DTIC ADA286943: Generation of Space Debris from Explosion of Spacecrafts' Fuel Tanks

75 Pages·4 MB·English

by  Defense Technical Information Center

K - -'%
A A12
t-
~if'~ FINAL REPOR
Project title:*
-
OENERATION, Qf'- SPACE DEB'RIS tfROM EXPOIN
OF SPACECR'AFTS' FUEL TANKS
]
Pigiicipal Iluvestigator:
ANickohty N.Sinmv. Prof, rS.Iii(Pa~..ah
I-
P
Investigators:
Vaicry F.'Nikitiiu. N.Sc. DubU OUiI~ is
-
Alexe'y D~islv.b.Scl~i. Fh.-h. )O 14-
Ir.iDa iitieuo.Ph.D. (Phy~kMat h.)
-t Dihi.Ph.D. (Py .Nlth.I
4
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'N "l - J - .. %i.. - - . .
Par 2.~ro~igti~i~fdetonladtio anid cleflagr-ation waves
Part3ii. odelof ynak Iadig. of the, Nwaillof sphlerid-al
or~yinrkl ~sel- for (liffereit scellleiO 'f-c omlbustion-processes
-3W kil 'tO vckt ofi~ woiali~ir i2
1~2. Lojadijiego~i ~doudatioumi 13.
3.3 . DV ii-tmic laigo ~ ~l i-mv frfeto n D-zn
fragfnenitato dthction o dya iiteriu pesue1
42. Chkt1i4tioii of fi nz~.<iilw
4A. T6gulii )raitv of thO 1)11)1il 21eu
Part 6. FrAgmients hcelerationi tby gacs clou(I propighting int6 v, cuum 2
5.1. Prbjk~i Statewliit alnd assiiiptiouis23
3.3. Obtaiijiig the~ appl)~xiti oinformula. 27
Part 3, 'test ca%s , i'esults and cliscussiionis. 29
0
Part 7. Concdlusionls 32
Referencdes 33
Figures :33
*~ d~ppeldix. r illistruatiOn O
2
i'~~ypeset by *-T.
W7A
"V-.: K •,,a ,, , ct.ood
the"p ossible sceiatio- cievelopment.o 'hei'il xplo.i'm, isi de a recssueltisns;gein .
f
abreakump 6f;the vessel *ere (0s:ribed11) in ieliinlinarIiport No. 1. It ,,as shownthere that t1.
type of rea]kup, thenumiber amd velitis ot fr-a enwnts are deterinined, by thi( rates.,r ene-v
release and wall pressure eolutioi.isidthe t lvessel. The results ofbreakup-- nass-and velocity, ,
distibutmons of fiaenei ts -would ldifer (nvatldye pening, CRltheCOiluotIn)oessiside
vessel-defiagration, ( tonation or de.flao lon.to, d tonatnio ralis"tipn ibT,).
STo,
obtaina d sed-fom sol.ution of theprol)lemli at 'a' i)( 'compared with the existing
expefiment WeII tto concentrate, out'effi)rts-i ,it solution. fu, simple ge.oietries: spherica".l
an+d-cylindricaL vessels. 'The.iniitial uloilitioiswillbe ailso introducecihi,,a foi.m'makingit- iossible
* -toobAtat.i ni ssolilnuit ins lo.,;smg .:sphe~iic oi! 'ic syiiiii0ti9,,.
esuitTs eo f deflagt'iaon to-denttition-transition hivestigations 1) show hat the process
,* is-c6fniplidated and, tie highest pressures and inteiisitie: of-detonation waves are reached int,h e
transition e,T his tone -is charauerizc, by -tile existece of ,overdriie m, detonation waves that
sl6WdW g .attuo,a syel f-s ustaznmig:regime and the itensitv of- the-waves decrease to that of - .
Iii case-of central ignition of sphei'i,.al r ,'yiindii'd voliin. of (C)oiibustible gas there may
occur different regimes that can-produce diffremt ,,all.loadinigs after the reflection. If the D't
pfocess does not take phce inside the vessel th 1res siure'riz, and thie wall loading is monotonous
aiid it is detrminelby, the ratC of combustion; If the DDT pro,'s, takes place inside the vessel
rather far from the wall'o r there is- a direct-initiation of (4olonation tie loading, of tlhe wall Is
determined by a refleted normal detonation wave. But in caser the DDT - process takes place
near the wall the pressure rize (Wall loadhig) is determined )yt he reflection of an overdriven
detonation wave that.produoes higher i),,ks of p'essmre, h isiecessa'y to produce a closed form ,
solution of the problem of wall loading, des'cribing the multiplicity of scemario of DDT-processes
that wouldn't be sophistidated and could be obtained within reasonable processor times. To
solve the problem it is necessary in adldition to the DDT investigations [1.1) that match the
iesults of the experiments in tubes [1.2: 1,3 to 'examime ,somne peculiarities of spherical and
4I cylindrical deflagration and (etointion waves propagtion. Part 2 contains tim results of these
I
The closed-form model of wall loading based on the r,. lif iikVest iatio)is co)nducted takin,
into account the multiplicity of scenario of DDT-processes is described in Part 3,
Part 4 describes the results of breakup modelling muider tie influence of internal loading for
different initial conditions and different scenario of comiibustion lrocess inside the vessel. It is
shown that the criterion of breakup can be determiimied from am independent experimental data.
The mathematical model of breakup takes into accomt the influence of temup.rature on the
behaviour of the main patamete's. It is shown. that the critical pressure of breakup clhamies
depending on the type of combustion: the higher is the raw of loading - the higher turns to be
the'critical breakup pressure. The number of fragments is higher for higher pressures.
4 Part 5 presents the results of numerical modelling of fragmnelnts" acceleration int he expanding
cloud of gaseous reaction products after the reaku). The minerical experiment conducted
made it possible to work out approximation formulae (cte'rmnining final velocity of fragment
depending on its shape. orientation. initial conditions atl thermodynamic )roperties of the
expanding gas.
3
Typeset by .MS-TEX
. .. A 9. .. ,. .
-iierP2ia ;r rod Aiaiop -of 'e,R i tcii an-aed eflAgrittion N~ys.Wt spierikal sYil-
At dAeflagraion oxr detonationV iiiation iu, pre~kxi ed hioiieonsx ". omblustile iiittWC.,. foi
-
eka~fiil6]:iyjjeafi o as a. xutok 11dcativedv siall wone (if raised -vueso f paranieteim
~,. 1o~aion~Ekpandinolg prdxCis of f4etonaxor e'xploio ss i anl irtoei -ile ofa
,16e n neaI~o OS0- '11snfV lpT1
Self;,sihilJdntv of thie,,pto le' i splee i.al detonlation wa s e ablikhcd and the-,orffary lif-
fetenti~lecfuaqq~tioiiq-ofc i sxbn~ the -flow Ix hindt wa~'e front were derivedi 1.heas
91LleA aO~lan'.Jbuet .9 terical dJtonatio ii was c(1L. iidrvd 'fo. tmpie in 1.. T£I e
"v C~qnpktO so self-imila 1)roblejii-for cch-Ss of weak and- stre ' de1toniationt and'
Ch~p~~.Jipgetwyes as obtaijiedi in [2A54. 'in the )sn pt ,lroilles offowlaaet
in Propagatp of, iieic 'Ia!s (if stroii~afnd- weak cletoaioii fo eaae~se fao:rfai
different, ve -ociie' ofjp' tnnotiqjx tt'oii Ahe iinitia-ti( ixzn r nvestigated -on the base of
* We us the moel whee a dcitpnon-anclarto wve is (ato ns I (IeLt-ob e a tliscoh-
'156p aiats With, a, constaint veloci : thov initial inix~tiii0 jild(1 rctioi podhicts are polytropic
gaQO- nst is1t he speiify -lat of renatioin; ('oniitil mixtroe js nionoxbde ahieadi of
the wavefs and entixopvof alljl)artiol s hws u-constant vdlue., -il ilitropy inerases for 'All patti-
'Ae yth a~ v lue, in. the detonation auve movinag wijth cvolistwit veloci tyU i.e. a flow o
dletoniationi'ilU.Clttctsw itini the co)itinuity donxainis lxonxoven pcEqtin f osato
one.idimensibfial mnotion -has the folowig, formi undinr tliese conditions: ~
Ot, Or /)Oi
-
jx
I~~~~~~~ ~_2u_~22 0~
Ot 0,1 Or
where ui p, p--gas velocityv. prl)W55UatnIo ncltsity rospectively. Under these assilltions equatiColis-
and boundary conditions alloW an1 existenice of self-similar hollutious. .e. "olutionis beiing function's *
only of one p~arameter' It . Pitssing to svlf-s11iir variable bN tile formutlao:
& d 0 1(1
Ot td -;, ;' (2.3)
we reduce equations (-1); (2) to the form;
duL (1/)
d d 01 0 (2.4)
Since the flow is hiomoentropic then (1p/d = -2d(l/(, whevre c is the sound velocity. Transforin
equation (5) to the form:
pd l~(+ )-(2.6)
Typeset by AM$4STC\
77,
Y
Ni hud entcdthat a conlstant flow j,1" (on'st) is t, a solltX)11~of ( (juatil (24) fo thle
geerl cs , a]. soh'ution-.fi cou nta,f to i e (1 -4 (A.q uicsel, state), is onlyjpossihle.
Letus~eekd on uvi~sef-iniiha:Tplho Iito li cv depe klent. VarhiiIes
ne~vyaljl~ls'tle s eni(2.2.7") rake s rlie orifi:(2)
~(z+ s- V + - 0. (2.0)
where r =..p- o cdto it -s r'= 1). wherc ^I = 1/cp i's thle polytropic
ex ~t esblviig (S), (9) for tiw 2leriva *tivesw, e oIbtai:
~(~~~v:~±['V(1 (210)
(2.11)
With thle help of the systeml (210D). (2.11) we- dervirinine , Ij Midad thenl wrife the
expression derivMd in place of eqttat ii (2.10). Thiis.,thle followiti, sys$t('i is (lerived:
5:
-2:z-V-i(yV'-1)(2.12)
(IV V 3 0{ V) 2
System (2.12), (2.13) can be sobrd sequientially: at first cquxation (12) Aives the solution
found.e teslto otie ssusittdino(3.fomweet _ ()cnb
found adthntsltino)1e i usiit'dit 1) fo hr ~ ~ )cnb
Define boundary conditions for sal 'ving the (puttion (2.12) in the lplane (1Vz,) . For fixed t thle
condition ~ ()is satisfied. C*!ider the zone itlwad of the detonation wave Bo < r <4
(Figure 2.1). In this zone V = (1o(i t quiescent zone). In this case changes from
.A= oo for r -- o to B0 = Wt U for~ r R = U~t. Value of :within this zone
changes from zero (A=0) for 00 to ZZ=I for/ Bo = U. Oin the plane (V, z) it
corresponds to at trivial solution, wvpin a state cliankges from point .4 (V =0, ZA 0) to Point
Bo (V =0, :o=c0U < 1) (Figluf 2.2).
_7.
- - %
PaautesIellc tedt l nwavt, on -tit 1b)t n te -d'efr~ ,n iv t ie eqqat1nit a
?f
i htiu_._(21)S. I -A
41iuta V; 11c S- 46ii;A:
+ A0)(11 +A); (2.16)
where (
It follows from (2A6)vmd (2,11) that for fixevd A(U ) the point B3, in the plane (V", ~)is oil thle
1 -A
At the Chapinan-14oimet detonation at oiative gas Nelocit. behind the w;ve is equal to thle
local sound veldcitv:
thus, the Chapinan-.Jouget detonation corresponds to ease A 0when te parab)ola (2.19)
takes thle f~eni
(1- (2.19(t)
and goes through the(- points (0,3) and (1. 0) (Figure 2.2).
For A2 > 0, U > LD1 equation (2.1S) haws rwo roots~: A> 0, N- < 0, JAl = A+. It follows
fromi equations (2.17) and (2,15) that 4t > uit > Fixe. the root A+ > 0 corresp~onds to
the ca.3e of strong dotonlat ion, and the root A- < 0 crcsponds to the case of weak detonation.
Solution onl the plane (VK -) canl be prolonged either to the a-zxis (ixe. V = 0) that corresponds
to the rest in the s ,ninwtriv center or. at presenice of at sphiical piston. expanding with rate iu
11
4 up to the line' V =1 as
v g
In thie zone between the piston and thle dtftonation mvave a shock wvave cani appear for which
* the same relations at a xliscontinuity stirface (2.14) are t-le w; for a detonation wave b)ut with
"77 7
y
condpit~~ins that there is no -enet ix rc lese dt o i it ) n tiitr
',viipiat
Cbiiei C~ase (if ro DUCU, r'
0). (,.5"_) "t < 1 < j!PIn. l(i a the
pitg0tbtjoitB l the plane ( z), otnurs hy ;a ,imunp (Figure 2.2); hti he case when a~s
in16,ihfnIetry 6centeri is at- mcst. (u "-0.) ppinit 1(Pigtr 2.1) inthe p~d e-(I '.~)is
_nthe a~~V =0: ;AV~ 0, Nu (j K .-~-K 1 i glr p)oint oif the
knhot-tj?'6elfo equation, (2 t1) 'Thu11s8 li spllutio t.CO1olctinpoilit P, I- th po~it Alf goes through
poiht.iK p in (mo-natrml), mnvvs oil. the phlne 1%t ) wI, Itthe Sounid velocity:
C.,.
i ~! ~ o ~~- 0.
i.e. there is a, characterisfic onl the- plane (r, t) which-Ib oinids the res t domnain (at this charac-
*
teristic derivatives have dliscontinuity). Solution froml poinit k to point If is rest-, as in ~point
K the conditionm: Uk 0 is Natisfied1 and i lpohit A! it'l = 0 dule to symlmetry, then thle rest
state satisfies -the equations and, the boundary mulditiols Itid dtie to niqueness it is a Solui-I
tion-between the! poits k anid U., On the plane (V..-) statps from K( 1 , Z/ = 1) to
0(, ~ o r.o teai rm1u o -,~( Figure 2.2). Thuls. a desire(! Soluttionl
9 behind the detonation wave co)nsists of two lritche: lit-uiK orsonst h rva
soluti6ii -V _"0, and brnnchl K CiB to the solittion 'of the general type for sy.5tem (2.12).,(2.13).
0 1
On the plane (r. ) it sollution behinld the dertonatiomi wame r = CO1) consists of the exandingA
domaiof gas at rest ((I 0,J) "_(o nst =,k 1an1Jd the rarefaftionl wave inl the dlomlainl betweenl
the characteristic C'h :. r l and thet( de-tonation wa ve r = Unt . This rarefaction wvave isn't
a soluti6n of a,,simple Wave. Along each axis =const valuies of flow paranieters are constant.
S Consider the case of stroi~g detontion (A\ A+ > 0, C' = C1 > UT0)) when gas goe.4 fromt
the state characterized onl the (V, z) plui., by point B I to the state characterized by point Bi
11
(Figure22)
In this catse point B1. chiaacterizing jparamnetm-s izimiediatelv behind the detonation w~ave is
situated on p~arab~ola (2.1 0) whichm is hitgher t iemi parabola. (2.19a), Coordinates of Point B 1
1
V = VII and zi I can be determined'from eqputions (2.16), (2.17) (Pigure 2.2). In this case
we can construct thle solution w~here a -oiiipr(.s(Ion wave propagaItes behind a dletonation wave
(it is anl analog of a constant flow in at planie case)('2.oJ.
Continuous solution B1 Th (Figure 2.2) reaches the linle V = 1. points of which canl be
considered as points corre -pondhig 'to sp~herical pistois expanding with constant rate up- Point
PI, wherein the inerlcurve coinlo, definles th ny the piston wvhich corresp~1 onds to
.ltiyo
the given velocity of strong detonation wave U1. With decreasig U3 paramneter A+ diminishes.
and A+ = 0 at Ui = LID, parabola (210) transfers into p~arabola (2.19a) and point B1 becomes
coincident with p~oint B1. Continuious compression wav~e is described by solution B1PD, the
lopiston yelocitV decreases' toot o nitie II1~Ir w. Vi -~.w. aO Stoiloltaul finV 1t t.. Chap)iniJilJaUt .
detonation 'wave aidth xcontlnuous, comlpre sion- wav ltlictledonin 'vernt to lh
-tpeirtchl bolhid)i dit'naio witvefr to
On ieducimbp-,the pi~ston velo!cit Vs daOtla i.stato ahevad. of the pistonl Wvoii orrespond topointW
Fgu~ wn ret thle solut in 'cltailig thle CTapinan. iouet detoinatioini Wave. the
4 arfatonwaetkansfouinin p oatiri ste.'bl ito, certaini state nil , curve ."i~OK.
1shociiav tranisfoning fioni '4-ite C, inito stteC oil, curve D.,[CI K. and-lhi econitilolt
1
coinpiessioii wive C, P-2 -Cie BICC 1A presents ial ot is of states onl the (. pljan~ Whidh clii
b~e olbtalinec 'Is Li estiilt of~ a siioock train'~a ifcifromn th states l'Vilig onl integral cuirve B1 CjK.
WVith. futher decreaisig thie piston vlocity It, *Coirdihate z-p of poinlt P -tows lip. In- this
4case there is asolution, wherein p)oints CI diid Cu move along ciuvesBB B Ka ant
apprpach, the point K. Foe up, = thiere is a~ solutionl c ritainiig Chanmann oget, Akonation
W~ve,xarefadtionwavle aiclrest zone. 'Undler the given lioun~dary vooditionls, at, the-pisto it up
aind at .infinilty) the constructed solution 'inl the-ciass of'strodng detonation ajid Chap)man4ouget
tshows
4 ~~Figure 2;3n omted iesulltsf'foi caeo deitldtntcn itre 0~ 10,~o
J /kg; -;jo =1.4; -~1.2 i the -form of -depondenice of pressulre -p/po o11 nibj-Climensicinii
eoordiat r-/R Curve B B c K cotrspondis to the soltution voinainling the Chapmanil-
11 01
.Jouget-detonation wave BoBI (Ur) = 1.G30 ii/s), 'the rarefacetion wave and: the rest zone a
U-; 0. Curve BoBCOC1 ?2 corresponds to the soluttiou containing the Chiapuan-Jouget wave.
the rarefaction wave. shock wave C0C6 1 anld (-oinpressioni wfive C1 P2 fit thle expanlding sphiericail
piston velocity up, " 1200 in/s; Curve Bj0B1 C'C1 P- is the pressure profile ait decreasig thle
piston velocity ip dtown to 1000 im/.It is seen from the figure that fither decreasing the piston
- velocity will lead to redlucing anl intonsity of secondary wave C0 C1. Curve BOB, PD correspocnds
to the solution containing the Chainaln-Joigef detonation wave and compression wave B1 PD
at pistoni velocity it, 1320r/.
*
I Let us go to consideration of wea- deto~nation (A- < 0). In this c-asellhe problem solution isn't *
unique. Ani unique soluttion can be obtainied, in thle wpak detonation class when a value of one
lparamneter on the detonation front is giveuv-inl addition to, thme boundary conditions 2.,2.5), for
exfinile- this piarammeter can be detonation Wvave voity U,.whtich defines uniquely A -, 0 and
2
p)oint B12 Onl the (VI Z) plane. Solution contnntis the rarefaction wave BIC 0.,K (Figure 2.2) andj
the expanding rest zone beinid the weak detonation ait 1111 0. Pressure ,profile for the solution
obtained at U) " 1750) ii/ i li in F l-2 bi cuve B, .W10. aoupineftw t of
a moving piston (up, = Co710 > 0) containls the rareffaction wave behind the dletonationi front
from B12 to certain point C11l2 lying onl integral curve B1 C 2 .the shcick wave transforming
the medium fromn state Cuj2 into -state C,j on cunrve B II CI'2k and the Continuous vOunl~ressioui
wave to state P (V = 1) ahiead of thev piston (Figure 2.2). Curve 01 CljK in Figure 2 is a locus
of states which can bi reached by a shock transitionl fromn States Onl curve B, I C02K.
Figure 2.4 showvs pressure lprofil~s for the case of sphieric-al wveak detonAtioni wvave prop)agationi
with velocity U-2 =-- 1750 mu/s. Cuirve B0Ij1 ,1Q.0C 1 corresponds to the case tip = 0 when
there exists a rest zone in the sv iietryv enter. Solutions with an expanding spherical piston
are represented by curves BoIB12C1'12Cj2P (tip =1300 nii/s) and B01Bo2C()2C 2P' (111= 1000
* n/s). As seen fromn this figure. withi iiierva'inig the piston velocity tip, anl intensity of secondary
shock wave C0CI grows.-
2 2
Solution of the indicated type for the ca:.e A < 0 exists not for all values of the boundary
conditions but only till the piston velocity u <I (f12)stepso velocity
1 (9)whr
satisfying the solution with a straing dletonation wvave moving with velocity U, = U2. Curve
4 B01BI2BiiPI in Figure 2.4 corresp~onds to a pressure profile for such limiting case ( For the case
considered up, u, (U-) =1500 in/s). At this piston velocity a secondary shock wave C CI-I
1 2 02
ivaveBoIBw iiioviing withI tle~s mie- Velocity 'C 1" n1i1 .F t er Vvl0Citv ot the pisto
1
U ,( '2~~U ~1 sluio cIiib on oiwlv in the vla'." f 1 (latloil. For this, case it
1
-neiec tv to z iadiltioiml palinee at, (htitloI ' liiitc ~5.8 ,)i
$outfo ~n ilaialp~b( 1-,oil de.tonlationi wmaes~ mlotionl with the additional effect of, a
Consider the-.p()1)l~ii 011 _s)heri~dl h l111w tioit p1opfpafc~ioi1 I' 111inl)Cint souico of, inlitiatioin
forthecase le.(WIfhl ti- oi osin t eoi dctonation. Defla ration, fronit is assumed to
Abe a. discdiitiuit V suirface. and, the .ii t iolltfiofn 'is 'bving. ',omflt Within "the scope of the samle
asumptionis as inl the.1p revious ptoblefi. ojsdrth aOwhn *~eakd,e flatii front
~~priqep'n~~tn~~ttgt~h~o~mt~~bsi riwni l Friog~ninlitial quiiescenlt Ii ixtu&'
aatlvqfs~th,&.)armnr. otira ... of......, . 'l~' i.. . iiisoI1i %'vt+h wI )Wa hel . ofa s th dLf fo t.
froniti therefore at soluition Arill Con tain, it spliial shoti w~-llovii, avdtf-flie tlt
To fidaslto 6i~~I1 StV)~ Z 1oilnn-comhu~isted gais 1ictWen shock -.vand
darition frmn t ysep(. (h13),,c1a n i-e sed with. correspoiiding bIoiudary conditioins
othe9o1 aeailoit dfiagiratioi-fh ont Wh ich-hiv the-followig-forii in variables (V.:)
1) n thed sh6 ekwNave- in vinlg' thr~oiiAhm-q1Isent nitimn );from-( 2.15)-(2.i7) wve hak'e:
fro here t is seta shock waves-ftrlunsforrnl,1h axis V =Uito points of paabola (2.21);
12), 6w-the lamie front.- behind wich.I i taerls the onl( at, rest.. pualnmetersiahead of the
(1-_ +4 ~ )~1-.) 2~V( (2.22)
IAe. going onto the axis 0( throuigh. he- flanio front is possible only from poits of curve
2,.22); In deriin hee uato (2.22) wNe iise coniditions. (2.10)i (2.11)a h icniut
sui~de and the relaticoii connecting a niormal-buIminog rte W' anudflanu '(e iy- Up
Parameter -'o corre-sponlds to the initial mlixtulre. 0co res ponids to the 'omnbumstion produict..
The zone ahead of the shock wave inages intod sth-miieit .4I1 onl the ( V.z ) plitte. where
coordtinates of lpoint Hit corresp)onding to the state ait-ad of the shock wave front are: V'i 0:
zo c2,/U2 < 1. thle state b)ehindl the shock wnvelc(( Wre$))olids to the( point Hi lying onparabola
z=:dV) (2,21) (Figure 2.5). Going from state HI into state Nahead of the flamne front lying.
on curve n (V) (2.22) is realized within at continuols compressionl Wave (Figure 2.5), and
trasitoninto state X1 b)ehind the flame frout (11 = 0. :1 = 'v/U/. > 1) ovecurs by at jump).
The rest zone behind the flamie front iniages ito ray XS3I. For this case at solution As un1ique
at weak deflagrzittit velocity 11i given.
Notice that points of curve (2.22), for which coordinate < 1 at point N-:1 cannot corresp~ond
to the leading edge of the flamne front. as that leads to supersonic Blairn' front vielocity with resp)ect
to gas lbehinld the front (strong deflalgrationi. For this, cam- the ,,olution call be , ,mitructedl that
containis the Chapman-.Jouget oleflagration wave and the ivireffictiomi wave following the first one.
State Ibehind the dleflagration wave shoud be situated onl parabola z (I - V)'!, and a state
ahead of the wave is onl the curve:
Thof 7i1 '10 (1 -_!O)*2Q ~ l( ),(.3
*. /I Y - ^, 0