ~è.-(1. Dactr i 11 c:Ae vo\~e~~tde~ocfas+uk~ ~+ ze-er\a.ff frek!Ale~~tfie-s ~w&kt c.uor~+ ,O.u.s WiLi.i, 291-1 voor(I.~Cf-S') 5oede be~o.deri ~e~ ·.
\;r.z-CJi. \(; ~-Kc~ 0 8b Bb
( 1.5".2)
(1.53)
AcntCJe:Vt.o~eVI wa~ dat {;)i.?~. UitO .. -~q)2iel<1 w~ kt WL/lëe s1,cÁu-:, dat).:~ VF Ofclcd ck ~~aderi~~5formul~ ~ldiB'Zs~ l'hoet\i)"4~v.,dus or-zs~ l'kiW~t ~od CJd de-~~t da+ 0/ 3:Z .
Bs de ~o3kek~"'te ekktrol.-1et-~ ~lve~-t ·,s fe//Lj·1~ i-swo3 veel3rokrzodat ole b~dro.~<Z' va~" iol1el;l \rert0Clctr looscl k-a\lt fAX>rckl.,. Het bdttJpva~;~ (iA~)e~-1 (1A5)t.uorc.H
de eerste-orde- diëlek+riSckiZ" kow=;Jawr~ é•)+(K"e,'o/ï.) ["zie 1.38):
{i:Slt)
r~ ~u)~ (f.<-e,\{ft.)~ 2 t;j;_ \(;-' rtXr- f.0i.l + ~i. 2 b t<e'3 va & 2Ke.2& ~~'!.
(1.5b)
U;t O .55) vd3t de reële clispersiekrovwrne. Oaar Ke klei~is, lS w.,9c::ecle btV~aderiL-13:
. - 3 + 2 1 We.::: I t- Ke
2
Ç · "-'-b \{TT' h 1. · y ~ .. t \{ff hv 1 ~1 rfJ) 0 L == ke~ V86 'L,..xp-.zke2. '0 ~ = ke3 '8 ~r- 2'ke'l L .J
I va() sö) is :3e::bn..t;k <JeVHaa.k+ VO.t-7 \-tet fel+ dat i<e i L1 ~eJ al~t:~<l'CL.t klei VI i '5.
c!us da.f iÄJc. ~ 1, Wi ~ 'f?. D~ cii5persickroWlrvt~l1 2jl1 kier-voorde Ver5c~ille111de ejelil.oV'~eerc.ltZ t reln.u~tles fjli~ve~;~ .Nor~l wordot voorde chsper~ierelahes V'O~ de ele::k+YOlll<lltt ~ol Vel/I de paraVKeter~ G)e <lk f~ ~ebru.ikt. Ü~d.a.t ~tV, de- vol~ c1z l.-t<:xfd <;;tukket.t d€ i 11kra k·tie Va.1.1 +LUfll! elck+ro~t-t~olvet-t e~;t e'é~-t io~-te::l-1'301} LUOrd~ kke~~~t VH.od-Cl.t w~ voor-beide 'Soortct-t 5olve~~t eev~ scorl-
po- ra IMe ter Vte ~et-~ . 1-1 i ervoon s 4:5 i 01 f i. ~~o"" eV!
2 De +~e-0rd.e bereke~tt.iK:J v\.a~~t e:e11t 'ila.sov-'Poisso~A Ç>la.sn.ta i~~t ee1..1 o\i ~e.u 'l>ie eL-?
~e +w~Uole.orde clië \ek+ri&kc kciA.~talll.te
J~ dit l-toold~+u.k word.t easf aa.w ~oV..ud wetk 5eva.l Va~ ~t-Bdf kot=feli ~<3 be: kek~.t <.\X> rel:/. '"Peva.ts wordt aa."'-Cjrtx:;fVdA o~.tcAer we\ kc o~ Shll\d.iBl-tedet.t de
OJlc:fVev. besckouco1119 <3eld'3is. i-\ i a na. worolt (u~w twud~ .or~e storiK~!) rt'kelti .._, U..lt 3evoercl e~ cz-~ tc.uade.ordc
c!iël~k+rïScke koK~ta.wht berek€~;~d voodt~ ~eval va~ ~olf-~cltkcppelin'J tu.sse~t~
olri ei ol-te11 ~ v~ eh voor he~ 3eval va"' Bolt-Bolt korpeli~ flLSSe-t.t fWq<Z elek+ro~eY~~Veln ~ e:e~;~ ioket..tBolf. Voor de-zt; beide Bcva\le~r~ c.vor~Ae~;~ belitad.e-ri lit~:f~-forl"'tul e-s
atlja::leid.
We ~aa1.1 wur~M* vo.vr k bohi~\oee vu~l0kivt5 va"' V\a.sov e-ltl d.tr
Ver cyl~ k, ~ Va LA- Po i ssow ; !ft éé"' .:Ai t\t~ sie .
\Ne bkj ke~.t ck i~ ferakhe Va1.1 tlrie-7)0\vet-t. Deze- Bo\ve~~t {,OOY"de-C~~ Bekee-t ~t door d4!"
reile ~olt ~+alle~ k, k' el-\ ku e~ door d.t ko~p\exe- frek~ties 't'· ~·el-4~11
• We 11eMet-t
Je vol3eltldq- oplo::.~i111.5 va1.1 L.et elekfti:)Ót vt&ld:
r /V tv 1 IV 11 ( J. I c.[x,t)-= t:. tE- tE: + ~o~E:;recrdete-n\itet-t)
= Eet)~xpi(lcx_ 'lft) + ~:*c+)exp-C:(k)(_<y*f) t
Ë '(t) exp t(k'-x _ '\jl t) + .t'*(l-) exto_C:(k' )(- 1\f''\) +
~''U)expi[k")(_~(f) + t="ct)e>q:J.c:(k"){_~ .. ~~"t) ~
(2.1)
H((:t t-u teraard rlli Ie dektrisclt veld is op~sp\;~+i~ fi.Uee elk~(toe-;5evoe8de ko~~eu dele~-t.De a~pl;·h.tt.k i';)Uw fuwktie V"Cl.l.l c\et~à ~aar~ Va~~tdc-plaats. t:>e fr~b.ue~hes ~· llY1 e~~t \V 11ZSH à.e ~ah.wr~kefrckwekiics; dah:Sh oliefrekLUe~.tties die voldoelil aCUt de eus+e .orde clisp12r5ïi~re1Q.ti~:
(z.:z)
(2.3)
Bs ve:ld ~ koor+ vervteli ~'Stu\llktie fs. 1~ f' koert f~' ~2. he+ beblp vo.w Poi:>~oL1 ko\11. ~~~~ziet.-~ oa.fd.fl OMplifu.de~ VO.~ oe urshz .orde ver5fori ~a-1-1 Of> de Vcrde\i~~t~.f:O
t(.A\Ilktle Op ae:ze~de ws~ Va.lll detjd o.fká.IA~IIr ctl-s ole daarb~ bel-tore~~tde ur<c>te .orde ferr.1 Vo.111 ke~ vdd, d.u~ tsdsa.f~"'k~ljk-. Moor ~iet ploatsa.+~w~k. \0e -s+elle\..1 lltM U\.1 reSofAaw·he vocY!Uoa rde op voor de 3o1f ~+oller,.,. !De ~eV'tte~
Stoppt:"' c.ue(2.1) e~;~ {2. 3)"1111(1.1) ev. ~e~v. UJe die +er-V\4el'l bSelkaardie docr
Mzelrde ~oltezJetal ~ekeiA~rkf r.o:Jrckt.t, da~.t ku.kkct.. eoc de Vol~ de drie
ve:rlJe~ k;~~"' vorn.en.é~ 6te~A de>i~~tckl( s~.:De ver~e~ k;~IA~Idett voor
itet.kr ~l+jesseort)
-o+ - (. '\f' t + t kV+ t e~ Ë oi=Cv) + (l.s (E llë.>~ I +EI oV')cxp-i.('\(t-1\V: "l')~o ol- 01-;, ov Wis ê)lr ov (
2 ·'-~o.)
_tlt_
Jt' -L Y't' + t.k'v f 1 -t e., E' èJFLv) -r n (E: af"" t E11*àt)ap_i(V: "V"t"}')~o a~ n.t~ ov ~5 àv ov
(2Ab)
Ot11
_(, '\V11f"ti.k11vf"t ~ E11 ()FCv) tQ(E- ,w• + e •*c)\.)exp-tt1'-l\f''~"V 11):; 0
at l"tl<j o v ()1~ ttv av (2.1-t.c)
Al-s~ cle Ver~l~ ~\iltt~L-t (2Aa._ c) vert'lltet.tiBvu.ldi <3€'1-1 11tet e~ t1s/f.o Cl-1 delet.t doo'(
resp _ k (kv--\t'). k-'(k' v _ '\V')eiA-k "{k"v_ "f) <Zk vev-vol~eu s i~te3ra~ overv, daw
\<ry~L-« ~ ~ bekulp va.~o~ Pois~o~-1
o<..: - ~s es folc(l(v_f\V)
A 1 ~ j l-Is e:s~ ( Jv cl [ ()~Lv)/Jv è/FCv)/ 'dv J
r s iEon.; k )(kv-"V) cJ.v Ck'v-"V') t (k'~v-.Y") Da.ctr k frek<.vel'tti~ "+1. 'V1
el'l "V" de ~a.tu .. to·Sk.e irek~ties"Z3H ku.wl'l~l.ûe~brwk ~ke;~.~ vaw {_2.2). bitBuft
{J.. 6 a)
{2.blo)
(1.6c)
De \fol<QDtd.e '5tctp is o~ 011derscl •. e1d te Vkakett h.t'5'f>~ ~IA'pOIM.f?Bdf Cl-1
itVee c.vi-::,selc.uer~t-tde sdve:w. S~eiL-tM de a.!Mplih.tcle- Vetlil cle potM~~olt V<ee:l ~roter c;la,Vt clie \fa~o~ cle- +w(lee LUi~se\!J..)U~de ~olvew. ]tt~ teite Zttllei-t er lt-1 ee111 plasJM.a al tSO. -zeer klei~ tluk+IA.a+ies '0 1.-t 'Zcda.k15 da:t b ~olt~+a.ll-et.. t ~tt re~oiA.al.(i ie -c9L-t IM.et e~"' op ~!.0ekie-tjolf 11te~ arot-e a""plituct.e-.
Als (()e het veld Val/I O.e poVkpCJolf ~t E QO.k?f-Vevt el!t Val'! de tUÎ':>'î.dtUet"ke....de so\n~.-t rwctl:'ev. E'', MIA ~~du.s:
IE/ /JIE'/) IE"/
(2.7Jh)
(2.8c)
De ui~oru.kki~'l.f- "t-'\('worc:Mi!-1 kdalge1\t.e~ de,.~is~oJcki~B" of .. o.a~~tpcc-o::.~i~j~touJ '' ~l-\o~d. On,tdat (2.7) ?flctf.u:ordt het produ_kt t:'t::11 Verwaar\oosd .l-teh+elselver3~ki~ (2.ba._c) Luor~r l.tus h-tdbe:Lwl.pv-a.,.(2.c?a-c):
at: -=0 af
bit std~l va"' ic.uee: eerste _orde J;~ere~,.Jia.atverB~'6ki"'-<:Je~;~ kav.o~5e7.6t LUorcl~"' ÎVI ee~ f~J.e_orde c;l;~~reV~t!a.a.l Ver-<jelS'Kikj. bi~erelllfieer(2. qb)
li nb e111 ruWs VIaart ev. ~a.a k <jebn.O k Va.k('Z.qc). We kr 8 'je111 ola1.1 :
Nu. i :s b' +.b'' * = _è( "i'_~~~*- '\j/) ti. (lf *_"V~ '\V"~~')= Uï(_ '\V)= 2Ir11 '\V Op cJ.a+ i 111 O.tt'U +couck -orde dïHere11hoal VerBeljkil-l~cle- koë+icie!-tleM
+~a-s OVlaP'taiA kelSk --rs\11. ~od selde~~~ ·.
(2.qc,)
(2.Lo)
(2.11)
De poMpCJolfc.uorcH veroVI.àersteiJ e:e~~r reële-l'lcAftUU'(jke frek<..Ue.t-\hete l.tebbe~-1.
Dt.« verolllders~elli"'9 \5 ~baSee:rd op de 1'\1~~ kl-te i d OWI vo~.t bt.Ut~~Aat e~er~i e aa.~-t de po~(-Xjdt toe h~·: lroe;v-c\4. Is ditKief -zo,da~.t hts* ~~~~lsel Ver<JdSk~k~ll"(;o orlc:>~baor. tx oplc~i~w 79!1td~ Vliet ~o.r\'W.ciA.i<;,d. twaarli~eair<Z" ko~bika.tles Vo.t.-.
Be~xltuLttkhe~. Op d.at ~e~ol ?~-t tüe it.t d.it vers~ H-ie-+il!t. -s~ e-l :
(2.17.a.)
Ë1
: é'exp-~~·t Ë. e'. ~··. çb'~ ~~~okct/-ka~~<b-Sk ().1.2b)
Va~~tt
Eli- lqll . r~.t•l - c <ZXf-lr r (l -12 cJ
t1iaiV1 "Z~Vl. rj' el-! rp 11 1rektveNt+i~ve:rSckuivi~ . Voer i~~t de euevlce-S~ freb.)Zt.die-s
~· ~· e:~tt r( Do.~~t ~~l&t ~atu.u.r~\c:
1" ty
{= 'o/' t ~·
~".: '\f'u t ~''
Deze- toerkdijke-frekwev•Jies v;. ~~ e11. ~''ZS~~td.us ck frek~he-s Zodlsolie i4
tY":>i':>c.~e <;)d·uati~ optredeLot . Vu.! (1.12. b_c.) in d.e 1rer~l§k;~ (2.q 6). ".Je krgs<Zlll dctk:
Het \ik kerlic! V~t-t (2.15) i:sc:Matb.,.ke:!~k vo.w i. Opd.a.t ook W re-d~ terlid
Ok at ka-Lt ke l0 k Vd.lil ti~, Móet lJelcle\11:
bus
{2.l3b)
be tuer~LS ke tr"e ki.Del,t +i <l'S -z.yw eksa.kt i~;~ tec;,o~l.t +i rç
OW\oai- I"" "Y =- o <jeld + ook:
k'.:: ktk"
(2.15")
Re'SU..~~ o bvt.~.-tet-t ~-ze~~"' dat als we kskeM ltlaar k i ~-t+era.kti re lra&.~ dvi e
~oiVe1.1. tu ctaro.a~-t U0c de Lbl~elftdc- Vocrtoa.artkt.. ste(lcz:k:
a: De po~f>Bolf lso~edetM.p1 .
b: De o..IM~··hule va"' de po~peott isveel3rotcr da.~.t die lro."' de
h.vtee; Wi%d c.verke~-tcle- gohre~~t. c: De ~olteetaU~ -z~~;~ ek5okt it..c rer>oKa~tt·tle
tuvt zsVl de kc;~plexe LUerkfl1~ke trekwewties eek e:ksakt i ~A reSol-1aktl ~ {vc\ ~eiAS 2.15). Voor k+ velol kehbe:4.t de OtJicssiLt8et,.t Oe vol~~otdevoriM:
/\. /\ 7f Ë(I,t) = E expi(k1l-~t) t t <ZXp-t.Ck-x-l1t)+
C: ll L' ,, ~f A ,,lf . Ik• "*t) t: exp~ K''i-~ )+ t Up-tc -x-~ -r t
_i8_
1\ À
t t~ ~~p t (k' )(_ ~·f) ~ r·exp-lCk._ rt't)
/\
r 11 ·ckll t•f) ~fuif - 11 '' L) t 1 -::,eXpl 'X-~ t exp-L(k -'1 c
Voor de: lcoV\iplexe d.IMf'r' tudes Va~~t dCl <Ze:r-stc -on::lc tetr~~ vaw ~et veld e-1-1 d.~ <2er~teord.e- ferMelil Va.~t~ d.e vudeli!ll~Stu~ktle ku~l'l.etA (Oe w.u ~tvo\~d~ s+else-1 Ver9e:fjk;Vt!Jd;t VorV~.te~.t- ( bedakj~ worO.~ lûcs~latfl11) B9 de re<:oiAat.die-ko~;+i~:
fk; k' t k" L~-=~'tlt"*
èCkv -\?)[s + ~ Ë 2> fscv) ;: o I'Ms dV
· ck• , .. ) 1 '" es ( t:.' ofsLv) .E" at: E* dt~' ) _ - t V- Vj T5 t - 1- t - t - _ o
h-ts ov oV ov IA'
è(k-''v _ v() ( ~· t ~ (ë" ot:~cv) +- E. è>h t-E '*di-s);: o r i'Yt;) ov ov av
koor+ heh:.td-se:l ver~e:lj ki 11t<jC'-' :
~Ckv-1)h t es t o~v) t'vl:c, ov
=0
(2.2oa)
(2.2cb)
(2.2oc)
('1..21a)
{.2.2lb)
ó:nb)
L~ ~ _ :.: o Fscv)/0v E } '"" Ï(k V- V})
t::: ~ ;J Fslv)/ëN E * r)'ls U kv -VJ•)
(2.J:J.'-)
k f , r 111 1" r,u,
Eek 'Scnv+ qeli.j ke wtJr cJ< 1~-t~ ~~ku ook voor .. I ts l t s ~110 t s . De-zr (.0orc4.c~tt , .'f l •1 t" ld vco Jt· lcw.h,k· i COl~ tenw~w Vo.&.o ( 2. 2o b -C.) e~.-t {2. 22 b_c.). De-iiZlll cJ. u.r
re<::>pL(k\,_~'*)of t(k'v-~1')e~,t t(k"v-~'') e:~A ver~t-tl~vu.ldir L-\(ef L1<;es/t"o
~ozlf VOCY (2.'lob-c.)
nses Ç'~ = hse~'2. t:'1 7>t~Cv)/ov +VIse~1 E~E11 [ ofse.v)/oV _ à~sC.v)/è>V]
fo i~o nts (k' V-~'') fo nts'l (k'v-~••)[(k''V-\1]") (kv-~*)
(213a.)
_ è>Fscv)/àv] {kv-?)
(2.23b)
, VIs es t's=_ nse~ e-' ots(v)(iJV _ nse~~ [;. E11 [ ofsCv)(oV t OfSCv)/àv ]
Eo t~oVMs Ck'V-I•f) fo~s1 Cic'v-~1')LCkv-~) (k11 1t-1f")
( 1.2/.ta.) nse.sr.)=-.VJ<>e'>-z.E~ al;cv)/alt +n~e{ c""E' [ (J'FsCv)jiJV _ ol='sCv)/dVJ to Lto~s Ck"v-~") ~o~$"lk"v-IJ'') (kv-~) (k'V-~')
{2.24b) A Is tUtt de Ve;rlJe~ ki ~e~~t (2.23 a_ .t) et-1 (2-2~a-~) likk~ ~t"t ve-e kt<:> ·,kt~Ujrerevt over v
~aaJ 0.0 i vtt~raal i L<t ..kf BNa.l kt .té i IMCü3i VJa.ire deel V~"' ~~ ~t, Vf1, VJ
111 ~~~~ ~~~ ~aox ~l~~to.cLdl doorolesil!tCjt.dierep~AM.+ew lt:: ~lkil1"/k- j ~'lki ~·~;~"/k. PI b fAJf:.. a.t5pr~ke...w d.at ~' ~~ e"' VI'' staan Voor- r~p l wt-, ~· +eli!IJ
11 t e~.-t J.(, ~·Jfstaa.t
Voo'f f ~k ~~- (2ie § 1.1.) kr-0~e:111 t0e voorde l~ttt~raa.l dej Lti-ste ui\\coiMs.\-
-2D-
c.tls we o~Jer de silii'Ju.liere pUA.t~e-lfl V-; '1/k; ~Ïki ~11/k b\SVe:~tt M1.aal""bovc.v. de
si~~u \i ere flU.Iiltft1.t V::: ~·Ik.) vr''/k la.IM;)S 300.1.-1 Zelfsals b9 voov-~td cJ.e
i ~aei VIW re d.eler.. VCHI ~· ~· fllll r/ klei k.t'Y do.'-4 ~.t.u.l ~1.1 lll-l ck ll\ta.t;}Î t.taire de~~.t VQ~.ot ~ 1, ~~~'i) roter ola~t~ 'vtu.l ~ltt zoals b~ voorbeeld 'n1 ti3 (2.-t) i~aa~:f=ljéVe'k
0~''/k
-v
A blOC (2.23 G(_b)~ [2.9Jia_ b; Of~ (..)Sze-iL,tfeBrere~ overv e~-t~Ut:Bebruik make~ \(UIII O.e stel\ i 1-{8 VO.L-1 Poi'5So\11 die il.t ~ ~alle~tt de vol~ole- VOfl1.1 heet/:
ïk'E'.:: 2 l'lse~ Jr~ dv S fo t
_ik' E'* ... i 11 ~es rr:dv S €o
il/c\ ~ ~e~ rr~cJv
bct\ll ~ao.+ (2.23~-b)over i1.1:
( 1 _i nses1. rots(v)/ov dv )'='*= ::~ fol'\t~k' J(k'v-~'*)
:::-l=.*ë~~~.n~e~3 l clv _Èr ;1f5Cv)/è>V
s L~or"l1s~k' )Zk~-~'*}:'vl(k"v- ~11) _ otsCv)/àvJ (2.'2.'fa)
(I( V- YJ*)
(t_i 'tl!,el roFscv)/ov àv)E1;;:-
s Eon-tsk' C.k' V-11')
=- E: e'' i n,. es."l r ov ~ [ ()fs.ü)/ov s GEorn~k' )Öt'r-?') è)ll (kV-tJ)
(t-l ns ~'l. f ot=sc.v)/àV cJv) E11= ~ Eo nt:.k'' ( k"Y-~u)
= E.li'E'i nc;, e.,.?. r J.v È.. r d~Cv)/ov ~ i~o,.,?k" lLk"v-vf') ov [[kv-~41 )
t ofsCV)/èJY](1..2ba) Ck" Y-1111
)
Hd likkerlid vaw {2.15"b) e111 (l-26b) kak volcoe~<s (1.1.2) ~~ltrevett r.uordev als E.u)t( k1
', v/'), ke~ I i ~kerlicJ va~ (1.1-ba) ctls ë•)+(k', ~').
Doo("VaL~~ [i..1'2) li~.tb el.t reckts Oe kohtplel< ~ko~-tju..w~de te 11e~ew 2Îe~.t u:x dat btli~.-tkulid 1/a~ [1-)Sa) <'1~c~reue~t ka"' t..Xlrdew als E'')+ .... (k: ~') E'*.
Hel- bekttip va~ pctrtiii~L itttre3ra.ti~ luüHl~v. dcrecltfrrl~d.ew vaw (2.25a.-b) eltl
(î. · 2b ct_ b) (lCI!I \Jou d_j [3et' ~e-S.ckreYe-1.1 (()oY"d.etA .,l)o.ar ei~ S-\ok+erttte~.t n..ul "2~ ~ krb9q-L.t <oe Voor( J.. 25" o.- b)
ba.ar voov- (1. :qa- b) ~elotf- da+ k:: k-' t~'', ~-=-~·· +~11 e111 ~'J= v; kaw {2.2ra- b)
~e-;,c\.r.re.vet.t LUon:!e~;~ als:
Ec.)(l<', ') E.•* =- t: .. EuZ hse~'! I è)rscv)/ov {2:Ztla.) '1 s i-Eont~'l. (kv-~)(k'v-.,•*)(k'\'-V'')
Dr pree-ie~ de-u-It de lUS -ze krSff M€111 vo~ (2.2ba -b )(lJ.lao.r voov- ~ldf k'-= ktk''·~·-= ~t~'' e1.1~/==~):
\fOCJr {2./.oo.) kv-s~w V2e Ltctfu.u.rtjk ec')(k.1)E=o,~ ~<ûZYke~ke}rdc~-fie:~ is ~e~ k aa~~t cle 1-taf<.w.r~ ~ tr~-he ~-
bef"•W~VL«. de ·tc.oade-orcle diëlc\ctriscke kC~.t~+a~.teo EC2.) die~~lfel.f~ordt doorde redA~t-kwet. va't,f(2.'28a-b) etA.tL-29C\-b). bu..s.voorb:zó'ct-b):
~Lil Voor(). 2.qa -b)
~C·) Ck', ~)E1
::- fC-z1t::k-.k',k; ~~ ~·· ~~~) Ë E (I
~ü)(k1:~•')E''= fl-z1(k,~,k·~~~~··~") Ë't=«'
We b.(.,llt~ Ë1ot E-'* <2t1 t'1 0livuit.tere1.1 t..Ut (2.3cJa- b)eJ-~ (2.31 a-b). \toor(z. "3oa- b)
kr9 ~ek tuz;da~A·. 1t
f 01 Ck', ~r) €(•) (k': ~")::- ~ ec~tk. k',k': ~~~~~")l2 1EI 2 {_2.?)}~
eC•î Ck',~l) ~o)(k'\1{)= _ ~ ë·1)(k,k',k'', 11 .~··~'')TlEI' (7.326)
Uit (2.~2a-b) kuJHte~~t do.IAVevsck,illelildt r~lahes 'àehaald (J.)Ot"de:~.t -zoals·.
a: 1~ ~· ( =- tW~ ~'')als tuiAktie va~" k, k'ot k" telt! de- Cl/l'Jpl:+ud.e- l.bl-1de
pow.p~olf!ËI,clu.:> d.e ~a.te VOl"' cÀeK-tpi\t3otaa.l.(woeit.t3 VaLt de:
hu4'&e LUi5sd cue;rke~.tde 9olvet1. b: ~· (ot f) als fuLA kt tb va"' k, k 'o{- k"e~~t IE/
r~ de Vori~e pctra.~ra.a.r {J.)aS at Beleid:
EC2)(k,k'.k';Vf·~·~~")=_i_hse!)~ f oFscv)/ov dv (2.33~ ~ (,~o ~!>-z Ckv-~?Jek'v, ~,., ){k"v-11")
ECZ)(k,k\k'',~·1'·~'')= 2 ~ .. e ... ~ f ofsCv)foV u\v (1.'3~6) s tro t\4s"' Lkv-"t)(k' v -~') (k''v -I?'')
be ~u koe de: ilitte3raal ~ir,àtrpre:l-rerol w.toehuordel.( Ï'50C>ki~~tde uorl5e
pa Y U 3raa f beb.IA del d . ~ebekS~ all<U:~.t (2.33b} (VoCY{2. "3Gl) i~ke+kieroplfol,elde ko.tLU.LrSkook
Va.~ toe:paSbii(J). h~+beku.lpra."' bretc.kspr.~r.t.~ c.uordt (2. ~3h):
~(2.) .:-i ."' ... e~3 [ lf ot<:,(V)/'dv av +_I r otsL.v)/ov dv+-' I otsév)/dVJ s L~o~~'l A Ocr--~) A•}Lk'v-~') f}z [k11 v-~'')
('). .-~4)
Hie:riL-t --zSlit d<Z koëfftcie~AteJA A.A.~ A-z·.
A= kk"(u._u')Cu-tt'')
Nu. ku~~t~~te~.t wewuv ~ebrwk~ketA Va1A de plasn.ta_d.is~r<i>ietu.lofklie Z(f). De vd'je:~Ade substi·hA!~ ku..vHteLt ~~ktc.uordeJ. [2ie. Hfd'htJ ~.2)
J~ èJtsCv)/àlt olv .=. 71
{}..") iJ è>ts(v)/oll olV.:: :l(}~) teV\Z. ékv-~·1) 2kV5
2 Ck'v-v{') .2k'v,.-z
Ll :: )'s Vs '.J1 j u' :: )<:.1 Vs '{2 e~~t te.''"' ~s Vs \fi!
He+ bek lp l{o.~ bt bc>Vlll.t'btaa.tA. de kv-y~et.t cue voor (2. '34)
rCl.)-= i V!se53
_1_ L z'tr~) T 2'{)~) + 2 t)<ö") l 5 Lro'fK'f," v'ó ~t kk' k" l{f<;,- }s'){)s.- t;:.") (~~·- ~ .. X'fs'->'i>'') C~..,'~}~)()\''-)~·)J
[2."3~)
hses ~ 1.
E'o f'lls"'.V:> 't k k' k''
Q\5 é(freciu.au-d 3oltc;,da.l Ke~e~~t toe <.uUY~ .tt~t-be~~ \ro.k (1.2o -25') ko.lf
{_7..3b) Cj~c.krevett word~ a.ls·.
[2.31)
{_2. 30J)
-1. bc o.; Me1.1ste va" t1 is~Sk aa"' ck&~cl.(~le- vo.u E . Nel-de?.t 3root~idR 61.1 d.e VeJds~erkte \(a~ cJ.e po!Mpep\f clf~~icbs ~VI.tc:takt (i)Crd.c:tA.
be{; vt•ct.-er Je 5aeGtttceenk 'fdd~erkte a.ls vol~+:
E == E J_ ~ (o 1
::: Ë .A {_7.'?:>9) '1 ka% Vle
RI~ 1'e tj~eYeNt i'5 ÏLt°K ~~ei~ aa•.Aa.llew ~ k~.tb. ~terdaiA is A:: G,Lt133 x ~o" Vïrre~~· (k~ ~ Yu..--z.p-') ~et a( 'S d.e CJered.uacetc:le Vta.tc.(.(.(.(~ke tt~ktilel,t+ief. lfocre~~t we d.e corcre<W.ccark
(;JeYke~'.cetrelc.ûe:Mties ~5 ,~'s0t V~iw(qs: ~/wpc..)· [-zie (1.'36)).
De ve:rco~ k; ~ (2. Y->) cuadt da.~~t:
E. Ke,Kerl<e,IA,IA,~ :: ~n., ~s ®s- es 1~ 1 }s (-z..)c+- t_· t ··- -( -11) ~ 3 2 R 1 ~(_ • ") 'l .L S J~ i fceke'~ f" r
(2./.jo)
k kootd~}ukt §3 ~IA eltti~ ~l.taderi ~s1-crV~MA-~s \[ct:r2.C'f{)?Jrc~~ltr. W(L>gaaL'I
ek-ze lllU ~cbn.Uke-V~ 6VVI bel1o.d.(tfikf]Stonw.tks \reu ë:z)hclJi ~.-tckk Voor kt <jélfal olar
eNe po~Pjo\t ~ d(L> +wu r.,Ji~c\lJJcrkwd~ Bdllet,. drie io~~~\re~it ~L-1 (Ä/1\roör
ketg~val dat de fO~""tt:J90lf ~tilt iiw Wi~ ~ke~ BDlt tU)ee ekktrohel.-1 ~olve~~~ ~v, e\.-f ck a~ttdere wis.std~Akr"lc~de 3'4-ttk ic»tew8ciT is.
De ~deri~sfcn•w.des voor "Zc))2Slit lroorut.t btpakt seb"ted9Zid\9. baaY d.tt beperkte 0ebied ook kd ty';>isdt ~+ rdeva~.t+~ ~iedis,z.ullek k2ll' bel-tod~~~ coku~.t volaoelitk bt~c'krSv·lkjlJevet-t. Aa.t..t~<&o~e~ U)ord~ Ja.+ (!;);_ > ~6 l';).
U n- (r.Lt~)) Li~ ux aa+ vcorae noJu.urS~ irek:we~~d-ïe ~;_ ,ó.ls 1\YiLlt.<jelJ+ kf~t ~ ke. Hel-i~ dus ~~tt~k dat <:te lij d.e (J)arkeSke-freku;c"dle~ ~." .~.,.'eJtt ~~· ~brd'lde- pctra.~tcrs )s. )~ e~-t) .. " '3eld+ dat }s~ }s
1::t)!;11
• ~.kiDerb k fu.lllkhct- SC)s. )s',}~')co~vefA ob::Y(2A'\)doaroiM ~~JQ.htaderu.t+. We- bc~ \<:e111 lttz.htol ~~de:
G (q, ~.c.); Fca) t Fe~) -t- t:"c!.) J (a-t) (a.-c.) {IJ-c.)C~-(1..) {c-o..)(c-f:.)
;)tel dct~ de t u IA k+ie Fel!) tll Sc k v-s VelA is als-.
N N Féx); 2:' t=""' (>t) = E o< ~"X"'
Vl=o 11'~ 0
N
bo.ltt i "3 S(.o., ~,c) = ~ 5~~t(a,~,c) . .befu~~tkties j ... (a, _g,c) ~k ~l'\ltctkkcS~ te bere~"'. VJe: kv~eelA·.
<jo(q, t,c.) = 0
c;. (q,~,c) =o
s2 (Ci.t,c) ~ o(2.
~3 (a,g,c) = ()(3 (a-rbtc) I
{1..1-f2.a.)
()..4'2~)
{J...I;-zc)
f'(CI, !, c): _!!_ ( ___!_ + _!_ + _!_) 'j abc a b c.
We Lttel'llte.w cw111 dat u::.t. UIA fJ6..SI'l-\a kebb 11.c.et ~ e~~tk~l ~r+ lokefA. De ele~e1.1 ~ iowzl..t wordeJA &Ju:ro.a~.t~e~'Ve!A Joer-elt it.tdiazs. er-esp i... Bo:. CUorJ~ (8 cg&oetlkcl,
Ós worcH b fZ-V1 ~"' = t.. (j)e = 1 efA P.:"" -1.
Jl;t cAe I;+IZrd+uurLuorcJJ ~~ i~~il.1aire ~eda:H-e Val!t se~.~··>") VU(J.)aarlocr..d(ctus.
ktrüle deel ltaw ~e-re)_ He+ c.vorcH ook kier w~Bela.tet-~.
Voor de 6~ d.e- elektrowet. ~dra.<Je 'oekora.tde parat\<.ettY}e <jeldf.: ~e<'<' 1
Voor- 2'L?'e) ~eldt ola.tA cle vdiJe:t-tcle reekso~.ttu;ikbliLt~f
h 7'(r \ . 1\J 2 2 3 "-- 'i K.e;L l~J.;; -2 t 4 We-- we + ...
3 ~ ~
B0 iedere terVIA va.w de r~e-ks<:Mtwi kkeli~.tj lfaw 7t')e), z(~~)ek ZC>e") is de
daar 6y6ek.ore~-td.e t€f"IM lfctiA s(~e, )p11 'fe-) te Vi~.tOeu ~J bekul~lt<lW {J-ij zet- 11)
V
Di~ljuf:
f.(f' I ") j, 2 3( 'l. 12 112. I 11 I Ie\ ) J ~e,'fe,')e ;: ï-j!T }e +)e t)-e t)e)et)l!')e +)ef~ .Ji-··· (2.~5
bet ar ?-e ,{_<_.i is de eer•rJe teriM lra.t-t ~ reckterli d al ett.1 fjCtëJ.fè bader; ~-t5.
Voor de 6~ Je io\-tetA bjdra~e beh.orötde pctrll~tr ~C: cyldt:)c: >~- Vcor 2'Cç) i5 cb.~~t Je vo\~eJ-tck> beMJerlkCJ 3esdikt·.
2' -\ I Re ()tJ; _ {J).l
•
De b~ Z'(~i), i(~~) et.t z'(~ï '') beko ra-tde fu~~tktïe 5 C~. 'fi', )i") w ord-1 da~.~ w.d- beblp Va~;t (:2.41-t)
(2A6)
-21-
f.'[~i.~i·.~r)=_!i_ h1<e.'k~ I~+ t<e' + l<e'') J '(, (02. 7ï. ~i.' ~i." t ~i. ~,· ~i.''
Met kku-lr va"' (2.~5')e~ (2.tq) hS~~ ~ lfoor(2. 4o):
baa(' ket i ~aBi ~ttaire: d~el Va!A ·i{~) VevfiJaarloo~d u.ytrd ffiJ.t (2.40)7owd voc'f c(7.)(t. t-.1t ... -. _,_,,)dl aCz)(r..t•t_k-. -••-11 D _~. c;. LK~,Ke,Ket~lo~i.,~i. SVoorc "!t~t~o~Lo~i.o~i)· 00\'!Ufl:C'll€"
-s;Juatie b~c~ouwe~~t WM.rby de (i)etbz-~ke tre-kc.ueJAfie kirlverva~.t de
t.-tafu..ur~ ke }-rek~ tie at l1:1t -z.a.l oLtBe'lfU:Y 5elde1A cJaJ ~i. ':è k-e, ~i.'::t i<e 1 e~~,··~ ~" .Dd: ver3eLjk•k~ (2.t-t8) ko.~-t Jo~-t Ver<let-tvoud.i3d cuorde(.Atct:
2.5b~i~s for~le VoorEcz) als de fo~f?f1?1j <leW elckfro11e1.1 crliS Cf..1 Oe~ct WiS~
~de '3ohre.t.. ~ rdtz~~ ~tet.t <W.f ioLtC~A tj?r ztjk
~ll' sklkk dahle: txtra~tas )s. ~.: IZll \<e klww. h~cie fe>!Mf~Olf, )s', ~/ <Litl ke1 6~ dvz wissdli:lU~de de-ktro~ 5olt <tk ~;'.~i' <tl-Ik-~' bg de to((.~o\t. Als~ QO.kllleiMeiA dctt de- wdk~ts~ freko:~z::~.ckö "tier ver va(A dq- ka.tu.urS~ \rekl.Uwti~ l10<:JZ"'. do.111 Vol~t t..tlt (1.52.) et.t ( 1.. 51) do.+ Voor 7t 9c.ldJ &at ~i 11 1: w-e
Dt Voor q;_ (l);l~l do.:i 9i. ~~i~ vr. Hd- bekt.tlp VO.i-1 (t. 3'-t _1.36) kl.ü'\~~ t:Aaw ~ Vol~~ Sc-ka. ~~k~ \rOOY' de paran.taJc-r~ }e' )E'
1
t ~e11 ,~i ·}<1 et.( }(11
~tt"~t c.ûolrde~t.
r~ =~·V?-~_!_ V®{ {)i))/1) fee l ke 7..
)•' = ~nre ~J:... ~r;,&, [ );.' >>/1) f<e' V1 l"<e' 2..
v;t ::: ~:'V e' ~ y eI (1~" :>4, ?o lferCMd.er<5teiJ. ~aa..- }i".(.(),') tee" :2. 1..
Voor Re; 2'C~) Lorta.lellt 1.00 C~fbruik VO.IA d.e vol~et-tde be\l[aderiltltjstorVtUt.les
,.._ ~2 =- + ... r~re'
{!Ao)
(1AS")
( U)$1 c:t~ e.~.< c.oeer ket i 11.tctjclla·r re
deel !oej) {2.4 cj)
\t oor s er~. );.'I ~i") Vi ~Oelit we (l€;1.4 hz- 1/U<.Uo.arlo"'Zel,. bSci.'la.,Q;". Voo'{ (z. 4o) vi HcJ.e~ ~lt" cia111:
(2.5o)
ba.af toe d.ie 'S~ ~u.a.tie- b~chll.t.We~~t bJaa.fb~ de- c.uerk~S kc tre-kwet.tiies L<.i.e-f Ver uaw
de l.<atw.v-~ lee- treki.Ue.ktt~ li g~ttt-, 'Zal gelelew dcc\ ~t ~ ~/-:t Va'. De- Vtr<3e-~ki1tt3 [1.5o) b111 do~ lje::'>ckreveM LOOrdeA~t al-s:
IV\ dit hoofd Stuk tvorde~rt oploz.si kljet.. ~'Zoc~tva~.< de- t~c- ort.tq- diSpe:r<i>ierdatiq'
Va~.t etz;t.t e.~;~ \(d VoLu:lilj flas~ ik iéw J,; ~~.t~e. s~"Zcxkt LUcrolt 11aa.r-0.i<Zoplos~i "t~/!1
t:ua.a(by à€ t..:U~~frekrMa,\.+tc ~·e-t +eva Vo.l4 dz l-tah.utr~ka- trekwet-ti ie- llzi· Vooro.\ die opla:.siwr, C.U<tar d.c+<.OU wi':l':lel li:Uke14d.e ~olv~ ~~ aa.~~~.~~ic~.td~ !èo\v~~~~ b!Sb +e~~t~, d.us waa.r Ht-~+o.61ifeit cpirrud ,t.oordck no.der belce-k~. ëer-~t c.uordt IZellt al~~~ o~O!):>I~ ~e~lleti, d.ie~abrui~baa( i-s Voc.>r Vt.tttKerie-kc
tM tUJerk, k~. be-u k:Orc:U do..~.r 3ebrwk+ o~ fki~e ~6 ~~ for~ lef> te VI ~deu.
Valit I.W- a.t Clelh (Oor ct~ voorcJe czrre:cW.carck ~rekwe~:hes ek. 0dt~etallet-t Z3eWCOI..J
~ e~t k ~~sckreve~;~ i u ~ao.Js vaw ~i ~ Ke. ftew k;e~ttl.UU wïllekeu_ri~e rülefrekwe~Attö 0
1el-Uu'' Jie ?oJa.wi37J<t>b2el-t ~llt Jo.t ~elctt:
~ = cu' + w'' {als k:: kt k")
o+ w' ~ ~ + w'' ( c.tls k'., ktk")
(~is reèq;! e11t <Je(ijk cta.111 ~· <:ïle kattw..r!ijke irekcuekt\~va~de-potc.c-pBol+ ). De-u- frek ~~~t+i~ 1.de~~t ü/ LUorJC~.t ck- 5\-ar+fr~hoeLttict<il :yl-toe~-t-td "S'te I
{3.1. o)
{'3.11)
(~.2.a)
{3.'2b)
VJrc, b.ütVt.e1A eldA parat'l(eter P< Ik Voe:Yt:JA ,tJie k Qttu~ki~ Vak k sfartirehue:~~tti~ +.o. V. Je ((.)erbS ke trelclilOt+ie re:pr~sett+eert:
1 .n' .n" 11 ( I k k' k") I)(:;- W _ L..:U..I :: IJL.. - W <:( ~ "' t
.n I ' .<I" " { I k I k kl') o<.:: 1-lL-W:: UL..-W O..'S ::: +
Het Voor Jed Va\1 kdîk voeve~t~ va.k •==Jar+trekwer.t+~ js, da.t ckzq 5otl(<;:, Z'od.a.~.ti 3 kW1.~~1.1 C.Uor~ ~cb2~ ,O.at ctcbcnz:b-L1il1.j Velo..hqf 0&Voc-0i:J Wor-J.+. Hef
belutlr \ra\4 cia-u s{artfr~lc.ue/A.ks. b<r~ekt.tew lö<r 1/W. e~ kornz-k-h<Z' ordic
trek(J.)el.die<;<-U·L Ro.~tt v\ereso"Whhe voorwaarole- bl~~ b~iedet"e-LX voldaa.l.t.
~e kv.vu.t.el.i Ec1)(k',cJ+cxtL(')e"' ~c~)(k",w'' to<ttr)oKtwiklcd~ it.teet.
lto..ylorrecb. ktefow.+wikkelikfjSpctr~~er{P<ti(). tw kootc-l~tu..k2. i'5<lfVo~clett (z.ie{:]..'3'1)
(3."3)
(~.3a.)
Ec')*"(_k', ~·) fCI)é\<'', '1'')::; _ ~tc2)1 2lEI' (ol":>k= ~'tk") {'2.3.za)
fc.1i(k\ ~'), ~CI)(k'.~)~~ fC.')[k~. ~") !Dorkw ~ i t.t C-et.t 'Pa.ylorretJks Oth.ul~\celd. HiabS Wordt (IX'+Ld')-zokleï..-t vuoKde-rshziJ , dat ECt)(k,w)Ol<«?fV(ter \i~.teaërvev-loopi. Dit \eictt tot re:spckhe-ve~k:
c0)[k'.vt')-= Ec.) (k',w'+D(ti(î:t
Re Eü)(}',w')t i~ é1)(k',(.)') t o< ;;~ ~~){~~.~·) +i'r aTZaç(•)(k~~U') Ol.c.)' 0 o(J..)'
f 1(
fC•) (k'. ~~) = fc.') (k', W1_ oe +<J )~
(3. 4a)
\Uz fC..)(k'. w) _i I"" é'Yk\ w)- ç( d l2e ECI)dcl,w') -l 'y Jlte é•) (k:w•) oc.v' o ow'
~ fü){k", wu) tiké>(k~wn)tcx ol<e Ec1)(k",w'~ +tv ol?a é')(k ':~ow) ot.u 1' o àw"
('3.4c)
• D~ VeKBe IS ki~t~c"' (3.4a)e~~t (3.4 c) koret.c bSde V«~ kc~ (2. 32 b), Je, lfUfJ€l[Jk·~·t., L~ Ab) e~ c~.~t.) b~ [2.32ct). ]~~~. (3. Lta -.::)is àl~ott E-'')(k,w)/ot.U {o~kt Ü1té1 )L~ l'?tz·E 6))
CJ~3~~Ia.l-e~.< o.ls ook a.\le ~~~re ctf-?flet"&et... A Is d.e I.Jevke ~-ke trekwet.1Ji~ dl okt b~ de- ka.+u.ur~ke trektUIZkties. l133etds lle ~·- w'-= oLo<.) e!A 1&"-l-tA:l''.: o(f)(.) OIM.da.t f(•)(k\~')::: ~(•)[k''. ~''):::o 2a.l tjelole-L1 ol(Át:
Voor k~f. 9e-val J.af w'e~Aw''k-id.e op;ok~jolllet.osloaL< i~~fbe.b.lrva"" (1..J.q)
te Vlk.:Á<Z"~-< c;ta!:
Als Met.. ke~ produ.k~ Wl"t~M~ lfa~-t (s.4a)<Z~-t (3.t~e) of-va"' (~.4b)ek {3-~c) koV\tet-~ er ck trol~e:okde terVHew waarvoor ~eldf:
o< 'dt2e f(•)(k'.l.V') 1(e ~ C•)ék'', wu) = 0 {. D('2. )
~w· ~1~'
p(, 0~ f''){k',w') ol2efl:r)(k'',w11) ;; 0 (~)
êJw' ow" t.o'1w"~
D~U? proclu.kt~ -zs~-t Vak ~rde ordfr'. Bei ole- ~oefe-k ~ kttZfl'~eko!Me-t...
(J.)orde"'. NWMt ~e"' ook k twade a~~eleide~-t ~e- olaw k-r59f ~ prodtA.kt~ \ro.111 k vovw.:
2
t( -z ~Re ë•)(k',w') l<e é•) {k'', w'')~ o( fX2. . ..!_) 'd w•'l. w' 1 tAJ• 3 w'
D;+ Soort prod.u.kre~A -z~ t.t d.u..s klei ke\'" a Is bovel.\ ~taat.tk als o<<(.Cw' et-t
D( .({. W11• 1-tetZeltáe ~df VooV' de prod.uk~e-IA k f e-w f kva~e-t.... f71sw 1
6~ <Ze1A loWë~-t~olt bod et~ w'' b~eet.. iotAe:tA~o~ istre~.t uoldoetAJc uoorr.oa.arde
oaJ ~ L< UJ11 ~oet-zs"'.
~ kul-tL-teM da.~~t v~fa.q"' rKef-de vol~ekde fe.rH.t.e:vt voor ktët li~-tlcdlld vcu. {2. 32.q_)
Re~ éd .. Ck\ c ... L o< t-C:J-) ~c.•)ék'', w"+ b<tt<f )1 =
r~ ~ f(t)lf(k',(.o.)1_ Q( tC:!) ~(l)(k". w'' t-1){ tC!)l~
_ ( ~ ~ EIA)") ~ 2 P< O + éi't I") D< +- (R "- ri)r t(I'R''- I "t<') 1 Cs.~b)
Re~ é•)(k1
1 t..>1 tot ti i) ~O)(k':t..>''to< +i( )l:
( ~ :0 c~·) ~ P<:. '/ ~ ~~< ( '< 11 + T<') _ y( I'+-I")-( r'r"_ R 'R")' {3.6a.)
[~ ~ é•) (k',r.u'+O{ti{)~C•)(k':w 11to<ttf)1=
( <t'w ~1.0'') ~ 2o< f + (r't I")D( + (tz"·.t2') 0 + ( r'r<" + l" rz') (~ 66)
E.llc.o = o Re €.6)(k'',V)") à <.u''
R' = l<e ~C•)(k',w')(è>~e-EC')(k',tJ))-1. ow'
t -= l2e Ec.')Cku,w'')(MZe~c·)(ku,c.u~))-1 ow''
r' .::0 I rn é•) (k I' l-J1) ( 0 r2e ë•) Ck',w') ri
ow•
L" = r~ €(1) (k«, w'') (.C:H20 fU){k'~w"))-1 OC.U1'
Ook ko.lft V\w.t Eez) o~A.twlkkele-~tt.•ke:.et.t IFa.ylorruks. eekter ctel~v(oetï{ Va.,~-t
d~ rrekwe..ktiekorrekt~ bveel kleiLe daw i~.t <êc•). Daaro~ l()ordew ck"ta: korrekties ~~al-ew. ~o.. kwa.dra.a+o..f$p\Jsi~.t.9 e1.1 kÄ ~ reke~.tt.Oerk \w..~Ke~.c de redc.+evledet.t
Val-t (-s.sa._ 6)01-1 ('2.>.ba- b) k.evscl<.reve.u l.Uor~ "locla+wecle-uol~~-tclevov~
krS ~e\..t Voor verBe-Skik3l2. 31.a.):
(()( + (R11
-R1
))2
- ('r t (L1·1-i"))'\. (I'-1 11)~ (R 14.R 1
)'l..=-+ ( ~C-z.)1'1l~ft 2. V 2 .lt lt , t~ E~
c~ gb)
t=Vt voor de lfeY~e~k·k.tj (';;. 32 b)
( (~u .. l<'))'2. ( éi 1+1 11))'1 cr'-fl')'2. (12 11_ 12')Z- r Cz)77 I eF ~+ _ vt _ ~ _ --~f 7_
1.. a 2 lt ~ ~ ~ e~·
(3.'fh)
bc:r oplo!>s·, t.t~:~f-~~~ va.IA k-za- stelsels ver~eSki~t~ -zs~~tde- s~~(.(Mte~t~ va.~~t-1-t.vu Dvtko~~.-tale ltyperbolew die ke-t-ze ~de ~ae.l pUMt '-teb~tA 0\.1 t.oaar va.~-t de
osyll1!tf.-\-otetA eeiA koek va111 45° vw.ef- elkaar nta.lce~t(~~2i~ fïec~.1))
. -. -.-. ti~~"~
]~-t, f-i0 [~.1) i~te"li€:1.1 dat er ftuu rtt~eoplossi k~ VCX:>'rïX eJA.i ~l.i hlc:tllfe a.ls: Q: r'= l" ( ~ldt voorbeide sf-el$d.; 'ler~(bkl~e~ (3.8'o.-~)e~('3.~C.-d) b: RI = T2 h Ce~t a.U<ëet.t Vor:Jtr w shd se/ C1. .(Ja.- t)
c.: ~'=-IZ"(~t- a11~ voov-brs~dst::l (3.Q"cct)
be k_y pe:r bocJ hA..er ck ctS y ~ptd-eu <2:llet1. ~ di5 o.o.~.t oe !X te1<1 r a.s i~ t:la.~r~
oktaarcllû~kr0 0:;eM d.o~~t UIA. oplo~-sit.tB voor o< ~ f~voorro~ ~ voory ek +ooa Voor ti. o.l ~.tao.r ~a.k~ de li ~91 WS va. 1.1 ck o.~.tderc ky perbool. 'i h bei cl.~ ~per bob Ol(taard -zS"' kr~ qe-J,. we 1.t cttuux ~ k V\otaar ee~A oploss .. 'r-t.3 voor (X~ (r· we bek~~ ~eue~~t c.tl\tt:L< ~+stelSel ver~ki~'3.èl'a-b) Dih:Jelsd kood- d.u.s b .. de: resoua.L<tie Voorwaarde k::. k't-k'', "~·· t-tzu)
tt CI'+l") 0 == d't 2
u
o(~2._'y4'2+ cr'-I)2 _ CR"t-R'Y'2. == rEc"2)/2.1E.I'
0 ~ ~ t ,E~E~ (3.toa..)
lf lf (I'-I"){R.11 t-R') o<o_ tt =o
bit ljre~ d.<é vol~etA.de v~r~e~kiwe voor ff lf
~*lf _ f* 2 ~ cr'-I''/ _ O<"+rl')~ r~cz.1~'1,EI~' 1- éi'-Iu)~l( 11+12')"2=o Jj lf ~Cc]~lo> 16 (3.11)
'S+e-1: H,-=fCI'-I")' _{R\1<')2 _,fc~)1 1 1EI7 7
l 4 tt ~~we~ 7
k~ = ér'-I")'lL!t +t<')'2. lb
Al~ ~h""l J. 0 kehb (.0(2; t:tl.s~i.~~itteve Opbssi~ voor r*~
(3.1~)
{3.14~
Al~ th=- o ka.t.tljt ke+ l!at.4 ket te~ Vaw \i~ ctr Lûe\k soor+ oplcs~i Lt~eu tM kv-8cve'W. [1-h. ::o c.ds 1 1= L" of als ~11 = _ R
1)
Al-;,.-
Ht < o
da~.t kv-s ~a. t0e voor 0 Je, vci~Ltde opbs<i.ilil~e"' :
(r,2 = - (l~tlu) !. '(H: {3.1bo..)
Voorket st-elsd \reV~c~b~.t~!Ut c~.3c::-ol) vukv-s~t ~" ckoplos'ii~ op
prv:i~ cku-lf-tJe (Jj~?e (b;h:hd::d Wc:ort b(..j de re~oua.~~the:Vcortoao.r~ k'= k+-k'' ·~·"" 11 t-~')
Bs d.it '5telsel '5hdl- WA.e:k:
~x~ y t cr·~r)
rf..* .= tX t (1211tt?.') 2
{i,= (CI'-r"t _o<"-R')-ztrfc1>1 2 1~1-z 7 ~ ~ lt lt ? E'w.> t''tv J
'l. 2.. - '2. (I I- I") 02"-12') ~2..:
lb
be 0 rl öSh i r.t~el.t ~9 1.1 OCU1:
Al'5 1-b :/. o; ~------------------------~
éf I +I") r = - 2
U<-u+ tz') ~~ D(, '2 = - :t V~ 1;,
I 2
{3.q b)
(3.1~~
{3.1oo.)
(3.7ob)
Lat~~-t Lûe de koë~i ciczLtteL< (,I'', R'e!A R11 0af kadet bekjket.. \De k kj kk ko-t vol f31Zk d a-
E(tk', w'- R'_c: t)~ ~ ~Ck', w')të.I~ ~Zk',w')- !Z' à l4z ~ck'.w) ow'
_ii' o~e.c•)ék'.tJ) ow'
Eu)(k',w'- R'-ii')~ Re ë')Ck\w')tC:I~EC•){k\w')-IZe é'\k~w')_t[~~·)(k:0
(~ i:':> c:k ~ k' beltorel.1de ~~~ahwx~k~fr~k<UetAhe) Voor R'~ I' <Zilt opde-z.e~ck wsu- ~crtl.UtfY.1kook Voor R"e~.ti''fYidt du~:
I R ~ w1-l2.e't{''
R" ~ u rz. "V'' -=w-e-
r· ~-LlM ~~
r ·· ;:'-t ~ t\i"''
We be~ke-111 desitu.cdie k'.:: k tk", w'=cutw" VJa.t lila.aer. l0exkv-SVcl-1
lte-+ -s+~Se·\ vex caeSkl!.{<3e-"' (3.'8c _ &) op~dll iets a~.tdere U)~ -ze op.
=0
{3.'l.7a-)
t~.l2b)
(3.7:lc)
{3.21.c1
2.
tEC2.1]~~(l,, = - ç6 2 (cfiis rcz:ëel. bit~ o~do.+ ë,)i~aaiKa.iris) cwé w i)
'
\ \
~I
I I I
I
I I
I
/ /
/ /
\
/
/
/
/
I I
I ~
I
\
\ \ \
' '
He-tis~e~akke~k ~~ te-zie11 Ja+ Qls H~(= ~·S7)i-o keftr-ajekt Vak de S ~ pUM h~~ \rO.II! de twee kyperbclet.t loop+ Via. P1, Pl, ~ .... ew p,, p.,, Q~ ..
(3.23)
(~.2~ib)
-zoals i 5 o.o.JACje <y:VeJta i"' fi 9 (3 .2)
H~ltl ka~ cle p (Al( \-(U( P. <Z:k (i) I (jg ~a kke~k ber-eket..elll V\.{et be-Lu.t.l r va"' (3.1~a.--t)e~.~. (?..lqa.-b) (IEI=o). he~cliekt ec~Jer~te Wettket-a dat
d.e oplo'5si t.t.co~ belwre:t.wl bj IGI ::.o IMeJ ~s~~ it-t ck fysisc~e711.1. Ë-r<s
datA r:;uu spr~ lfa.IA a~-t po1M.ro~e" ~wis~ (()t:Y" keude e=Jvet~. Het ~kuip Va.k t'-9 ("3. '2.) is liJd +e-ueu da.tvooro.\le (,.)Qa(~ Vo.~.t cf ~oU:
-4o-
-l'L )'/. -1" (al~- I'!.- I'~akd.e~-I"i ( L- I')
Het bek.u.lp v-a~~t (~.22 G- c.t) kau cA~ t ~c'5c.kreve~~t (LU" kt,.. als:
Al'b H2:. o krj~ek LUe als ~3e:L5ke <uoaràek 'fa.~;r O H. ( o ) 0 = _ a;r") ~ Ct~tt'\t;I"' V'')
H )o ; Do."' k'~'b~w ~Ut!Oeer (~.zs') ol-C~.2f.)
Da.ar !.O<l:- lè-1.4 p\ a.5 Vlta ~~c~ow . .uetc (a)C{ar ~e ~~tat u..u.r lj \ce tnz=-k~he-~ ~ ktz:ljoJ..ie~ i ~a~i k.ait· tr~kl.öe!A.+ïedttel ~ebbe~.t (~eéde~~e: Z)ol~rew) ka~-t c;{ lLS r vo\~e~.cs. ("'!>. 'Zb)~ (3. z1) I.{ ooit ~,a+erot tjCSk a.aL.c \A..ul !Uord<èt... Dtê tl.Vtt I.Vi '5~el IOer~kdt- ~lVt1.4 -zsllf d.u~ bij~ veso!Aa.~ot+ie \!ocrLUaarde
k'::: k + k'' i 'rf.1= 11 + 'f a\~0cA ~e~~KA-pt.
Voorde rc-::oKa~the VocYtOc:tarde- k.:: k' +k'', ~(.= \f'*'t~ 11 i".:> h.e·huzl ~~e- ~l cAaJ Ó'" Brote.v-o}-~k Qa~.t ~.utlwordt. tJ~ ekt~ fe-ke~.t~IA ka.IA ~1.1\/Ü~d.e~,o (k..b.lf{3.1'2Gt-b'J,{~.I'tc.t-b)eM(3.'Z3))
da..+ a.\"ö [ / o , Ja+ clc:nt M.toef ~e-Jàe~.t
rf,2> r (I.'t:ru)"2+(R.'t12h)"t1 I ' " ~ r --er, ~r"), 7
be Va-<jdjki k~c~:~ó") ~co» cltt~ de- cl.re\1\tpelll.:)a.arck voor)ËI orc.la.t er lk~to.b",l;teit k(1.1,4 o~red.E>M
{?.:zO')
De re-ël<e cli':>~ckYoMAt\te Voordf: ~h.w.rL5~ tre-kwet,d\e.') LAX>ro+ r~lf!F"fök door(1.4B). Ao.111 cle lrcnt\11 t.tierval-t i~ t~Li~ do.+ l.<ooi+ ().a.k bel '-*e rtZ:'5Qka&.c:i-fc
VOCK<.Uaurtie~A k= k'f-k"eiA 12e V~ f2~ \V'tl?.e\J'' f~~k te\fd~ i~. l~Ve\Ac.o Voor k'=- kt-k'', ~ \{:: l2e \Vtl?e '\V''). E=ri~duS:> a.lhjJ. <teM aa~~~Lu:j~to~tt.
4J' ------------- -· -·---- -------·-·- ·."f / /·
0-'\oll / u" --·-----·-·--·---------- -;;;/- ·-,.. ~
w' _ c.u'' _ ~ _ t - - - - - :-;:::::~
k' k" k Yk"l \- t •
{?.'2':}J
b({; rec'k.te \91-1 ~oor Je cot'5'fro~<O€~.c kef p1.4.~tc7.k) i~ Jï~(?.-3) 9<U>~ lt<ëtverba.~ + us~L4 w' ek k' <é~ tiA~~A w'' e~.c k''. w' e~~t cu" -zu.llec.t óH~sda.l- ~J.Je'k, k'<é~-« k" ?eev-kle.il.1 kie~l-4 d.ickt b~ ~ 'Ve~ h 1\y''' li~9e-k. I~~t fi ~('~:?.) is&e kro~VIti1<9 St(l;ric over drevet.. UJecYo/èOfveiA.
Ik de u- pctro.~ro..a.t tuordJa.l-:, a.rSpro.a.k k';> k11 tQdtoH-löt.]..., par-~~ra.ct~3.1 wa.s a.rco(l;l~d dat de oplos~ 1.1-t:~~ voor .x e~~t! liB~~ op~e kyperbool ?Jfl.?Jr;;vev. door (~.(?J) vol5e1.ts +18(~·2). :Deu ky~erbool kak 0k"5a.ld oepaa.ld <.Oord~ J.aa.r O.e a~y~ptokk e:t-tcie- pu.k~ ~. ~ g~, ~J~~-r [ I:>e p ~.t.kk1.1 P, (l;l.t c; '5k IA.t'l+u..urSk C3ec~A ~ckte oplc<i.s.i~<r.t.c (()a.Jtt lt=l is dd~.t~).
t-t~+ beku.lp Vat.t (~.13 a__ b), {3. ~J) ek (1. ''1 -lt 0') VIIA.d.tt.t we Voorket-pu.nt~ :
[
I '"'" \~ w" tt '{;;- u t{ttl k u v-r, = _ =- ~~- _ ~ __ w ~ -v:::--a 86 k11 3 86 8ó
..t n '' "'- n- '1. 11 '' u v-.p, ~ - ~ ;;;: lU:: 'f -<.u
(3.3oo.)
(33ob)
v r.~, = - t ·~- ,rrr w•lf ~- \/rf w' ~- (.f k" 4 -e. r-ar k' ~ 'u as (3.'?.1 a.)
o< 2, = -1<12"'1Ze'o/
1-W1
Het b~ktJp Ya.l-1 f i~(~ .3) ka.l.4 M(C.I.-t ?.i elA cla.f D( p, ) o ~ 0< S2r C:. o
o~ t!aJ k".( k I i "I> OP•? f'i•. Zo ku~.-t~-te:f;f cue de;; k.yperbod ':JefY-VCA door c~ .~ J ) te k.~~el.1
B6 tO<z:;~tJlMektÀtt o.~plt h.c.d~ Vll.~~t o.~ po~p~olt ~i3l-k~H·r~ekt Vak cA~ Opl o'5~l~.t5c.1.< VOóY D< t.~ r op ole àik aa~.t ~~tZV~ '5t1A. kket,. Va !.i &e k.yperboc\ iKtiB(~.Jt). NctarMa.t~ tie oplossi~otBc~;c dlckterb9 Proi-9!ko~<t~-tte liWk L woJ ov~r~ ko~.«.f Met ShutdS klelkeY tUoró.e&uJ.e \~l)krg ~k de q:>los~lt.tC()e"' Of ezt.t 9e~ev~ ~oKte:l.t t CjU.IA }ysiscke behdctk\S ~er. 'voo~ ket dik ljd·eketA.dt: <J-ttk li~~tks Vak _12.1 l\lt +i~(~-~) tjddJ:
f- l' .q. L- a;r')
l CJ<. L- R'
Het Lvw ~'&I hit~''), f2.e ~~ == w' + tt ~ lle ~".:: l.U11
+(1( kr·s~ Wt ~•teH:leku.lpva."' l~-30 (l._b) c.lar VOOY f2€J(1 lliA I.._. ".f
1 O~~eV<tet" <j~ldf:
_ lftt w' 1_ I'M ~·1.. _ '(!; t w' +w11)
Vif 8ó 2
Re ~ 1 .( Tl~ 1.y 1
{
- [I~+I'') L. '0 1. _r''
(I( /-R"
Voor In.-r Jt1 (:: L~ot-t ~")I((~~' ~k ~~~~ Be!d.t actl-1 o~~eveer:
_ '(.;; {_ w'+fAl) ~ Iht ~· (_\/ft' w .. 86 2 '~
Re~~ / w'- w'' + Re"f"
(3.'33a.)
R<Z ~I I f"l.e ~11 , r~ ~~ ek rll\t ~·· ~(s T (,tj,t ktie:~ VAA \Ê:.\ ka-bbtët..t Bra.ti':lc.h de- vo l~eVOV'lM
~~-.:=-:-·- ------ -·-·------·----
,, k"V 1-·---·--- ·-----·-·-· -·-·-·-·---·---./""._......-=:..~----·---·-·-----------
~J'~w'+~"t' lleifll L._ ____________ -====+-...,IEI
'Z-VJf;cJ' r-·-----·-· -·---·-·---·-·- ---86
~-~ V~'(r,.J'f-w")r-·-·----- -·-- --- --- ---·-·rrw~'=Irkl( 11 -:;:-1]:~>'~-·- ----------
8f L------------------------==~1&1
T?.t7lf I
--~--=-=--=-=-~----~-=-=-=-=· =--=---=·---
!Ze-I(''
~ ~" '-=-==,.,-::::. =-=-:-:. =--:-:=--:-=:-::. -=· =-::--.:-~-=--=--=-=-~-=-=-=--
"·~I I
~Vjiw" ·- -----~-ll[;:(tJtw') ·-·-·-·-·-·-·-· _____________ r~~';r~~,,
2 Bó
~-'({i-~' ·-·-· -· -·-·-----------------·---L-----------------------------~==.1~/
BS de~ re~~~~hevoor(J.)a.af0.~ i5kt I.Ue! ~~elijk O.a.t- de it\t~i~a..lre rr~bo'lkti~dfl:lttk V~ cÁe Wrz..rkd~kttrekwe~Anecs lJYOtUOt ~'ijle Cla\of l.uJ IU:)fde~. 1' wu ~1/a.\l e-111 <OOr de~ aad.er be-kd,ea.c
,e: D<e rela~le tc.t~~l4 k '( otk') e.~-t k (ll.t de a~f>li+l.t.dt veuc de- pc~ 10 ~Ij- l~J 1
wo.ar-b8 '(Lo (:sta.'bllikit ff :: o [HA-av~i kGtle ikc;Jc1bilcldt) r )o I ~t~abi 1, teit
z~ De rela.+ie hts~ <X~ tËI e~ ttt";>~~.t / ew l'ê-1 waarbij k'= k".
De -s-tavftrdcc.u~ti05 w'e~~t wv ~ de CJdr3d-allek k' et.t k"C~.::Crdek o.k vc~+ ~e:<&:-\c.rev~ ·.
l tJ = cue v2 1" (f")
k' -=- k ( 1/z r rr)
r W11 c (.o ( y# -er)
t k11 -= k {Yz -(/"")
1-tierüt ~yddt &o..t -'1-z c( <t'L Y"l. . ( IUela.t~ t:k a.~rra.ak k '.> k''cuU:rvavallek)
lie+ ~lt:'v~(t.tq) fèl{l.(3-Ta-b) kY~~ew 10e uoor e'coé 1~:
tl~J b0ktc.lp lfa.~~t {~. 1>1} ku.~~tkeu v.:>co f 3(l~l/~ Gioor (~.2.:~) berekat {.<.)a.ar b~ C.Ue Voor é~) ~ ~c!derl~ ~~c;,tev~ door (2.48) ~e'bn.U\cek.
D;t- 0(C,0~:
~7.-= _ \ E('z)r .'t:'~ ~ lbC/4-<T')'ë.''t Et..l~W
( §. = E. ~ -zie z:a'1)
(~·30)
He-t bekttip va.~.t (I."']- 4U),(~.2.~) C~-7 Q._ J.) e~-t ('3. ~6ct-b) \cu.k~e~.c ook 'R.\12"
!' e\llL" berek~"'~ Lwrdc111. Dit ~ee*:
_4b_
Voor-~== o (~arBi H.ale it.tc;,to.bi l;teif) kau ~tkl.utlp v-o~ C3. 28) (~. '39a.-b) ek
{3. ~o a_ b) ~Vo1-1deu {A)oroe~ hd ~oefBelcl.eu:
lb k'2.('14 -~.,_)lêl'l.=_!!_ k~ (';"-cr"1.)t~kf{ 1/~t-lf't)~ {3.'11) 8 6 l.t
Ee~~t oplossi ~~~~va~;~ (~.35) is (Y~t -u-Y~ t~;d.~ <J'=! Y-z. bi+-zjn .k ljke~.t k~ k (l;k k'=o o} k'': o /ZI!I k'= k. NeJols cle oplossil43 CU.:oo -z:jt.tdd·~eu eekte Of>lo~~Îkf3t:.l.4· Ë"r iS cktw ~eeLt sprGtkc Irak UI-< ra~p~clt~ teeu oj-tc...::;u (.ÜI''SSel <U er lc.el.. de ~ o\ v ew .
be lu-tckre oplos~l~ vol'Jr c.Ut:
~kb ( y"- !J"t)' = I bI ËI2-J!:. k.,_ ~ 8ó
E:r -z.s~ oplo$j l<ijö voor !f al~ ~ldt:
lb 121'-~ k 'l / b dó -
De o plo 5~1 1.1.5 Voor <r i<ö ck.u .. :
o-:,'1 = :. V v~ _..!:_V lb lËI' _ k :!!:·' 3k~ tló
be> oplo!>~i ~ ~e~ve"' doe4' (3.44) el.t de S WeM k-'.::. c:., k'= k 6akeke1( eeL-t
driekoe;k; B làebieJ a.t (7.ie fi3(~.6)) Voor(A)a.at'cÁ~ vaw k~ k' -zoJaL-ti3 dattUe i~ ~~'Jeo.rwerc::le '3ebil2'ol Vttl!t ti 5C~. 6) bevi~d.et.t Is à' )o. !>~C +(A)(l;e" CVi"S'Se\ k)erkei.t.de- <Jolvew ~Lt G(aw
t(.U(m; aa.~A. ~ro~ teuk ~vet...
k' ... k
r
~ ~ (3. "3tt) lrol~} llt.~~t ~oort ab~kt~Có d.re~M. pel1.0a.ard.e- voor I~\ Opda.f- er
i~ sta blli tei ~ ka~-t optreelew. Voor I ê / ~oe.t G!cAJ.< ~el~·.
Wc ki{.V\.VI.{l\4 ook ck s;tuaht kbj~ dGl~ k' ... k'' (er.::: o). ba~ ie::. R'-= R".::: Ra.~;~ r'== I''=r Voor H, e~A ~1: (CQ~tleveM door(~. I'Za- b)) kr~<i}~ toe 1,4.(.e~bélw.\plr a."' {_3. 3Ö)01A (~.'5q ct-h):
2oc:tl<s op bb. 3b isaaklj~~ l.tak<jt kefsc:orl op!ossi~5~L4 dct.-1-wevoor
p( illll. f krj~e.L< a.t \(Ok ket ~eket-t Va\4 i-1, Als H, ~ o, d.u.s a.\s
"= _ (L'rr") =-I~ _J.. \f"ill k a ~ 2 Y96
\r,2=- CI'~I'')-!.W. ~ _yn 'k ~v'-tiEI-z._.1_-k6' o, 2 u ~
(~Soa)
E "'- vo or- f>( t:Ae o rlo~i L-t~r
(~.s-ob)
1 k t( 'j (3-f) i s_t (i)eer "JeCQc..IJe.tA als +u.~~t.kti o Vd.'"" k <2.1.-t voor- vu~c~l\et(d<Z" waa.r k~~t VCl.lit IE /.
~ 'o
I j
-+ I J_ I !
~ :g >< ~ r+-" 'dl ' f~i
--- --+ ! !
_ _L __ ·--I
i !-
t--_ ... __ _._ ,. -
3.3 De 'Sif.~.tati(& da.+ o.~ po~p~olt~"' ié~.c wis~~we:r~~f)olt hJ<l0 ~e~ml-1~~ol Yet.t 1:..SIA et.t dJz a~Adcte- Wi ssd IOa-kekde~cl\- (l.IZ:kiOk.tek30itis.
Daar- ~<t: \a1Ada.u._ clcw.pikS Val.( ttl-t <2.\e\d-rokel.t ~olt Voor kle~e k (k~C1) Zeevk\ei!AiS,be~~to.d.ert dil-d.e csi+Lta.ti(l"Vttl.( ~~ ctt~Ld~pt0 po~p[)o\t Vla.hw.rl5k \tflel bekt- da~-t d~ Wis<::.~ ~ki t-t3 +us~ d.vie iOIA.eu~vew. ]\-I Jeu situ.a.ft<r is ktz.+ Lu~ I ~o8ll"lt:;k da.t "ZoûJel d.e ~olr9era.llet.t dis. J~ re.ile dtz::leL-t \rOk cÁe Vla.hw.r(jk.t t~kfu~ti~ dcsct\ct Ik re~oka~-t.he '"ljl--1. be potM.~,~o\tLVordf tueer 5rzb~er-k* door- ke+ 50~~~tal k ~de trektûet.tfie ~~ de tuisse\(,.)C_r-bk e\ek+ro~ett~olj-door k'e1-1~ 1 e~., de iovtet.t 5o~ d.cor k"ek ~~~ be :5to.dtreb.vet.tties W 1 ek w'' t.uorck~~t -zoob~.ti B $je ko7e:IA dat
t</:;; w'-~ (alsk:::k'+k") w":; ~-w'["-ls k:k'+k'')
Hoe~AJel er ve.rsc~i.II<Z~~~de ko~ b; l-ta+ia:s ~~e~k --z.S\.1 be~~ lû<t ot.tç
tot k, ~ eu w'' a.lledrie po<:.itie+
Om hei\- r~k<z:~Ae.IA 5eK<ta.kk~t.ske~+~~~ wszi9ek (AJe Vercve'5'kik~ ( t. 4 7) doe>r te s+ellet.t ·.
ba.af" 11-)v àe 3olt 3eta\\et.. CeeY k\et!.. kie"U1-t {k, k\ k''~.(1') Ztc.\\e1.. cl.c
vUt ko ~ ~+e~~t vev-k~ec;,ew ~+ k~l F lfa.l.f (?.. 52.) "l(ley ~i ki~ Y/l-rçc.k"l U e-t..
~t cke, die nte-1- bdtulp Va.JA(I.I.fr) (Oordek V~rkr~CJew. Lt;+ (5. S-z) vol~t dc::d- TZe ~11 : k".
_ót_
Voor de ~~k\-rok~ t}o\Ve!A Is diG ~at:J.er\~ Voor l2e'\( ~e:~e-IJ~ZM door(l-~J) ~oed., d.a.at" k "ta;lr k\eiKi s
~ =(l t; k'l)'{f
~\V':( lt ~k't)Vf 2
(l.'if-)
Voor Re Ay'11
( = Re \y'- ~ c.tls k':; k ~k'' j ~ ~- rze "V 'al':~ k:- k 't k" ) 'fkod-dctli
~a: I de;t.t
tle\{1 11 = ;V"fck'1.-k"t) (alsk'=ktk'')
lte 'o/ 11 .. ~ Vf ( k1
_ k'7) {o.l~ k:: lc '~k')
2 {3. S""3b)
J~ bei d.<Z" Bev~llek ki"~ ~fèk 1.0 e, a.ls de ~df~t~llet.t et.< ck r~0le dele~.-t va.~ ~ 1tt cc.+ Lu,.r-lb ke fre k<.Vck+ics eksc:tlc+ 1 w resc~a.k 1-it: 'lS"', de ~~elget.tolerela.-1-ic tu.~~et-t k et.t k' ( l?(i "V''." k"~ L 'k'_ k) o\- ( k-\c' ))j
A 1~ lDfl.- 01":> berer~L1 totd-at ~cbicd IDoafb~ "Zoi.Uel cle ,. Mi~VKa!-c~ o..l.s k'l lltiet a.l te Bra::A-zjvt krEBet;t IUe- tocM.fYVtt:rket~bied cla+
OV\.tf{et<.SJ word.t &oor O.e s+i ppel \~t.r 'uo( tï3(3. 8)
Ie'
/
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//~k=k' /
// fi3(~-~)
"k+k'·3~
2. k: 3'fr
IVt ~ehite<:! 1 isk'/ k. bi~ ~ebled I.UorJJ b~cl.d0Vtlk c.loorde rtt!)ohGKhe_
VOOYI.UMI'"d~ k':: k-t" k-'1 j lz 1 :: ~ t ~'1 • ""J.-.. ~~bi~ol2 ÎS k-'.t k e1.1 IUorc!f· ~<t$ck~Vek JGbr de ~oltt.cuA·Ievoo~ï.va.cA.rck k = k' +-k" i ~= ~,,_ + ~11 • ba.a.-- kLt 1. i.s r~ c<.')(k', tAJ') Zll~rk\eil1 {-z.itt l.s-6'). t>~ctro~ kcu .. ~n.c.'i>~
(./
'j<Z'f>~eld ~rlÁe-IA da.t r~ EC')(k'. '->').::o ér~~CI)(_k~'-l') f.L. r~e-'{k': w'')). 0Md.a.t b~ e:k5a.kte r~oH.a.t.t he de rcz:ë\e dele\.t lra.M àe kai-w.c.rlb~ tr<l.b.u~htts OjC s·k ~~ cta~.- .de ';:>tartfre: kc.ue~.thes {:z\e(}:ól'J) Is R~e")Ck~w'):. R:e ~c•)Ck''.w")~o. 7.o kr~9e.Lt t0e d-(f; 1td3eH.d.e l.ûa.a.Y"d.et.c voor12',R",L'aci";
R' =o
I r =0
c 1 " 4 ~ ~ c. w E 1.û = - 1::: -
c.:>'w"""?. w"!J.V{; (3.~6)
[ H iefb9 i 51A:l<ler"~~brc.U k ~~a.a.kt Irak ke~ }ei f dat voor kie-ir.-te k' w'~ 1/i"') A Is (A)e cJ.e 'bek<:t.o\erl ~voor tr."l) 9egev~ cJ.oar (2.s-1) ~brui'ke~-t 1.11~-td.et-t W<t \foor 1"7 (3.'2 ~)
Voor d..ro u,ltc:!r~.t.kki ~CJe~< -H,. Hz,~~ H.c ~lt~eVe~.t doorC~.I? o.- b)e~-t {~-IO'o.-6)
Vi Lil~ we ~~+ b~kttl F lfaw (~.~~a._ at) ~114 (~. fil}
H I=~ k111 +- Lf k'' V? IËI"'l 3lf
k"). = o
_53_
Op dt- ISk k tk'= 2/3Vf' ~aJ doov- '3ebied zloopt(t-1~ '3.8). da+ d«!:. ~5~kre.v~ oloorot r~So~o.•·tt·ievoovï.ua.ar~e k= k'tlc"; ~=~,". +Jt 11
, v\H-olel-l
~tbelw.lp v~ (?.. lbct-b) de opl05sik'j~IA vcor- ~ ea.. y. Voord' kr~ty:~.t 10e oa~e[H-t ç.o):
Ö'' 2 = _ _!_IJL k" ± y (..!.'{!; k .. ? +- 4k''V7U~r~· (s.6oo) I 2'86 2 8ó
()( ;:; 0 (~-bob)
be reë\<2- à~\efll vo.rA O.e <.uerkeSk~trektUeM.tles. bl5veM..c4t.~ tjtSka.a.~,., Je: r<ee le à<Z:Ier.c va."' de ~~tctl·(..t.UX ~k trek~t.ttieS>. Voor k''=o z.ic~.t we !.Ut c~.6aa.) do.t o=-o. Voor·h:~.ke~c.le t.Uaar-k va.~.~ k- 11 e~Jdt Op ~k ?éker ~o~t.Jdctf:
(-1 ~ kil)'// 4 k11 fil El'l 2 Bó
De ver~e ls\.c~ ~ (~boet) ka.l-4 dak 'Je5c\.trelftk worcl~ a.ls·.
'v, 2 ~ ,....}_ l{f; k'' ±..!. y tt, ku { 1 -r_l ~ k"VftËI') 0 I 2 8ó 2 8ó :z. (...!..\fif' k") 'Z.
2 '8! be Ot=>lo~'î>\ "tl()tiA Voor-r 3a..a.&il ~ VOO'r Vo\~oet.to ~ro~ k" ~a.c:tt·:
().6~b)
U.i~ (?..boa)!~ktkef also~er- Vooriedcrelöaardevaj( IËÏ cr~~.t \<'osdteve
orlo-s.sj k~ voor~ is, d.tt.-:. d..a.+ ik lt)i:»<ZitOtrkettdt ~ol\1~ cta.k ~roeie:kd '"Z91.i. \NlP kebbet.t e~ d~ kr.t.(.pi ~sta~ Vaw k elektroh.tfAfld4 VeriUaarlco~, Aaav- d.e:U? ka:~t e'r9 kle; kis . ] 11. te-ik di"'"lt ~ d~T4" terM.« ik t0t ~e-kot'«f'liaertk
IMl Glrl.\.k'KI~t~ i 1t ke-f- re-c\J-crlid V au {~. 6oa..) te I.Oordat op3(2'1(oWKeu etc. we-l ~<Z"t
<2~1.1 Wc~tittr t~"', 'tod.a~ 161 ~ ~i ~ti Kc.(.UA.t Waa\"de ~ef. ~bb~ opda.t
'(/oi'5.
Op de Lij 11t kt k '= 2/3 \J? d.a.l d.oor'àebt'eà tloopt (ti~~ .8), J.at du.~besckV"e-llf--t.t (OorcH Joor de- re~l-faü+icuoorl.c)o.ark k'~ ktk"j '! 1
::; l(+~u. Vi~cl~ LU<e ~el
klw.lrvo.IA (3.2o ({_ b) (l;~~t (~.21 o._b) ole oplo~i K~eLc voor- ~G ~ (-
Al~~. <. o, du.-s.als (-ziq 3.5qa.}.
r = -fVl; k''
~ltt rtte+ be~lpvcu. (~."Zo b) Voorc< :
r·-t = -~ 1lrrk" tV{~ g.k 11)
1- 4k"'/?1Erz'
l 2 '86 2 16 L3.6{a.)
e~ ~ [~. 21 b) voor o(.
(~brb)
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Ü!A.d.<lY cÄ.b voorwa.ark (s.61) Ba.aLt de- optos~ik':FVtVOO'r(~e-'tJeUeiA ~ ("?..bra.) ko..av-:
'-'· ';;.:_ -'{i;. k .. +~à '{f I ë I'L ~- 'ffi k" I ~ó rr 86
(' '>\:. -4 óff. IBI't
Ot.td.er- voorc.vaardt (:?.6~b)-ziew <.Ue d.aJ ck re0k<k\et. va.~-1 de~~ tra: kli>~~es flit et 3eS k ~l.t a.at.t d.e r<è0le dtlet4 vat.t d.e Lta.tu.w-Skte trek~~~ifl:s.
\-tehsooki~~e:re~-x.uA o"" a.~ verk0ltdi~.t3V<U-t ó..~ au.trl;tt.tdet..c Yo.k clt
t~.V<tc wi§ei vJ~kel-\ck gol~.rev. IE='I ett/t11}fe- ~~!.-'VOO"" d(~ k" liJaarvoor
(3. G•) ~d.t 1 cAu.s uJoaV' O ~kaJ(l;r+ kaLt!.û:>r~doov- (~.b3a._b) e« {'3. (,~fQ_ b)
tlerkk.u.IP Ya."' (2. ~oa.) o} tz ~o..t) I (1A8) cu (.1. ~6) b Gle:w 'V0r'ko~.tdit.t~ bz:rckcL-td lûorc.le:Vt.
Voor de:u- V<Z:rko~.tdik3Vi~dttti.Utals kfZ'f iK.ta~il'lctirfv-tkw0t~i~O.eel 'ó~'ó~ve~ LoorcH door(~ bs 't) ~"' c~- 6~a..)
IE''/ I \f'(Tl k"' ~ = 2-Ó ru 1ël
'voor k\ei ~ IE/ {A)oroH I 1:-'l kl<éi 1.1 i 1.-t ver kot.«:!i ~tot IE-"}. Ook Ltee~.ttt ( 6'1 (.(+ tel!\ o~ -z.ic\.Je Vo.~-t li=''l a.lcs k11 toetMeelAAf.
Al'b ~t- ~~~il.1air~ tr~k~{it'lkel6 ~ev(/AA l()orJJ O.oor (~ 6'!>b)o.(."3.68b)
L Ol!tder-<k VOOt'VJaa.r ck k"' k't k"; ~= Lt 1 +-it'' 3ceeff (~. 6~\> )clt.W. aawleid i IAcj tot
hve-0 a.a."'-Broda..de8olv<êk) V'ih.J.~ ~voorde-ze. ver-1-totA.diu'j:
Voor~ k\ei~Wo.ctrde1.1 va~A-IËI i~O.elA.~pl,tt.tde\ra.tA oleio~el.t8olt k\e.iM tct..~ op-zlclic(!Q~..t ck aw.t~\,"tuck, lfo.~.t ck lA.>ISselt.oer~"'de ~ldt+ro~ ~l+ . .De '5·rJ.~Aahotij'kt tktk ~IJJcx;,.. ep-hu«: clekfv-ol.tcu~o\veu Xökd.ercce~.<~ io~~-tcoolt eiA duc; -zok.da eltc~e M<ukbcue. wisSelt.Ver-kin~. Opva.ll~J ,·s
cl.a.\:: deu lrer bot.tdi lACj tO<Z..~(lewt t,taor M.a..-1-e ka ~olh~etttl k" ~ rof-er fJ.) o n::lr.
_5"'b_
Ik f•tj{~.q) is r· -zoa-15 O.au ti)ov-O.tslé~Ve... Joor(~6~6), ~lstu~-tk+ie Vaw
k'' ~r3e:Beve~ l(oor versckill(l~.-td.e c.ucu.tr~ va.~.c '~\. Jlit }i~ L~-l.o) k~t-relrd<L'" ~a.avda.~~t voor'(Zoals J.e>u worcM- ~e?J~"fl:J/t d.oor
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~~ b~~koLLIUIZLo <tll~ d.e feX>ka.rA.tie voortuaàfJe> k::k't-k'', ~ =lf 1 ~~"t"('~ Voor
ck tl~-t de-re ve~ ~.ta.~otli~ voorc.oa.arcl.e ~aa..+ ke-t o~ de:zeij-de w.ta..Uer- ook.
]Lt pc;ti'CA.wCÀ.af~.1 curu. ~e-Vokdet.t d.a.~ d.h H-r c~- 12b) fO i~ de> oplos~i l<.'je-t.c
VOOY f <.ie Vol~[..t~ VorH4 hflbhz.~t:
Yt,l.: - (['+I") :V:_ H, + .!..VH'1 t't H~·' 0 2 l 2
(i..( i er( k v.:>orde"' H, e~.e 1-11 ~~vw. doot (~.1'2 a- b) ). 1ief I 1: o (~o ~e:<:J-elà) J2~
RI= 0 L door d.e btA.~ va~ Je s~arttrekwcl'k +ie cv') Ba..a.t {~.l'to..) 0\"t:t i~:
(=3.1~a)
ka~ de ter~ d;ld.er ~ef r<Z"cktse Lûor~elte-kc~.t klila.c!erJ euorJ(Ck ~l- be-blp v-at..t
J.c reeksot.t~wikkeli t.-t3 ( 1 t -x./'-= 1 + "'1../2 _ -x,/8+ ... De Voor-Waarde&,. r-z) is Beldi3 Gtls k" vo\d.oekcle coroohs (l~ ~roof..)ot a.ls. c1e .. w.t.i~kt,rkl/ volcloekdv t~{od is(~" Broot).
voor r kr~~ek wc d.a"" Je bekaJenf.{~r
D~d~ d.e-ulf de Voorwaarde- C~. r2) ka~.t {r. -13) {()<t~r- be~aaercl cua--detc.
Di~ Bee~(_ we. l-t.e.~[..t all uk de pef:>itievc oploss; k3 lfG't\.( r):
l""l 'y "- f' (~l Lt) 0 "" (t'l ~ ~11'1)
Het be ku.lt:' Va.~t ('3 .52) vî ~a~ we Vodf 12."
~· = ~'fo (ktk') k" 2.
(3-17-)
w" k•' Door ts ·1 ~)k sieJie"' kr~ Gt~Z'-' we VOor _ ek _ J k" IA>''
I =-
(lt-~)
- '2. I 1:: I (S. à'o)
]k fi9 {1-12)"1-s r/ (~ dVó'ht IEI"l u.itl3~a1sf-u.kktte Vo."' 2.1~'16 r. Dit ko~t OVf:YfWII 141.~~~ sd-uah~ cJ.a.f kva.sthjtewk' e~vav\~crol tUcrdJ. H~t I'S clw a~~ k tia+ b~ ~kctktc rük V'CSOkcuilc (vul ~rotu l'S cAa"' bS k(eJ e-kakta- rr.:owaw+i~. be t10a; wisSel lûer~dc ~o\vet. 9roei1Zw olu.s.
~~ <i;hzr-k~~a.a~-e. al~ kuL-t V'a.ele cki~L4 va"' kttw ~a.hu.ur~ ke>fra-kwekh<t.s. ~k-sakl- ht v-esol(akfie -rsl-t. l Llit3- <fo b~H da.l-- (r 1-tl(l'f--zst..t l'K0.X1H.W~ ber-eik~ Voof<l ;O, ~a.cwvooy- 0~ ~lbÓ.
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Er i~ ook u~ proB ro.~~a ljt: ~aa.kt o t1.i cJ.e ko~ ~ex~ LUevk.t,LSI(e, t re hue.~A. he~ t~ ber~k~~. {.4ieri JA Ï5 u.lt9eBaa~ vaw O.e- be-Mdàcr-i kf)S [ov-nw.l~ VOOY" Ec')(k,<c.J)
0Jtcyvew t!(X)r( 1.1-tb) ttw (I Stt)e~A va~.t tk bewak-ri~c;tov-KAult>b voor ~c~)
'3fL-r:JLVetA door {2.1-t'K)e~~~ [1.5o). FteJ cld-· pro~ra.~VKa kunul.t de koVKplà:e c.v~'ij~ [rek{A)et.tties.bere.~~a
wor~w VOCJr i ule-re- ke~A k' t.ua.cH· voorele be~~taden~sforH.W.~~ ~dl~fCSl-1. DC2" resu\to.tet.1 -z01.1 i «A d;t ~oo~d.~·hc.k ~rcttisc~ U:>eer~fL-~VelA.
lt.t HJ pro~ra.ltkWI.ct
ka~ pro~ro.~ Wta lo~~ d<l" V~r~e(Ajki~ek lfOOr ll( e~ ~ Df>"loa.k. ck:--ra-lk rara.~raa.r "3 .1 ~e 3e\reV1 '!.~\ft) wa.arw~ ket reë-le I! IA. ~f i ~it.w.ire deel Va~~ WeKiceLsktë trek(.U(A{tit2:~ \rcJ.c~{ ~t reële deel Va!A. de LOer~k~ trekweM-ties U:Ordt 0oed bekahrd &oor b~ ckstor~trekiOOA.t(e.ç, &.e trekLUe~.-ttle lrerscb V( LtgeLt op te. kli<Z"t.t ~IA <:keu "Zo ~vooO.<Z"w f-rek<.ceuHes ll.)or ckVt cAa.~~t C.OU::r Ot:>~~uc.uctls Star~fr~k~ tleb ~brt.t.i k~. Dc-zekorrekhe tûordJ ki ~t 'JWaalf1 voov- ke+ i ~a~ i ~i re trebJwh c deel '"Cc;&a.t
kt resu.l ~cwt o.\lUM ~t-t verb~+tZril..'J is als (LL oe. Door ck IIJGtarJ(è TT' YP { = 1 of o) it-tte voevew W'eke"'t ke-f proeraiMM-ta cle
koiM.ple.Xe t.ûeYbS6: trek<.U<tM.tktS als r).e drie ~olvew drie io~.te~.tgolver...r
-zy~ ot +cve:o dckfvo~5olv~ ttt.t eet~ io~~dt-De resulh:ttet-t <j~ ctif-aeds ber~t Voot- ar90L4 {O; 10" x 10'~~
LI~ 0005 '8EGJ!'-J' 0010 'r<EAL' ê\(v;l-'. PI, UEL, DEX.~NU.; DE Tl, DET2" ~RT; 0015 'r<E:AI-'~-<••<'1 ,I<O"X"XI .~XOJ cr120 'REAL'Al"A2"A3"A4,A5"A6,~7"A8"A9J 00?-5 ':-<E.i\L 'f31 "B2,83"B4,85"B6"87 ,H8"89J 003U 'HEAL'Gl,G2"C3,C4"C5,C6"C7,C8,C9J 003S 'REAL'~,L,M,I,H,J,STAPJ
0040 ':-<E{\L ''r'.tHEl .tkE2, IMl, IM2:.P,Q; 004S 'lt'iTEGEk 'MAI\L, KEER.t VEEL" TYP J 0050 ':~EAL' 'Akkt'.\Y'GAM[l :4l"ALI'[l :4lJ OO~:ó 'kEt'.\L' 't'.\HKA Y '1'1'\X [ 1 :12] J 0060 'I.\1 fEGEk' r<, SJ 0065 '~EAL' PLIP,PLAPJ 0070 'PH0CEDUkE'rkEDCXl.tXO,Y>J 0075 'VALUE'Xt,XO,Y.J 0080 '!<EAL 'XI ,xo, Y J OOR5 'HEGI.\1' CXl90 Cl :=C~O*KO+l )/CKO*KO>- C2-NU)/CXO'!cXO>; 0.19S C3: =CA.!iSC2*DEX*XO/CKO*KÜ*KO> > >*SIG\lCXO> J 0 l no CS: =2* C 2-i'-JU > / C XO*XO*XO > .J 0105 'IF'TYI-'=1 'fHEN' 0110 •c-1EGI'-J' OllS C2:=CKl*K1+1)/CKl*Kl>-<2-NU>I<Xl*Xl)J 0120 C4:=CA~S<2•0EX*Xl/CKl*Kl*Kl>>>*SIG\lCXl>J 0121 C4:=ABSCC4>*SIG\jCKO>J 0125 C6:=2•C2-.\IU)/CXI*Xl*Xl )J 0130 C7:=4*{4-3*''JU>*CKIX+Kl/Xl+KO/X0)/CX*Xl*XO>J 0131 C7:=C7-4/CK*Kl*K0).J 0132 CK:=K/X*Cl/Xl+l/XO>+Kl/Xl*C2/Xl+l/XO>+KO/XO*Cl/Xl+2/XO>; 013S 'E:\10'; 0140 'IF' TYt-J:.O 'THE'\1' 0145 'REGI:'.J' 0150 C2:=-J*Kl*Kl+2*Xl/CSQRT<DEL>>J 015S C4:=0J 0160 C6:=2/CSJr;TCOEL> )J 0165 C7:=4/KCH 0170 'EI\ID'; 01 75 kU :=Cl *C2- CC3+Y*C5>*<C4+'r'*C6>; 01 RO 11Vi1 :=C2*<CJ+Y*C5 >+Cl *CC4+Y*C6> J 01K1 RE2:=AMP*AMI-'*CC7*C7>J 01 f~5 .1\1 :=G5*C6JA2:=C5*C2+C6*C1 JA3:=C3*C6+C4*C5J 0190 A4:=Cl*C2-G3*C4JA5:=C3*C2+C4*ClJ 0'205 kE2:=C7*ChAMP*AMPJIM2:=0J81 :=HE2.J 0'210 'E\lD 'FkEiJJ 0215 'Pk0CEDUkE'S0LVC0,L,M"KEER>.J 0220 'VALUE'0"L,M,~EERJ
022S 'REAL'0,L,MJ 0230 'DITr.:GI:.:K 'KEEkJ 0'235 'f3EGI\l' ~240 ~K8:KEEk:=KEER-lJ
0'245 Cl:=A2/Al;C2:=A3/AlJC3:=<A4-81)/AlJC4:=A5/AlJ 02~5BK:=CCC1*C2-2*C4>*CCl*C2-2*C4))/16J
0260 R9:=-Cl*Cl*0·25+C2*C2*0•25+C3J
_65'-
0265 ~LFCJJ:=-89/2+0.5*SQRTCB9*89+4*88)J a210 GAMCIJ:= B9/2+0.5•SQHTCB9*89+4*88)J 027~ 'If'~Lf(J l<O'THEN'ALJ'tt li•OJ 02HO 'If 'GC\M[l J <0 'THEN 'GC\MC ll :=OJ r:Y28~ ~LF' U J : =SQRTC ~LF Cl J HALF C2 J: =-~F' [I l J 0290 Gf.\M[l J :=SQkTCGAMC 1 J > J~C2l :=-GAM[ 1 lJ Oë9S ~LF'ClJ:=ALF[lJ•C1/2JALF'C2J:=ALFC2J-Cl/2J 0300 Gt\M[ 1 J :=GAMC 1 J -C2/2JGAM[2J :=GAMC2l-C2/2J 0305'IF'~ER>O'TH8~'
0310 '8EGI ~' 'If 'ARSCAL.F'C 1 J > <ABSCALF C2l > 'THEN' 031 ~ 'BEGIN '0 :=0+ALF'C llJL:=L+ALF'C ll J 0320 032~
0330
F'KEDC0~L~O>'END''ELSE' 'REGIN'0:=0+ALF'C2JJL:=L+ALfC2lJ fkE0C0~L~O)J
033~ ·~~u·;
0340 'GJT0 'BAkH.J 034~ • E.~D I ;
03j0 A1:=0iAë:=Li 03~S 'El\JO 'S0L V; 036lJ PI :=3.141 ~<J26~JDEL:=18*1834JDEX: =!::iOkTCPIIC8*DEU H 036~ :\JU:=t<EAI.HAi"'P:=kEADJ 0370 lYP: =r<EAUJH~ :=t<EADJ83: =READJ84:=READJH5:=READJR6: =kEAû.JR7: =r\[·''i);
0374 •\JLGkJNLCtONLGfU'JLCRJNLCRJ 037~ NLCHi·\JLCt<JNLCRJNLCt<JNLCrU 0376 H:=OJ 0380 SPACEC3>Ji-'r<INTTEXTC 'C 'AMPL·= '> '>JF'L0TC4~2~AMPH 038~ SPACEC2) ;PRINTTEXTC I ('NU= I) I) JF'L0TC2121NU>; 0390 NLCR.JNLCRJ\ILCRJ 039~ 'F0k'K:=82'STEP'R3'UNTIL'B4'00' 04JO 'RH"3I:--l' OLl_ll 'If 'TYi-'=0 'THE•\1' 'BEGIN' 040?. 8~:=8~-83.JB7:=B7-B3'END'J 040:, '~~J~ '"<1 : =RS 'STEP '86 'lj\JTIL '87 '00 I 0406 'HEGIN' 0407 11:=H+1; 0410 'If' TYP= 1 'THE\1' 'BEGIN' 041:::> X:=KIC.SOkTCK*K+l ))JXl :=Kl*XIKJ 0420 'E'JD I; 042:, • If 'TYP=O 'THE.\1' 'BEGIN I 0430 X:=3*K*K*SCJkTCDEU*O• ~J 043:, Xl:=3*Kl*Kl*SORTCDEL)*0•5; 0440 • E:\10 ' ; 044~ XO:=X1-XJK0:=1<1-KJ 04~0 'lf'Af3SCKl-K><O.OOOOOl 'THEN' 'BEGIN' 04~S SPACEC3HPklNTTEXTC I ( 'K = ') ') JF'L0TC212110 J 0460 SP~CEC3>JPkiNTTEXTC'<'••••••••••'>'>J'JLCRJ 046S SPACEC3HPI-<l>\JTTEXTC. ( 'Kl=.) I HF'L0TC2121Kl )J
0470 SPACE ( 3) J Pt-< I NTTEX'f ( • ( I 0 ••••••••• I ) I ) JNLCRJ 047~ SPl\CEC3> JPKI~TTEXT<. ( 'KO= I) I) IF'L0TC21210) J 04KO SPA CE ( 3) Ji-'fü\JTTEXTC I ( ••••••••••• I) I) JNLCRJNLCfO 04K~ ·\JLCtONLCkJNLCi·U 04'10 I Ei"JO' 'ELSE' I HEG IN. 049~ ft<EDCXl ,xo~o> JMAXUJ :=Xl JMAX[2J :=XOJ osoo S0LV<M~X( IJ 1Ml\,l([2) 1010) JAlF(3] :=ALf( ll JALF'C4l: =ALF'CëJ;
050~ MAXCIJ:=Xl+ALFC3JJMAXC2l:=XO~LFC3JJ
0~10 FKEDCMAX(lJ~MAXC2J,Q)J~LVCMAXCll~MAXC2J~O~J~)J
0~1~ MAXClJ:=AlJMAXC2J:=A2J 0620 FkEDCMAXClJ~MAXC2J,GAMtll>J 0625 MAXC5J:=RElJMAXC6J:=kE2JMAXC7J:=IMlJMAXC8J:=lM2J 0530 A9:=A~SCIM1-IM2)J 0535 rkEDCMAXC1J~MAX[2l~GAMC2l>JGAMC3J:=GAMClJJ 0540 'lr'A8SCIM1-IM2)<A9'THEN' 054~ 'BEGH-l' 0550 MAXC~J:=kElJMAXC6l:=~E2JMAXC7l:=IMlJMAXC8l:=IM2JGAMC3J:=CAMC2JJ 055~ '8\JD'; 0~60 MAXC3J:=X1+ALFC4JJMAXC4J:=XO+ALFC4JJ 0665 FkEOCMAXC3J~MAXC4J,O)JS0LVCMAXC3l~MAXC4J~O~l5>J 0570 MAXl3J:=A1JMAXC4J:=A2J (b1'::J FREDCtvlAX(3),MAXC4J~GAMCl J); O'::J80 MAXC9J:=RElJMAXClOJ:=RE2JMAXClll:=IMlJMAXC12J:=IM2J 0~85 A9:=ARSCIM1-IM2)J O'::J90 FkEDCMAXC3J,MAXC4l~GAMC2J>JGAMC4l:=GAMtllJ 059'::J 'IF'ABSCIMl-IM2><A9'THEN' 0600 'BEGIN' 0605 MAXC9J:=RElJMAXC10l:=kE2JMAXC11l:=IMlJMAXC12J:=IM2J 0610 GAM4J :=GAMC2J J 0615 'END'J 0620 'lr'ABSCK0-~1><10tC-5>'THEN'
0625 'REGIN'FKEDCXt~xo~O>JS0LVCXl,XO~O~O>J
0630 MAXClJ:=Xl+K*ALFClJJMAXC2J:=XO~LFCllJ 0635 I"'AXC3l :=Xl +k*ALFC2JJMAXC4l :=XO~LFC2l J
0640 GAMC3J:=GAMCllJGAMC4J:=GAMt2JJ 0645 FkEUCMAX[lJ,MAXC2l~GAMtll)J 0650 MAXC5J:=RE1JMAXC6J:=RE2JMAXC7J:=IM1JMAXC8J:=IM2J 065'::J FKEûC.VJAXC3J~i'•1AXC4J,GAM[2J) J 066fJ i"JAXC9J :=."(El JMAX[ lûl :=RE2JMAXU1l :=IMl JMAXC12J :=IM~J 0665 'E\JD'J 0670 'F0h~'t->:=l 'SIE;.>t1'U\JTIL'2•1'00' 067'J 'YEGIN' 06~0 YACEC2HPt<INTTEXTC 'C 'K = ') '> JrUHC6,2,K )J SPACEC2); 06K5 PrW..JTTEXTC'C'X ='>'>JFL0TC6~2,X>JSPI\CEC2H 0690 1-'r<I:\JTfEXT< 'C 'Dt:MP. =') 'HFL0TC6~2~ Gt\M[P+2l >JNLCRJ 069::> ::)i-'ACU2> H>rHN fTEXfC 'C 'Kl = '> '>JFUHC6~2,Kl >.J SPACEC2); 0700 Prü\JTTEX re' C 'Xl = '> ') JrL0TC6~2,MAXC2*P-1 J )J
0705 SIJACEC?->JFU.fl'C6,2,MAX[4>tcP+ll>J 0710 SIJ~CE<2>JrL~TC6.?..MAX[4*P+3l)JNLCRJ 071 ~ .SP~CEC2> JPkl\JfTEXTC 'C 'KO= '> '> JFL0TC6~2~KO> JSPACEC2> J
0720 1->kFHTEXfC 'C 'XO=' > '> JrL0TC6~2,MAXC2*Pl) J
072~ .SPACt:C2>JFL0TC6~2,MAXC4*P+2))J
07JO SPACI::C2> JFU1TC6,2~MAX[4*P+4l > JNLCRJ 073:, 'Ei'..JD' JNLCrUNLCKJ 0736 'IF 'Af.1SCH/6-ENTIEkCHI6> )<0.0001 'THEN' 'BEGIN' 0737 H: =CHNLCr<J:\JLCR J
073lj 1-'kl:"ll fTEX re' C' • o o • ••• o. • •• • o •• • • •••• • •••• ') ')J 0739 PklNTfEXTC 'C' o o o • • •. •. • o •• • • • • o • • o. •. • • • ') '>; 0740 !\ILCrH\ILCRJ.\JLCi~JNLCRJNLCRJNLCRJ 'END' J
0770 'END' J 'END' J 'END' J 'END' J 'END' J 'E)\10' J
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