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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

â€¢' "â– -"^â–º.rsf.t fllar.cntLr.i:: :;i6d â– 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.

RESTRICTED

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 Â£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 (Â») 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.

RESTRICTED

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.).

RESTRICTED

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 Â£"â€¢ ^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 â–¼elocltlea 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 â– 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 ,

RESTRICTED

Â«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 Â«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 Â«ave is called "rarofaotlon wave". (=0 .

â– >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, 'â– '.-.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

â– '?>â€¢ 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 aÂ»>

ponent "i such that pjÂ» = Poji ^" **^* equation of

state for adlabatlc changes, (^ =1.4 for the usual case

of a diatomic gas, t' ~ Â§ ^'"^ gnnatOElc gases and

- * for polyatomic gases). We define

3

(2.1) ^ r 3^ â€¢>â€¢ 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= ^â– [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)

RESTRICTED

Obviously, 5v^P^ *â– ' * 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. ^ - â€” â€”

, then

fk^P) >fh^Pi fÂ°^ 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^

RESTRICTED

velocity U^^ j, g,.,,^ ^^

(2-11) u

â– 1 â– ' 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

- >Â» 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

Â«^-lS5 /-.CP,) =-t^,(p,, ,

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

Some useful properties of the function )^Jip) Â«?â€¢

/'k^P-'T oÂ«=Â» , as pf oo

^Ij(p)^ O . as pToo

Â«"^ 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),yÂ£/(p) for P - ^ "

As bafore, the curves

(2.18)

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

will be ceiled R- and B- curTea throuSi k Â°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).

RESTRICTED

Graphically (â™¦) 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

â– i

Finally wa not3 some releticna between the functions

^ (p) and y/ (p) (n-.-.-.

(2.19)

(2.20)

^k(Pk) = flip^) . â–

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.

(Â«) Branches not s'lown here would correspond to

contraction waves.

RESTRICTED

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.

WÂ« 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.Â°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

RESTRICTED

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 lÂ»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

{

Interaction of Shock and Rare-

faction Waves in One-Dimensional

Motion.

by

R. Courant and K. Friedrichs

New York University

m

r-

O

os

u

O

IB

01

c

o

c

M-

in

11

T3

o

Qi

â€¢rA

V.

>

â€¢o

re

c

x:

o

z

E

u

â€¢â€¢.t

â– i

-*-Â»

c

^.4

cc

u

o

OS

IS

â– rt

c

r-

!^

AJ

o

^H

-^J

01

u

â– Â«.!

1

c

â– u

no

U)

CD

fO

c

^4-

c

z:

I.

t-c

01

01

â– ;':;\;jrl; :;: .. l^'O. 'J, ^. :C.

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

â€¢' "â– -"^â–º.rsf.t fllar.cntLr.i:: :;i6d â– 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.

RESTRICTED

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 Â£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 (Â») 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.

RESTRICTED

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.).

RESTRICTED

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 Â£"â€¢ ^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 â–¼elocltlea 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 â– 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 ,

RESTRICTED

Â«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 Â«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 Â«ave is called "rarofaotlon wave". (=0 .

â– >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, 'â– '.-.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

â– '?>â€¢ 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 aÂ»>

ponent "i such that pjÂ» = Poji ^" **^* equation of

state for adlabatlc changes, (^ =1.4 for the usual case

of a diatomic gas, t' ~ Â§ ^'"^ gnnatOElc gases and

- * for polyatomic gases). We define

3

(2.1) ^ r 3^ â€¢>â€¢ 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= ^â– [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)

RESTRICTED

Obviously, 5v^P^ *â– ' * 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. ^ - â€” â€”

, then

fk^P) >fh^Pi fÂ°^ 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^

RESTRICTED

velocity U^^ j, g,.,,^ ^^

(2-11) u

â– 1 â– ' 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

- >Â» 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

Â«^-lS5 /-.CP,) =-t^,(p,, ,

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

Some useful properties of the function )^Jip) Â«?â€¢

/'k^P-'T oÂ«=Â» , as pf oo

^Ij(p)^ O . as pToo

Â«"^ 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),yÂ£/(p) for P - ^ "

As bafore, the curves

(2.18)

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

will be ceiled R- and B- curTea throuSi k Â°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).

RESTRICTED

Graphically (â™¦) 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

â– i

Finally wa not3 some releticna between the functions

^ (p) and y/ (p) (n-.-.-.

(2.19)

(2.20)

^k(Pk) = flip^) . â–

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.

(Â«) Branches not s'lown here would correspond to

contraction waves.

RESTRICTED

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.

WÂ« 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.Â°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

RESTRICTED

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 lÂ»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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