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Richard Courant.

Interaction of shock and rare-faction waves in one-dimensional media online

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Interaction of Shock and Rare-
faction Waves in One-Dimensional
Motion.

by

R. Courant and K. Friedrichs
New York University



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D nr.i t-i vr.e r^e. -.iirr-r.-ia r.'fjrrsc t; '.-.herciir: ). !'r-c:n
c'lcl- cci: UilcTTs not pnJL:r^_i;icc'.c5 ti;-;. cl3c I'ait-rnc-Jicri

•' "■-"^►.rsf.t fllar.cntLr.i:: :;i6d ■ ca- r.ja-ilc :'' :.
?c> cVtalr a complete ur-ior-jr.crjcilrr nf t>:e ponalbj e

tC cir^-.>r ilniiitlsa In a linear
.^obion of 2 cc-nrcEi.ib.'.s ;.i3. It senna therafcfe
natural to consider ahooka, rarefaction waves end - -
contact discontinuities or the same foatlnr, i.e.,
to atudj the effect of the Interaction of an^ two
or more of thocu The present -lomorandun is con-
cerned with the aysteniEtio study of such posa'ltle



(-:>) A "contact ilacor.tinulty" cr "contoct Bjr*"ace
occurs v;hen two inyers of caa in contact have the
3a:ae prosjT-re and velocity, tat .different densities
ard (naturally) different entropies and te:iperature3 =
Tlse laportonca of these contact discontinuities for
the understandlr.p of the ohen:-.cna was clearly re-
cognized by von Ilauinann.



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Interactlona In one-dl-ncnslonal notions of com-
pressible fttses or fluids.

Phenonona of 'his kind may be produced by
lettlnc two piston- act at both ends of a tube
filled with £03. he effect aay consist In shocks
or rarefaction wa' >s aovlng toward each other or
followlne each ot. art or. In cose of "contact
surfaces", shockc or rarefaction waves passing from
one layer Into a. jther. As soon as two waves or
shocks meet, or ne crosses a contact layer, a
rather cosapllcat id cas-flynamlcal process will becln.
However, In luut.- cases this complicated process of
"penetration", e fcher Immediately (») or after some
time, results 1- a nuch simpler "temlnal" state,
characterised ty two (either shock or rarefaction)
waves which HOT s steadily awny from each other and
are separated ty a region of constant pressure and
particle velocity. In this intermediate region new
contact discontinuities may occur. There are eases
in- which such s simple description of the final



(e) Such laaaeiate separation takes place If two
shocks meet or s shock crosses a contact discontinuity.



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state la lot adequate. However, In the present In-
veatlcatlon the aasuaptlon la expreasly made that a
simple terminal tate as described will resiilt.



Under this aasu: itlon it Is pesalbie to find the

terminal atate Ithout analyzlnj; the process of

penetration, 1. . without intecratinj; the differ-
ential equation of co3 dynanloa for this pi-occis.

In 3030 of the interactions considered oui" as-
sumption liaplle a nev; phtinonenon: sir.pie contact

discontinuities nill not occur but rather " conta ct

zones "; l>a., o lunns of constant lancth, (jEovlng

ivith constant v loolty through the tube) over which

density and ent opy vary continuously. In these

cases our basic isausptlon of a slraple temtnal

state can serve anly as an ap.iroxlmatlon, Trhlch has

to be Justlfiec ind Improved by >. more complete

solution of ths Jlfferentlal eqiiatlons (e).

It must be amphaalzed thtt the study of one-

dlnenslonal mot >n can bo considered only aa a

preliminary att >pt at understandln- the" greater



'.-:i) This has b n carried out In a tyolcal case.
(See pace t.).



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variety of antloeo^M phenomena In three diners Ions.
'Hie results of the present r.eiaorandum must further-
more be confronted more throufhly rlth experimental
evidence.

The InveatlGQtlon will te based on a convenient
form of the Ranklne - nuc"nl-o- conditions for a
shock and In a parallel way conditions of transition
across a rarefaction Trave. Furthermore, the occx;r-
rence of contect discontinuities makes It advisable
to Introduce as basic varlaMea not, as Is customary,
pressure and density, but rather the pressure p and
the particle velocity u, thera\:y avoldlnc complica-
tions by chances of density and entropy. The re-
presentation of possible states and tranaltlona in
a (u,p)- dlacram makes It possible to analyze the
phenomenon by cfaphlcal methods (which of course
have to be supplemented by numerical procedure to
obtain precision, not only qualitative results).
In this report we assume a:i ideal £"• ^8*
the method can be applied to substances with a dif-
ferent equation of state.



&ni^^^



1. Shooka. Rarefaction 7avea. and Contact Dlacontlmiltlea



The motion ahall take place alone the x-axla.
Throughout ▼elocltlea ore ooimtod poaltlva If In the
direction of the poaltive x-axls. Particle volocitlea
are denoted by u, ahock velocltlea by 0. The prosaupe
la donoted by p, the density by f , tho specific volume
by "C ■ P -1. '.Ve asstuaa that a
vol) aeparates tno aectlona (
having constant velocity, pressure, and density. The
regions x < x^ , " x > x^, or generally tho raglona of
constant p and n on tho loft and right are character-
Izod by the subscripts j^ ("all") and , respectively.

Suppose first a ahock transition takes place at
tho point Xq. V.'e call It a "forward shock", S , if
the nartlcles cross It from r to 1 , a "backward



ahock", S , If the particles cross It from 1 to

r . According to tho principle that all shocks

are compression ahocica, one has tho Inequalities



(1.1) Pi > Pr



Pi < Pr'



?1 > ?r
^1 < fr



for S ,



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«hlla for the velocitlos one has In both coses

(1.2) VLi > Ur for ^ ^^ ^

(a remark quite useful for the later dlacusalon. )



A continuous transition froa a region of con-
stant state into another region of constant state
con ba affected only by a "glmElu «ava". I.e.,
zone m the tube for which one sat of character-
istics in the (x,t)-plar.3 is straicht. If those
charactorlatlcs divorce as tins soos on or xvhat is
equivalent If the particles move apart from each
other the «ave is called "rarofaotlon wave". (=0 .

■>Ve apaali oi

wave R or R according to whether tho particles
pass throuGh the wave Bone from the right to the
left or frojn the left to the rlpht respectively.
For pressure and density on the two sides one has
(1.3) PI < Pr. fl
Is always positive for forward waves end negative
for backward waves.



titsiwOT



':^ixc-ll-j, '■'.-.ontnct stirfaoaB" sliould be dlafcln-
5ulah.;df de:er. linE upon which aide of tha disoon-
cinui';y has ir^ c^oatar density, A "contact dla-
oontir.uf ty" vlll ba called "T < " if f>^ < P^ and
■'?>• if f._y Pj. Tho equality of prasauras on
ooth oices of tha discontinuity will lamediatoly
r.aad tc incqualltios among quantities that are de-



tjy pre J euro



Dlntly such as



.VoEolrte tent'ici'.-.tur'o 8, entropy E, and spoed of
sound cj i.i)..



(T>) pT V, ^j, :



=1 < Cp. ej < 8,, Hi < Ey



It may "lie rotod that tha two atates separated
by a cor.tact Eurface are represonted by the aama
poln1;5 ir. tho (v.,p) diagrajn.



RKTWCTtO



We aasumc an Idaal gas with the adlabatlc a»>
ponent "i such that pj» = Poji ^" **^* equation of
state for adlabatlc changes, (^ =1.4 for the usual case
of a diatomic gas, t' ~ § ^'"^ gnnatOElc gases and

- * for polyatomic gases). We define

3

(2.1) ^ r 3^ •>• 1 (y = 6 for diatomic gases),

A, RalatlOT-S for shocks .

If the regions 1 and r ore separated by a shock
with Taloclty XJ, we obtain from the Ranklno-Hugonlot
condition the relations

(2.2) Jp /?i = Ti/Tj, = Uvp/Pi) where



With the gonaral notation [f(x)J for the jvoBp

t{x-y) - f(Xj.) of a function f(x) if x^ and Xp



approash the point



of discontinuity from the left



and from the right respectively, we have further

(2.4) ITK^1= '[M-f

(2.5) TIpi = u^ -rr^y= ui .-Ti Sk3

again as a consequence of the Ranldne-Hugonlot conditions.



RESTRlCTtO



Now we express



In terms of



using (2.2),



then. In view of (1.2), we obtain from (2.4)



(2.6) [u3= ^■[Pj



Pi + Pr



for S end S ,

It la eonvenlant to Introduce the function



(3.7). ^^(p) = (p - p^) y J^-T^^
which depends upon two positive parameters Pj^ and ^^
assigned to the state k. ^(p) represents the differ-
anea of the normal particle velocities across a shock
line as a function of the pressure on one side of the
line when the pressure pj^ and the density ^^ Is given
on the opposite side. Consequently

(2.8) ^^(p^, = ..p^lp^) ,

If the regions r and 1 are connected by a shock.
More generally. In virtue of (1.2)

(2.9) u^ . u, "j^l(Pr)| =1 $*r(Pi)



f ^cf^tp)




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Obviously, 5v^P^ *■' * monotone Increasing function

of p and Its derivative ^^(p) Is a monotone decreasing



function of p , symbolloally

FtartharmOFe It will be useful to make the following sln^jle
remark ooneemlng the dependence of ^(p) on k :
R^.T.v 1. Tf a ^ f^ end p. ^ - — —



, then



fk^P) >fh^Pi f°^ P > Pj,



The curves



(S.IO) a => -Ujj + *^3j(p) ai-'d u = vij^ - f^ij,)



Tflll be called the S- and S-curvea through k or the
curves S^ and ^j. raspeotlvely. ( In dlasrama we oailt
the arrows since the positive op nefjatlva slope of an
S-CTiPve la sufficient to Indicate that It refers to a
forward or backward shock respectively). A graphical
representation of the possible states u

::-Pp_. If state 1 Is prescribed (or. If state r la
prescribed, the possible states u = "
Is shown by the diagram^



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velocity U^^ j, g,.,,^ ^^
(2-11) u



■1 ■' Pr and



Elus if S s 0 a..d hence V, > u, > u.



For peptiol



Rarefaction W=-,



33 aoTlnj aoi'oas



a rerefactlon wave



W3 have a.laha.lc ch^^ges o. ,tate an. hone, t.e relation
(2.12) f / p ,j,

It la decisive that for rer3faction -^aves al analogue '
to (2.6) foliowa ., inte^atlon of the aifferentlal
equation of notion , .ith the notation y^ z H. ^-Ij



the change fuj



- >» P Of the



REST!]ICT[D



Telocity u across a slmpla wsvo may be expreaasd by

(2,13) [_u] = + /

the plus sign referring to a and the sinus sign to
Wa Introduce the function



(2.14) /;^(p) -. ^f?TT ,ti rt*"S (pt - pj, .

vhlch also depends upon t^o positive parametars p mil

6 X *^k (tho latter la easantlally tlia entropy); then,
coiTeapondlns to (2.9) ae have

(a.l5) [u]= u^ - u^ = -j/i-^Pi) 5 =-V^i(P^' .
F.iaaLi.y analogous to (2.8), thuro eiiats the relation



«^-lS5 /-.CP,) =-t^,(p,, ,

1. 1 end r are connected by a rarefaction wave.

Some useful properties of the function )^Jip) «?•
/'k^P-'T o«=» , as pf oo
^Ij(p)^ O . as pToo
«"^ i^(p) approaches the ^-exls tangantlally at

or ^R , u ;( ^ - iT c 2
const, (of. Panel iiamorandum) or u = ^^( tf^**- DTf r

const. In vie* of (2.12) we have '^* pi- 1 -vi i- "2 j

r p P

honce (2.13) follows.



RtSTRlCTM



(2.17) j^ = -^^2 . D^-^Pk = -(>r- 1)



y'^^^cp)



Ir. addition, wa also havs



Renerk 8 . If P,j< 1^ and 2'u Pfc l"^!!. Ph " ' *^"^
Y (p),y£/(p) for P - ^ "



As bafore, the curves



(2.18)



= ^k+ /k^P^ ^'^ * ' ''k -\f^^^



will be ceiled R- and B- curTea throuSi k °r *ho
ourves 9^ and R reapactlvoly. (In diagrams we omit
the arrows since ths positive op negative slope of tbe
H-curvo iB auffleleat to Indicate that it refers to a
forward or backward rarefaction wave respectively).




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Graphically (♦) the possible states u = u - -

If state 1 Is preaerlbed (or, If state r Is prescribed,



the possible states
the diagram.



p-j_ ) are shown by



■i



Finally wa not3 some releticna between the functions
^ (p) and y/ (p) (n-.-.-.



(2.19)



(2.20)



^k(Pk) = flip^) . ■



C. Contact Discontinuities.

Fop a contact discontinuity, there are no transition
relations between the values of velocity and pressure,
since across such a discontinuity these quantities remain
continuous while only the density or entropy suffers
the discontinuity.



(«) Branches not s'lown here would correspond to
contraction waves.



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Rlamann' a Problem

In hla claaaloal paper on "Luftwalien Ton endllcher
Sohwlngungawolte" Rlamann dlacuaaad a problem closely
related to our topic: At the time t = an In-
finite "linear gas" colunn along the x-axla la divided
by the point x s o Into two constant states 1 and rj
It Is required to determine the aubaequent atate of the
gas. Rlemann showed that the Initial discontinuity may
resolve In either two shocks moving apart or two rare-
faction waves (of the special character with charecter-
latlca meeting at a singular point) or one shock and one
rarefaction wave. However, Rlemann' s solution Is not
complete, in fact. In Rlamann' s theory no contact aurf acoa .
are poatulated since only two shock conditions are used
which can be satisfied without Introduclne lines T In
the (x,t) plane.

W« ahall now give a complete solution of Rlemann' a
problem by a method that will likewise be applicable to
all our problems of Interaction. The aolutlon consists
In showing that we can always determine uniquely states
l.°Vf' following the Initial situation , such that the
Intermediate constant atate n^^ la connected with 1
by a backward wave, with r by a forward wave, each of
which, according to elrcumatanoea, jaay be either a shock



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OP a rarsf action wave. (Under certain clroumatanoes no
Intermediate state will result, aee below p. 19 ) .
To reproduce euch a solution, we realize what
possible backward waves can connect a state m_ rlth
and what forward waves can connect m with r , (In
cur general scheme ei^_ win rirgc plav the rola of r,
then th3 role of 1), liow, fron cur previous geometrical
oonalderatioa it is clerj, that in e (u,p) dlajji-tun all
points raprassnting sach 3tai:as which can be oormacted



with 1 oj a baclrjiari wavo



, will be on a



"left transition curve" Q , consisting of oa -jipper
branch cf an ^ curve and a lower branch of en ii.
curve. Simllarily we have a rl.^ht transition curve fr
connecting the point r I'ith other points representing
those states, for which the transition to r la effected
by a forviarG wave R or 3 ..



Now wa simply marlc In the (u,p) plane the two
points 1 and r corresponding to the prescribed
initial states, and draw the two curves M. and K



mmv.^



In the diagrams.



tf^^^^W^^^R



Wo 3aattsr where the l»fo points 1 and r lie, as the
baal-j prcpertiea of 'l and 'c ourvsa show, there will
al-jaya bs (a) one and only one point of intersection
except when

(S.l) u^ - (^. 1) cp , uj^ 4. (^_ 3L) c^

In which case the curves 'l and "r both reach the
u-azls tsngentially vrlthout intersecting.

According to the positions of 1 and m and to the
values Of the parameter f^, p^ and f^, p^ the points
of intersection m^ may 11a within the S or R branch
on either '1 or Ir and thus four posalbllltles arise.
(In the exceptional case when relation (3.1) holds no
intermediate state m^ will result, as the process of
penetration continues indefinitely. Nevertheless we may



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Online LibraryRichard CourantInteraction of shock and rare-faction waves in one-dimensional media → online text (page 1 of 2)