2007_Salas_The Curious Events Leading to the Theory of Shock Waves

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Shock Waves (2007) 16:477487DOI 10.1007/s00193-007-0084-z

REVIEW

The curious events leading to the theory of shock waves

Manuel D. Salas

Received: 15 January 2007 / Accepted: 15 January 2007 / Published online: 12 June 2007 Springer-Verlag 2007

Natura non facit saltus.

Lucretius Caro (98 BC-55 BC).

I will tell you these stories, not in the fashion of those

textbook writers who manufacture historical notices so

as to bear out their own views of how science ought

have developed, but instead as they really did occur.

Clifford Truesdell (1919-2000).

Abstract We review the history of the development of themodern theory of shock waves. Several attempts at an early-theory quickly collapsed for lack of foundations in mathema-tics and thermodynamics. It is not until the works of Rankineandlater Hugoniot that a full theoryis established.Rankine isthe first to show that within theshock a non-adiabatic processmust occur. Hugoniot showed that in the absence of viscosityand heat conduction conservation of energy implies conser-vation of entropy in smooth regions and a jump in entropyacross a shock. Even after the theory is fully developed, oldnotions continue to pervade the literature well into the earlypart of the twentieth century.

PACS 47.40.x 01.65.+g

Communicated by M. Onofri, Guest Editor, ISIS 17.

This paper is based on the invited lecture that was presented at the17th International Shock Interaction Symposium (ISIS17), Rome,Italy, 48 September 2006.

M. D. Salas (B)NASA Langley Research Center, Mail Stop 499, Hampton,VA 23681-2199, USAe-mail: [email protected]

1 Beginnings

The period between Poissons 1808 paper on the theory ofsound and Hugoniots fairly complete 1887 exposition of thetheoryof shock waves is a periodcharacterized by many inse-curities brought about by weak foundations in mathematicsand thermodynamics. In the early 1800s a few British scien-tists had read the works of Johann and Daniel Bernoulli,dAlembert and Euler [3]. Truesdell summarizes the pre-vailing current in England thus: The mathematics taughtin Cambridge in the early nineteenth century was so anti-quated that experiment and mathematical theory had turnedtheir backs upon each other. In order to set up a mathema-

tical framework general enough to cover the phenomena oftides andwaves andresistanceanddeformation andheat flowand attraction and magnetism, the young British mathemati-cians had to turn, finally, straight to what had been until thenthe enemy camp: the French Academy, where the mantleof the Basel school, inherited from Euler by Lagrange, hadbeen passed on to Laplace, Legendre, Fourier, Poisson andCauchy [34]. However, to be fair the truth is that themathematical apparatus needed to effectively deal with dis-continuous functions1 did not exist anywhere and the long

1

Our understanding of the meaning of a function has it roots in theacrimonious debate between dAlembert and Euler over the solution ofa vibrating string [18]. Eulers view employed the notion of impro-per functions which allowed for the representation of discontinuitiesconsistent with the physical observations of D. Bernoulli [19]. The dis-pute declined with the passing of both protagonists in 1783. The issueof the regularity of solution did not resurface until it was forced onmathematicians by Riemann and other physicists dealing with discon-tinuous waves in the latter part of the nineteenth century. The first stepstowards a theory of generalized solutions to hyperbolic partial differen-tial equations were taken only at the beginning of the twentieth century[17]. The theory reached maturity in the works of Sobolev [29], 1934and Schwartz [27], 1950.

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478 M. D. Salas

established attitude among British scientists to ignore thework of scientists in the Continent was equally reciprocatedby their peers in the Continent. In addition there was a gene-rally held belief that nature would not tolerate a discontinuityand this, compounded by a lack of appreciation for the sin-gular nature of the inviscid equations, cast a fog of confusionover the problem.

The main events that follow unfold as a tale of two cities,in Cambridge and in Paris. In order to appreciate the eventsin Paris leading to Hugoniots paper of 1887, we have to startby considering the contributions of Gaspard Monge (17461818). Monges work is unique in that he approached thesolution of partial differential equations by means of geo-metrical constructions. In 1773, he presented his approachto the French academy [20], in addition to other works onthe calculus of variation, infinitesimal geometry, the theoryof partial differential equations and combinatorics. His workon solutions to first-order partial differential equations esta-blished the foundations for the method of characteristics

which would be later expanded by Earnshaw, Riemann andHadamard. But beyond his technical contributions, Mongeis also important to this story for his role in the creationin 1794 of what would become one of the most prestigiouseducation centers in the world, the cole Polytechnique ofParis. The political turmoil of the French revolution from1790 to 1793, culminating in the Reign of Terror, resulted inthe closing of all institutions of higher learning and left therepublic without a much needed supply of civilian and mili-tary engineers. The cole was created to meet this demand.As Dickens observed about Paris in 1794: it was the best oftimes, it was the worst of times, it was the age of wisdom,

it was the age of foolishness . . . Four hundred students, thebest the country had to offer,were enrolled in the first year fora 3-year curriculum in revolutionary courses in mathema-ticsandchemistry[9].Overtheyears,theschoolwouldcountamong its faculty and students Lagrange, Poisson, Fourier,Duhamel, Cauchy, Carnot, Biot, Fresnel, Hugoniot, Navier,Saint-Venant, Sturm, Liouville, Hadamard and Poincar, tolist but a few. The schools great success, particularly inmathematics, must have been in part due to Lagrange. Thegreatest mathematician of the times is reputed as the greatestteacher of mathematics.

Simon-Denis Poisson (17811840) entered the cole at

age 17. There he was trained by Laplace2 and Lagrange whoquickly recognized his mathematical talents. As a studentPoisson had troubles with Monges descriptive geometrywhich was a requirement for students going into public

2 Laplace consideredsecond only to Lagrangeas a mathematicalsavantwas not a professor at the cole Polytechnique, but an examiner. As anexaminer he traveled to cities throughout France administering public,individual, oral exams to aspiring students. Laplaces influence on theschool was considerable.

service, but because he was interested in a career in purescience he was able to avoid taking the course. Soon aftercompleting his studies, Poisson was appointed repetiteur3 atthe cole. Appointment to full professor was a difficult pro-position, but in a lucky break a vacancy was created in 1802by Napoleon when he sent Fourier to Grenoble in south-eastFrance to be the prefect of Isre. With the support of Laplace,

Poisson took the position in 1806. A year later, Poisson deli-vered his lecture on the theory of sound, which appearedthe following year (1808) in the coles journal [21]. Theopening paragraph begins by giving credit to Lagrange andcontinues with: However, at the time of their publication,very little was known about the use of partial differentialequations on which the solution for these types of problemsdepend. There was disagreement on the use of discontinuousfunctions which are nevertheless fundamental for represen-ting the status of the air at the origin of the motion: thank-fully, these difficulties have been removed with the progressmade in the analysis, whilst those which persist relate to

the nature of the problem. Without further reference to dis-continuous functions, the paper proceeds to prove severalgeneral theorems for the solution of partial differential equa-tions governing the propagation of sound waves. In Sect. 24dealing with disturbances of finite amplitude he introducesthe governing equation for the velocity potential and par-ticle velocity d/dx :

d

dt+ a

d

dx+

1

2

d2

dx2= 0,

here a is the speed of sound which is assumed constant.Poisson finds the exact solution for a traveling wave in one

direction in the form,

d

dx= f

x at

d

dtt

, (1)

where f isanarbitraryfunction.Poissonsothermajorcontri-bution to the theory of sound was his derivation in 1823 ofthe gas law for sound waves with infinitesimal amplitudes,p , later called Poisson isentrope4[22].

Twenty years younger than Poisson, George Airy (18011892) graduated top of his class at Trinity College,Cambridge in 1823. That year, Airy was awarded the firstSmith prize,given for proficiency in mathematics andnatural

philosophy. Among Airys examiners for the Smiths prizewere Robert Woodhouse and Thomas Torton, both former

3 A repetiteur is a professors aid who explains the lectures to thestudents.4 Although the isentropic relation is credited to Poissons 1823 paper,Lagrange discussed it in 1760 [7] and Laplace developed it in a shortnote [16]publishedin1816,inthesamejournalasPoissons1823paper.Laplace wrote:The real speed of sound equalsthe product of thespeedaccording to the Newtonian formula by the square root of the ratio ofthe specific heat[s] . . . That is, a2 = p/.

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Curious events leading to the theory of shock waves 479

Lucasian Chair holders. Only 3years after graduating fromTrinity College, Airy was elected to the Lucasian Chair.The Chair paid Airy very little5 and thus in 1828, less than2 years after being appointed Lucasian Chair, Airy took amuch higher paying job by replacing Woodhouse as PlumianProfessor of astronomy at Cambridge and Director of theCambridge Observatory. In 1835, Airy moved to Greenwich

to becometheAstronomer Royal of the RoyalObservatory atGreenwich. In his position as Astronomer Royal, Airy publi-shed a long article on tides and waves [1]invol.3ofEncyclo-

paedia Metropolitana. In the article he makes the followingreference to Poisson and Cauchy: . . .[he] does not compre-hend those special cases which have been treated at so greatlength by Poisson. . .and Cauchy. . . With respect to these wemay express here an opinion, borrowed from others writers,but in which we join, that as regards their physical resultsthese elaborate treatises are entirely uninteresting; althoughthey rank among the leading works of the present centuryin regard to the improvement of pure mathematics. Airys

remark captures the views of his colleagues at Cambridgetowards the works of Poisson, Cauchy and other leadingscientists in the Continent. In the Metropolitana article, Airystudies waves of finite amplitudes in water canals and makesthe observation that the crests tend to gain upon the hollowsso that the fore slopes become steeper and steeper. The signi-ficance of Airys observation is understood by James Challis(18031882), a colleague of Airy, who is best known for hisrole in the British failure to discover Neptune [28]. In 1845Challis was director of the Cambridge Observatory, the sameposition previously held by Airy. That year, John Adams, ayoung mathematical prodigy, approached both Challis and

Airy with calculations he had made based on irregularitieson the orbit of Uranus predicting the position of a new planet.His calculations were ignored until late in June of 1846 whencalculations by Joseph Le Verrier, a French mathematician,became known in England. After seeing Le Verriers pre-diction, Airy suggested that Challis should conduct a searchfor the planet, but by this time it was too late. The disco-very of Neptune was snatched from the British by the BerlinObservatory on September 23, 1846. Challis record in fluiddynamics was not impressive either [3]; however, in 1848 hepublished an article in the Philosophical Magazine [2] basedon Airys observation about the behavior of waves of finite

amplitude in the Metropolitana. In this article Challis writesthat if we consider a sinusoidal motion,

w = m sin2

(z (a + w)t),

an apparent contradiction occurs. The contradiction comesabout because when the velocity w vanishes at some t = t1

5 The Lucasian Chair paid Airy 99 per year compared to 500 peryear as Plumian Professor.

we have z = at1 + n/2 and since at the same time t =t1 w reaches its maximum value (m) when z = at1 +n/2 + mt1 /4, and since m, t1 and are arbitrary, wemay have mt1 /4 = 0, in which case the maximum andthe points of no velocity occur at the same point.

George Stokes (18191903) graduated from PembrokeCollege, Cambridge, in 1841. Like Airy, he was first of his

class, was a recipient of the Smiths prize and was appointedto the Lucasian Chair. While attending Pembroke College,Stokes had already crossed paths with Challis and had quar-reled with his views on fluid mechanics on several occasions[4]. Thus it appears that Stokes seized the opportunity toembarrass Challis with a comment on the following issue ofthe Philosophical Magazine entitled On a Difficulty in theTheoryofSound[30].UnlikeChallisandmanyothersofhisBritishcontemporaries, Stokes had both read and understoodthe works of Fourier, Cauchy and Poisson. Stokes begins hisarticle by writing down Poissons exact solution to the trave-ling wave problem, Eq. (1). He follows with an illustration

that shows how a sinusoidal solution satisfying Eq. (1) wouldchange in time. He then describes the motion as follows: Itis evident that in the neighborhood of the points a, c [thecompression side] the curve becomes more and more steepas t increases, whereas in the neighborhood of the points o,b, z [the expansion side] its inclination becomes more andmore gentle. He continues by finding the rate of change ofthe tangent to the velocity curve,

dwdz

1 + dwdz t, (A)

and remarks that:At those points of the original curve at which the tan-

gent is horizontal, dw/dz = 0, and therefore the tangent willconstantly remain horizontal at the corresponding points ofthe altered curve. For the points for which dw/dz is positive,the denominator of the expression (A) increases with t, andtherefore the inclination of the curve continually decreases.But when dw/dz is negative, the denominator of (A)decreases as t increases, so that the curve becomes stee-per and steeper. At last, for sufficiently large values of t,the denominator of (A) becomes infinite6 for some value ofz. Now the very formation of the differential equations ofmotion with which we start, tacitly supposes that we haveto deal with finite and continuous functions; and thereforein the case under consideration we must not, without limi-tation, push our results beyond the least value of t whichrenders (A) infinite. This value is evidently the reciprocal,

6 Stokes should have said that the denominator of (A) becomes zero.Equation A representsthenewtangentofinclination,when1 + dw

dzt 0

the tangent .

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480 M. D. Salas

taken positively, of the greatest negative value of dw/dz; where, as in the whole of this paragraph, denoting the velocitywhen t = 0. After finding the breakdown-time for Challisproblem (t = /2 m), Stokes explains that Challis paradoxoccurs because he considered larger values of times. Stokescontinues with:

Of course, after the instant at which the expression (A)

becomes infinite, some motion or other will go on, and wemight wish to know what the nature of that motion was.Perhaps the most natural supposition to make for trial is,that a surface of discontinuity is formed, in passing acrossin which there is an abrupt change of density and velocity.The existence of such a surface will presently be shown to bepossible, on the two suppositions that the pressure is equalin all directions about the same point, and that it varies asthe density. . .even on the supposition of the existence of asurface of discontinuity, it is not possible to satisfy all theconditions of the problem by means of a single function ofthe form f{z (a + w)t}.

Stokes then proceeds to find the jump conditions for massand momentum:

w w = ( ), (2)

(w w)

w2 w2

= a2( ), (3)

where is the speed of the discontinuity. We immediatelyrecognize Eq. (2) as the conservation of mass across a shockwave. However, Eq. (3) does not look familiar. ConceptuallyEq. (3) expresses a correct balance of momentum acrossa shock wave. The reason it does not look familiar is thatStokes has replaced the right-hand side term which corres-

ponds to the pressure difference across the shock with thedensity difference using the prevalent Newtonian theory ofsound: Boyles law and constant speed of sound (isother-mal flow). Stokes then writes: The equations (2), (3) beingsatisfied, it appears that the discontinuous motion is dyna-mically possible. This result, however, is so strange, that itmay be well to consider more in detail the simplest pos-sible case of such a motion. After considering in detailthe motion of a shock moving with a constant velocity and finding no contradictions Stokes writes: The strangeresults at which I have arrived appear to be fairly dedu-cible from the two hypotheses already mentioned. It does

not follow that the discontinuous motion considered can evertake place in nature,7 for we have all along been reasoningon an ideal elastic fluid which does not exist in nature.He then discusses the effects of heat addition and friction,

7 Stokes is echoing the popular adage attributed to the Latin phi-losopher Lucretius Caro: Natura non facit saltus (nature makes nojumps). Leibniz expressed the same thought in his Nouveaux essayssur lentendement humain (1705): Cest une de mes grandes maximeset des plus vrifies, que la nature ne fait jamais des sauts, and Darwinquotes it seven times in his Origin of Species (1859).

concluding that: It appears, then, almost certain that theinternal friction would effectively prevent the formation ofa surface of discontinuity, and even render the motion conti-nuous again if it were for an instant discontinuous. Thefollowing year, Stokes was appointed Lucasian Professor ofMathematics at Cambridge. He retained this chair until hisdeath in 1903. Stokes note on Challis paradox goes unno-

ticed for many years.Lord Rayleigh (18421919) entered Cambridge as a

student in 1861. There he was a student of the mathema-tician E. J. Routh8. As an undergraduate, Rayleigh attendedthe lectures of Stokes and was inspired by Stokes approachwhich combined experimental and theoretical methods.Rayleighs first volume of The Theory of Sound was publi-shed the same year of the following letter to Stokes. The let-ter is in response to a conversation Rayleigh had with Stokesconcerning Challis paradox [34]:

4 Carlton Gardens, S. W.

June 2/77

Dear Prof. Stokes,

In consequence of our conversation the other eve-

ning I have been looking at your paper On a diffi-

culty in the theory of sound, Phil. Mag. Nov. 1848.

The latter half of the paper appears to me be liable

to an objection, as to which (if you have time to look

at the matter) I should be glad to hear your opinion.

By impressing a suitable velocity on all the fluid

the surface of separation at A may be reduced to rest.

When this is done, let the velocities and densities on

the two sides be u, , u, . Then by continuity

u = u.

The momentum leaving a slice including A in unittime =u u, momentum entering =u2.

Thus9p p = a2( ) = u(u u).

8 Routh is best known for the Routh-Hurwitz theorem which can beused to establish if a polynomial is stable. Here, stable is in the sensethat the roots lie to the left of the imaginary axis.9 The second equal sign was replaced by a minus sign in [34].

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From these two equations

u = a

, u = a

.

This, I think, is your argument, and you infer that

the motion is possible. But the energy condition

imposes on u and u a different relation, viz.

u2 u2 = 2a2 log

,

so that energy is lost or gained at the surface of

separation A.

It would appear therefore that on the hypotheses

made, no discontinuous change is possible.

I have put the matter very shortly, but I dare say

what I have said will be intelligible to you.

In order to follow Rayleighs argument, consider theenergy balance across the shock:

1

2u2 + e

u

1

2u2 + e

u = pu pu,

where u is the velocity relative to the shock wave and e is theinternal energy. A change in internal energy is given by

de = TdS +p

d

,

where T isthetemperatureandS istheentropy.If,asassumedby Rayleigh, the shock is stationary and the flow is reversible

and it obeys Boyles law, then

u2 u2 = 2(e e) = 2

a2d

= 2a2 log

.

Stokes replies to Rayleigh [34]:

Cambridge,5th June, 1877.

Dear Lord Rayleigh,

Thank you for pointing out the objections to the

queer kind of motion I contemplated in the paper you

refer to. Sir W. Thomson10 pointed the same out to

me many years ago, and I should have mentioned it if

I had had occasion to write anything bearing on the

subject, or if, without that, my paper had attracted

attention. It seemed, however, hardly worthwhile to

write a criticism on a passage in a paper which was

buried among other scientific antiquities.

10 W. Thomson (18241907) was named Lord Kelvin in 1866.

P.S. You will observe I wrote somewhat doubtfully

about the possibility of the queer motion.

It is apparent that Stokes does not have the determinationor confidence in his position to defend the convincing casehe had presented in 1848.

Years later, in a letter to W. Thomson dated October 15,1880 [36], Stokes tells Thomson that he is reviewing hispaper On a Difficulty in the Theory of Sound for inclusionin his collected works. Stokes reminds Thomson that bothhe and Rayleigh had pointed out years earlier that his ana-lysis violated the principle of conservation of energy. ThenStokes argues: The conservation of energy gives anotherrelation, which can be satisfied, so that it appears that such amotion is possible. Two weeks later, on November 1 [36],Stokes changes his mind and writes to Thomson: On furtherreflection I see that I was wrong, and that a surface of dis-continuity in crossing which the density of the gas changes

abruptly is impossible. In this letter, Stokes gives no detailsfor this conclusion. The details come 2days later [36] whenhe writes: I mentioned to you I think in a letter that I hadfound that my surface of discontinuity was bosh. In fact, theequationofenergyappliedtoasliceinfinitelynearthesurfaceof discontinuity leads to

p

2d =

1

2

w2 w2

,

where , are the densities, and w, w the velocities on thetwo sides of the surface of discontinuity. But this equation

is absurd, as violating the second law of motion. In this waythe existence of a surface of discontinuity is proved to beimpossible. On December 4, Stokes again writes to Thom-son [36]: I have cut out the part of the paper which relatedto the formation of a surface of discontinuity. . .The equa-tion I sent you was wrong, as I omitted the considerations ofthe work of the pressures at the two ends of the elementaryportion. . . And thus, in the volume of his collected worksof 1883 [32], Stokes adds the following footnote to his 1848paper at the point where he said that such a surface was pos-sible: Not so: see substituted paragraph at the end, andremoves the entire section dealing with the description of

the discontinuity. After claiming that he had made a mistakeby considering only conservation of mass and momentum,he says: It was however pointed out to me by Sir WilliamThomson, and afterwards by Lord Rayleigh, that the discon-tinuous motion supposed above involves a violation of theprinciple of the conservation of energy.11

11 What is so odd here is that by 1880 both Thomson and Stokes werefamiliarwith Rankines paper of 1870, yetfailed to understand itssigni-ficance in relation to Stokes 1848 paper.

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482 M. D. Salas

Thedifficultiesthattheseprominentscientistswerehavingoriginated from, as Truesdell has succinctly put it, the insuf-ficiency of thermodynamics as it was then (and often stillnow is) understood [34]. Through much of the first halfof the nineteenth century, particularly in Great Britain, theNewtonian theory of sound, based on Boyles law,p , constant speed of sound and isothermal conditions

was accepted,even though it clearly contradicted experimen-tal observations [7]. For a reenactment of the painful birth ofthermodynamics in the nineteenth century read Truesdellsplay in five acts [33].

The first step in improving the thermodynamics was takenby William Rankine (18201872). Rankine attendedEdinburgh University for 2 years and left without a degree topractice engineering. Around 1848, Rankine started develo-ping theories on the behavior of matter, particularly a theoryof heat. In 1851, before taking the chair of civil enginee-ring and mechanics at the University of Glasgow, Rankinewrites: Now the velocity with which a disturbance of den-

sity is propagated is proportional to the square root, not of thetotal pressure divided by the total density, but of the variationof pressure divided by the variation of density. . .[23].12

Samuel Earnshaw (18051888) studied at St. JohnsCollege, Cambridge and later became a cleric and tutor ofmathematics and physics. In 1860 Earnshaw submitted forpublication to the Philosophical Transactions of the RoyalSociety a paper on the theory of sound of finite amplitudes[6,14].Stokes,inhiscapacityofsecretaryoftheRoyalSociety,asked Thomson, in a letter dated April 28, 1859 [35], toreview the paper. Thomsons review stretches through sevenletters to Stokes, from May 11, 1859 to June 20, 1860 [35].

Thomson is decidedly against publication. In the paper,Earnshaw develops a simple wave solution in one directionforgasessatisfyinganarbitraryrelationbetweenpressureanddensity. Earnshaw works with the Lagrangian formulation ofthe equations to find the relation between the velocity and thedensity for p Like others before him, he observes thatthe differential equations might not have a unique solution.He remarks: I have defined a bore to be a tendency to dis-continuity of pressure; and it has been shown that as a waveprogresses such a tendency necessarily arises. As, however,discontinuity of pressure is a physical impossibility, it is cer-tain Nature has a way of avoiding its actual occurrence.

Thomson has trouble with Earnshaws Lagrangean formu-lation and feels that his aerial bore is a rehash of Stokespaper on A Difficulty in the Theory of Sound. In his lastletter to Stokes on this subject Thomson writes [35]: Onthe whole I think if called on to vote, it would be againstthe publication. . . On speaking to Rankine I found the ideahe had taken from Earnshaws paper,. . . was superposition

12 One step forward, two steps back, see footnote 4.

of transmn vel. On wind vel.: & he thought it good. Thishowever is of course fully expressed in Poissons solution.

Bernhard Riemann (18261866) received his Ph.D. fromthe University of Gttingen in 1851. After Dirichlet died in1857, vacating the chair previously held by Gauss,Riemann became a full professor. Most of Riemanns paperswere in pure mathematics and differential geometry and they

have beenextremelyimportant to theoretical physics. Uniqueamong his contributions is his more applied paper on thepropagation of sound waves of finite amplitude published in1860 [26]. The paper is very easy to read with notation verysimilar to that used today. Early in the paper, Riemann intro-duces what we know today as Riemann variables which hedenotes r and s. For an isentropic gas he writes the governingequations as

() + u =

k + 1

2r +

k 3

2s,

() u =k 3

2

r +k+ 1

2

s,

where k is the ratio of specific heats and = dp/d = a2.Shortly after introducing r and s Riemann describes howa compression wave would necessarily steepen leading tomultiple vales of at one point. Then he says: Now since inreality this cannot occur, then a circumstance would have tooccur where this law will be invalid..., and from this momenton a discontinuity occurs. . . so that a larger value of willdirectly follow a smaller one. . . The compressionwaves [Verdichtungswellen], that is, the portions of the wavewhere the density decreases in the direction of propaga-tion, will accordingly become increasingly more narrow as

it progresses, and finally go over into compression shocks[Verdichtungsstsse]. He derives the jumps in mass andmomentum for an isentropic (reversible) flow andestablishesthat the speed of the shock wave, d /dt, is bounded by

u1 +

(1) >d

dt> u2 +

(2).

Then he discusses what we know today as the Riemann pro-blem, i.e. the wave patterns corresponding to various initialconditions with jumps in u and at x = 0. Riemann, likeStokes before him, failed to understand the true nature of theshock layer. The problem is one of physics, not mathema-

tics, and its solution must wait for a better understanding ofthermodynamics.

2 Resolution

Rankine makes his main contribution in his 1870 paper onthe thermodynamic theory of waves [24] published in thePhilosophical Transactions of the Royal Society of London(Fig 1). Previous papers by Earnshaw and Riemann shared

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Fig. 1 Photo courtesy of Glasgow University Archive Services

some similarities; not so with Rankines paper which wasmore focused on thermodynamics.13 He begins with: Theobject of the present investigation is to determine the rela-tions which must exist between the laws of the elasticity ofany substance, whether gaseous, liquid, or solid, and thoseof the wave-like propagation of a finite longitudinal distur-bance in that substance. Here elasticity is the eighteenth-century term for what we now call pressure. Later he writes:It is to be observed, in the first place, that no substanceyet known fulfills the condition expressed by the equation

dpds = m2 = constant,14 between finite limits of distur-bance, at a constant temperature, nor in a permanency of typemay be possible in a wave of longitudinal disturbance, theremust be both change of temperature and conduction of heatduring the disturbance. Therefore, Rankine by explainingthat the shock transition is a non-adiabatic process, wherethe particles exchange heat with each other, but no heat isreceived from the outside, resolved the objections that hadbeen raised by Rayleigh and others concerning the conser-vation of energy. He goes on to find, for a perfect gas, the

jump conditions for a shock wave moving with speed a intoan undisturbed medium with pressure and specific volume

defined, respectively, by P and S. He writes:

13 In papers written in 1850 and 1851, Rankine developed a theory ofthermodynamics which included an entropy function [33], but it willtake another 15 years for R. Clausius to coin the term and fully developthe concept.14 Rankine denotes by s the bulkiness= 1/ , and by m the massvelocity =u, where u is the velocity relative to the shock wave. Inits discrete form, we call this expression Prandtls relation: [p]/[s] =m2.

m2 =1

S

( + 1)

p

2+ ( 1)

P

2

,

a2 = S

( + 1)

p

2+ ( 1)

P

2

,

u = (p P)

S

( + 1) p2 + ( 1)P2

.

Thomson in a letter to Stokes dated March 7, 1870 [36]writes: I have read Rankines paper with great interest. Thesimple elementary method by which he investigates thecondition for sustained uniformity of type is in my opinionvery valuable. It ought as soon as it is published to be intro-duced into every elementary book henceforth written on thesubject.

Pierre-Henri Hugoniot (18511887) entered the colePolytechnique in 1870. That summer, France declared waron Germany15 and patriotic feelings ran high among thestudents, see Fig. 2. In 1872 Hugoniot entered the marine

artillery service. The artillery service turned Hugoniotsattention to research on the flight characteristics of projec-tiles. In 1879 he was appointed professor of mechanics andballistics at the Lorient Artillery School and 3years later hebecameassistant director of theCentralLaboratory of MarineArtillery. He returned to the cole as an auxiliary assistantin mechanics in 1884.

Here, in the course of a few months, he completed hismemoir On the Propagation of Motion in Bodies whichhe submitted for publication on October of 1885. The publi-cation was delayed because as the editor explains: . . .Theauthor, carried off before his time, was unable to make the

necessary changes and additions to his original text. . . Thememoir appears in two parts. The first is published in 1887[12]; it consists of three chapters. The first begins with anexposition of the theory of characteristic curves for partialdifferential equations of which Hugoniot says: The theoriesset out herein are not entirely new; however, they are cur-rently being expounded in the works of Monge and Ampreand have not, to my knowledge, been brought together toform a body of policy. In the second chapter he sets downthe equations of motion for a perfect gas and in the thirdchapter he discusses the motion in gases in the absence ofdiscontinuities. It is the second memoir, published in 1889

[13], that is most interesting. Chapter four covers the motionof a non-conducting fluid in the absence of external forces,friction and viscosity. Here Hugoniot analysis is very simi-lar to that of Earnshaw. Finally, chapter five examine[s] thephenomena which occur when discontinuities are introducedinto the motion. It is in this last chapter that Hugoniot writesthe famousHugoniot-equation relating the internal energy tothe kinetic energy. It appears in Sect. 150, not in the usually

15 TheFranco-Prussian war lastedfrom July 19, 1870to May 10, 1871.

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484 M. D. Salas

Fig. 2 Photo courtesy ofCollections Archives de lcolePolytechnique, Paris

quoted form,

e e =1

2(p + p)(v v), (4)

but as

p + p1

2=

p1 p

m 1

1

z1 z+

p1z1 pz

m 1

1

z1 z, (5)

where m = , the ratio of specific heats, v = z + 1, and e =pv/( 1). Of course, (4) follows easily from (5), but theelegant way in which (4) connects the three thermodynamicvariables, p, v and e,islostin (5). Equation (4) states that theincrease in internal energy across a shock is due to the workdone by the mean pressure in compressing the flow by anamount v v. Figure 3 shows the three pv relations usedby various authors for = 1.4. Boyles law and Poissonsisentropeare constitutive relations, while the Hugoniot curveestablishes what states are possible across a shock wave.

Later, in Sect. 155, Hugoniot explains that in the absenceof viscosity and heat conduction the conservation of energy

implies that p/m = constant, but that across a shock thisrelation is no longer valid and is replaced by

p1 = p(m + 1) 1

(m 1)

(m + 1) (m 1) 1

.

It is Hadamards Lectures on the Propagation of Waves[10] that brings Hugoniots work to the attention of theCambridge community. In the preface to his Lectures,Hadamard explains that Chapters I through IV were prepa-red during the years 1898 to 1899, but the publication was

Fig. 3 Courtesy of Cornell University Library, Historical MathMonographs, page 183 of P. Duhems Hydrodynamique, elasticite,acoustique, Volume 1

delayed.HealsoacknowledgeshisfriendPierreM.Duhem16

for pointing out the theory of Hugoniot: Dans le cas desgaz, on est, au contraire, conduit la thorie dHugoniot, surlaquelle lattention a t attire depuis quelques annes, graceaux leons dHydrodynamique, Elasticit et Acoustique deM. Duhem. However, Duhems lectures [5] deal primarilywith Hugoniots treatment of waves of small amplitude, see

16 When Jacques Hadamard (18651963) entered the cole NormaleSuprieure in 1885, Pierre M. Duhem (18611916) was a third yearstudent there and the two became close friends. Duhem is well knownfor his work on themodynamics and history and philosophy of science.

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Fig. 4 Opening page toChapter IX of Duhems lectureson hydrodynamics [5]. Here hepresents Hugoniots Method forthe treatment of waves of smallamplitude

Fig. 4, not discontinuities. Hadamard writes the Hugoniot-equation in Chap. IV, Sect 209, of his Lectures:

(p1 + p2)(2 1)

2=

1

m 1(p11 p22),

here = v, and he attributes it to Hugoniot: Telle est larelation quHugoniot a substitue (66) pour exprimer quela condensation ou dilatation brusque se fait sans absorptionni dgagement de chaleur. On lui donne actuellement le nom

de loi adiabatique dynamique, la relation (66), qui convientaux changements lents, tant designe sous le nom de loiadiabatique statique. The adiabatique statique, Eq. (66),that Hadamard mentions is, of course, Poissons isentrope.

3 Acceptance

By 1910, all the principal players, Stokes, Earnshaw,Riemann, Rankine and Hugoniot, had passed away. Thusthe review article by Rayleigh, Aerial Plane Waves of Finite

Amplitude [25], is intended for a new generation of scien-tists.Rayleighdividesthereviewintotwomainparts:Wavesof Finite Amplitude without Dissipation and PermanentRegime under the Influence of Dissipative Forces. The firstpart, aside from a review of the work of Earnshaw andRiemann, is a rehash of his letter to Stokes of 1877. Onceagain he states: . . . I fail to understand how a loss of energycan be admitted in a motion which is supposed to be subjectto the isothermal or adiabatic laws, in which no dissipative

action is contemplated. In the second part of the paper, hereviews Rankines 1870 paper calling it very remarkable. . .although there are one or two serious deficiencies, not to sayerrors. . . He also reviews Hugoniots 18871889 memoirs,thus: The most original part of Hugoniots work has beensupposed to be his treatment of discontinuous waves invol-ving a sudden change of pressure, with respect to which heformulated a law often called after his name by French wri-ters. But a little examination reveals that this law is preci-sely the same as that given 15years earlier by Rankine, a fact

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486 M. D. Salas

whichisthemoresurprisinginasmuchasthetwoauthorsstartfrom quite different points of view. Of the Hugoniot-curvehe says: . . .however valid [it] may be, its fulfillment doesnot secure that the wave so defined is possible. As a matter offact, a whole class of such waves is certainly impossible, andI would maintain, further, that a wave of the kind is neverpossible under the conditions, laid down by Hugoniot, of

no viscosity or heat-conduction. Rayleigh makes two smallcontributions in the article. He shows that the increase in

dQ/ (the entropy) across the shock, for weak shocks, isof the order of the 3rd power of the pressure jump, and heestimatesthat theshock wave thickness forairunder ordinaryconditions is of the order of 13 10

5 cm.We can conclude that our understanding of shock waves

was hampered by three factors: first, a lack of understandingofwhatisanadmissiblesolutiontoapartialdifferentialequa-tion; second, the incomplete knowledge of thermodynamicsat the times; and third, as is evident in Rayleighs paper, thelack of understanding that the shock wave manifested itself

in the inviscid equations as a singular limit17 of the viscous,heat conducting, NavierStokes equations. As a postscript,consider how Lamb perpetuated the folly.

Horace Lamb (18491934) was student of Stokes andMaxwell at Cambridge. Lamb was a prolific writer whoauthored many books in fluid mechanics, mathematics, andclassical physics. His texts were used in British universi-ties for many years. In his Presidential address to the BritishAssociation in 1904, he provided the following insight intohis writings: It is ... essential that from time to time someoneshould come forward to sort out and arrange the accumu-lated material, rejecting what has proved unimportant, and

welding the rest into a connected system. His acclaimed18book Hydrodynamics [15], based on his brief 250 pageTreatise on the Mathematical Theory of the Motion ofFluids of 1879, was first published in 1895 and was thenrevised andexpandeduntil thecurrent 700page 6thedition of1932. In it Lamb discusses the conditions for a discontinuouswave in Sect 284. Lamb mentions the works of Rankine[24] and Hugoniot, as described by Hadamard [10], but hesides with the StokesRayleigh way of thinking: Theseresults are [the jump conditions for mass and momentum],

17 The conceptual leap needed is to see the inviscid shock jumps as theouter limits of the viscous shock layer as viscosity vanishes. Curiously,Stokes made a significant contribution to asymptotic theory with whatwe call today Stokes phenomenon [31], see [8] for an overview.18 The leading treatise on classical hydrodynamics, MathematicalGazette; Difficult to find a writeron any mathematicaltopic with equalclearness and lucidity, Philosophical Magazine; . . .it has become thefoundation on which nearly all subsequent workers in hydrodynamicshave built. The long-continued supremacy of this book in a field wheremuch development has been taking place is very remarkable, and isevidence of the complete mastery which its author retained over hissubject throughout his life, G. I. Taylors eulogy to Lamb, Nature,1934.

however, open to the criticism that in actual fluids the equa-tion of energy cannot be satisfied consistently with (1)[mass conservation] and (2) [momentum conservation]. OfHugoniots result he says in a footnote: . . .the argumentgiven in the text [referring to Hadamards book [10]] is inver-ted. The possibility of a wave of discontinuity being assu-med, it is pointed out that the equation of energy will be

satisfied if we equate expression (10) [ 12 (p1 + p0)(u0 u1)] to the increment of intrinsic energy. On this ground theformula

1

2(p1 + p0)(v0 v1) =

1

1(p1v1 p0v0)

is propounded, as governing the transition from one state tothe other. . . But no physical evidence is adduced in supportof the proposed law.

Today, the Cambridge legacy continues to resonate withHawking, the current Lucasian Chair holder, who writes: Itseems to be a good principle that the prediction of a sin-gularity by a physical theory indicates that the theory hasbroken down, i.e. it no longer provides a correct descriptionof observations [11].

References

1. Airy, G.B.: Tides and waves. In: Encyclopedia Metropolitana.Fellowes, London (1841)

2. Challis, J.: On the velocity of sound. Phil. Mag. 32(III), 494499(1848)

3. Craik, A.D.D.: Theorigins of waterwavetheory. Annu. Rev. Fluid

Mech. 36, 128 (2004)4. Craik, A.D.D.: George Gabriel Stokes water wave theory. Annu.Rev. Fluid Mech. 37, 2342 (2005)

5. Duhem, P.: Hydrodynamique, lasticit, acoustique. Tome Pre-mier, A. Hermann, Paris. (Available online at Cornell UniversityHistorical Mathematics Monographs)

6. Earnshaw, S.: On the mathematical theory of sound, Phil. Trans.R. Soc. Lond. 150, 133148 (1860) (Reproduced in: [14])

7. Fox, R.: The caloric theory of gases from Lavoisier to Regnault.Clarendon Press, Oxford (1971)

8. Friedrichs, K.O.: Asymptotic phenomena, cole Polytechnique inmathematical physics. Bull. AMS 61(6), 485504 (1958)

9. Grattan-Guinness, I.: The cole Polytechnique, 17941850:differences over educational purpose and teaching practice. TheAmerican Mathematical Monthly (Available on line) (2005)

10. Hadamard, J.: Leons surla propagation desondes et lesquationsde lhydrodynamique. A. Hermann,Paris (1903). (Available onlineat the University of Michigan Historical Math Collection)

11. Hawking, S.W., Ellis, G.F.R: The large scale structure of spacetime. Cambridge University Press, Cambridge (1973)

12. Hugoniot, P.H.: Mmoire sur la propagation du movement dansles corps et plus spcialement dans les gaz parfaits, 1 e Partie.J. cole Polytech. (Paris), 57, 397. (Transl. in: [14]) (1887)

13. Hugoniot, P.H.: Mmoire sur la propagation du movement dansles corps et plus spcialement dans les gaz parfaits, 2 e Partie.J. cole Polytech. (Paris), 58, 1125. (Transl. in: [14]) (1889)

14. Johnson, J.N., Chret R.: Classic papers in shock compressionscience. Springer, New York (1998)

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15. Lamb, H.: Hydrodynamics, 6th edn. Cambridge University Press(Dover Publication, S256) (1932)

16. Laplace, P.-S. de: Sur la vitesse du son dans lair et dans leau.Ann. Chim. Phys. (2)3, 238241. (Transl. in: Lindsay, R. B.,(ed.), (1972), Acoustics: Historical and Philosophical Develop-ment, Stroudsburg, Pa., Dowden, Hutchinson and Ross, 181182.)(1816)

17. Ltzen, J.: The prehistory of the theory of distributions.Springer, New York (1982)

18. Ltzen, J.: Eulers vision of a general partial differential calcu-lus for a generalized kind of function. Math. Mag. 56(5), 299306 (1983)

19. Luzin, N.: Function, In: The Great Soviet Encyclopedia. 59, 314334 (Transl. in: The American Mathematical Monthly, Jan. 1998,available online) (1930)

20. Monge, G.: Mmoire sur la construction des fonctions arbi-traries dans les intgrals des quations aux diffrences par-tielles. Mmoires des mathmatiques et de physique prsents alAcadmie. . .par divers savans. . .7, 2e, 267300 (1773)

21. Poisson, S.D.: Mmoire sur la thorie du son. J. cole Polytech.(Paris), 7, 319392. (Transl. in: [14]) (1808)

22. Poisson, S.D.: Sur la chaleur des gaz et des vapeurs, Ann. Chim.Phys., (2)23, 337353 (1823). (Transl. in: Herapath, J. (1823) Onthe caloric of gases and vapours, Philos Mag 62, 328338)

23. Rankine, W.J.M.: On Laplaces theory of sound. Phil. Mag.1(IV), 225227 (1851)

24. Rankine, W.J.M.: On the thermodynamic theory of waves of finitelongitudinaldisturbances.Phil.Trans.R.Soc.Lond, 160,277286.(Reproduced in: [14]) (1870)

25. Rayleigh, L.: Aerial plane waves of finite amplitude. Proc. R. Soc.Lond. A84, 247284. (Reproduced in: [14]) (1910)

26. Riemann, B.: ber die Fortpflanzung ebener Luftwellen vonendlicher Schwingungsweite. Abhandlungen der Gesellschaft der

Wissenschaften zu Gttingen, Mathematisch-physikalischeKlasse, 8, 43 (Transl. in: [14]) (1860)

27. Schwartz, L.:Thorie desdistributions, vols. I, II.Hermann et Cie,Paris (1950, 1951)

28. Sheehan, W., Kollerstrom, N., Waff, C.B.: The case of the pilferedplanet. Sci. Am. (2004)

29. Sobolev, S.L.:Mthodenouvelle rsoudrele problmede Cauchypour les quations linaires hyperboliques normales. Mat. Sbornik43(1), 3972 (1936)

30. Stokes, G.G.: On a difficulty in the theory of sound, 33, (III),349356. (Original plus revisions reproduced in: [14]). (Availableonline at the University of Michigan Historical Math Collection)(1848)

31. Stokes, G.G.: On the discontinuity of arbitrary constants whichappear in divergent developments. Trans. Camb. Phil. Soc. 10,106128. (Available online at the University of Michigan Histori-cal Math Collection) (1864)

32. Stokes, G.G.: Mathematical and physical papers. CambridgeUniversity Press. (Available online at the University of MichiganHistorical Math Collection) (1883)

33. Truesdell, C.: The tragicomical history of thermodynamics, 18221854. Springer, New York (1980)

34. Truesdell, C.: An idiots fugitive essays on science, 2nd Prt.Springer, New York (1984)

35. Wilson, D.B.: The correspondance between Sir George GabrielStokes and Sir William Thompson, Baron Kelvin of Largs, vol. I,pp. 18461869. Cambridge University Press, Cambridge (1990)

36. Wilson, D.B.: The correspondance between Sir George GabrielStokes and Sir William Thompson, Baron Kelvin of Largs, vol. II,pp. 18461869. Cambridge University Press, Cambridge (1990)

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