Theodore Von Kármán, 1881-1963 - Royal Societyrsbm.royalsocietypublishing.org/content/roybiogmem/12/334.full.pdf · Theodore von Karman received his secondary education at the

Theodore Von Kármán, 1881-1963

S. Goldstein

, 334-365, published 1 November 1966121966 Biogr. Mems Fell. R. Soc.

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TH EO D O R E v o n KARMAn

1881-1963

Elected For. M em . R .S . 1946

1. IntroductionT h e o d o r e v o n K a r m a n died in A achen on 7 M ay 1963, four days before his 82nd b irth d ay . H e was a m an o f g rea t in te llectual pow er, a leader in m any fields, especially ae ronau tica l engineering, fluid m echanics, an d solid m echanics. H e was a vigorous p roponen t o f in te rna tiona l co llaboration in science an d a successful an d m uch loved orig inato r an d p a rtic ip an t in its o rganization . H e h ad a genius for friendship an d w arm personal relations.

V on K arm an was born in B udapest on 11 M ay 1881. His fa ther was M oritz von K arm an , an d his m o th er’s m aiden nam e was H elene K onn. His fa ther was Com m issioner o f the M inistry o f E ducation , founder o f the m odern H u n g arian secondary school system, an d a teacher an d professor a t the U niversity o f B udapest. O n bo th his fa th er’s an d his m other’s side he was a d irect descendant o f fam ous scholars and rabbis. For exam ple, on his m o th er’s side he was a g rea t grandson o f Moses ben M enachem M endel K un itz , the au th o r o f a w ork on the Z ohar. P robab ly the m ost famous o f his ancestors is Y ehuda Loew ben Bezalel, the ‘E xalted ’ R ab b i o f P rague, a fam ous six teenth-century scholar, who am ong o ther things was an astronom er an d a friend o f Tycho Brahe, an d who is also credited by legend w ith the creation o f the G olem o f P rague— a splendid ancestor for a renow ned engineering scientist.

2. Budapest, Gottingen, the First World WarT heodore von K arm an received his secondary education a t the

‘M in tag im ndzium ’, w hich was a special dem onstration secondary school founded by his father. I t is said th a t w hen young he was a m athem atical prodigy, an d th a t it was under his fa th er’s influence th a t his interests b roadened and also th a t he studied engineering an d n o t m athem atics.

A t the age o f 21 he g raduated as a m echanical engineer w ith highest honours a t the T echnical U niversity o f Budapest. His thesis, on the m otion o f a heavy rod supported on its rounded end by a horizontal p la te , contains the theory o f the ch ild ’s toy w ith a w eighted spherical base w hich returns to the vertical w hen disturbed.

A fter his g raduation , he stayed a t the T echnical U niversity of Budapest (except for an in te rrup tion for m ilitary service) un til 1906, working w ith

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Professor D onat Banki as an assistant professor of hydraulics and also a research consulting engineer doing theoretical research for G anz and Com pany, m anufacturers of m achinery. D uring this period he wrote his next published paper, on the theory of buckling and compression tests on long slender columns. This paper contains the suggestion th a t the m odulus to be used is th a t corresponding w ith the com puted stress a t failure. This is a subject to w hich von K arm an was to re tu rn w ith significant and successful results.

In 1906 von K arm an , apparen tly a t the urging of his father, left Budapest for G ottingen— the G ottingen o f P randtl, K lein, and H ilbert. He first had a two-year fellowship, and then accepted an invitation from P rand tl to work on a new w ind tunnel for airship research. He obtained his doctorate a t the U niversity o f G ottingen in 1908, was appoin ted Privat D ozent in 1909, and rem ained un til 1912.

In his doctor’s thesis von K arm an was concerned w ith the stability o f struts under compression, loaded beyond the elastic lim it, and he carefully developed the well-known reduced-m odulus theory, which takes into account the unloading th a t occurs over p a rt of the cross-section of a colum n as it begins to deflect under constant load. In this m anner he corrected the earlier tangent-m odulus theory of Engesser, which ignored this unloading, unaw are tha t, in fact, Engesser him self had m ade a sim ilar correction following criticism by Considere of the tangent-m odulus theory. In a d ram atic re tu rn to this subject nearly forty years after he wrote his thesis von K arm an carefully assessed the startling announcem ent by Shanley th a t Engesser’s original tangent-m odulus theory was, after all, correct. W ith characteristic grace von K arm an acknowledged the great im portance of Shanley’s work, and in a few incisive paragraphs th a t were p rin ted as an addendum to Shanley’s paper he presented a m asterful exposition of the new concept o f buckling under variable ra th er than constant load to which Shanley’s work was pertinent.

D uring his G ottingen period, von K arm an m ade other contributions to the theory of stress and strain, including papers on the strength of corrugated tubes, the deform ation of thin-w alled tubes, and the stress distribution in plastic and g ranular m edia. He also m ade the first experim ents on plastic flow and fracture in which specimens were subjected to external hydraulic pressure; these were am ong the first o f m any experim ents by other experim enters directed towards discovery of the rules governing the flow and fracture of solid m aterials under com bined stress. In addition there were, as can be seen from the bibliography, two articles (one w ith L. Foppl) for the Encyklopadie der mathematischen Wissenschaften and four for the Handwortebuch der JVaturwissenschaften. In the article on Festigkeitsprobleme im M aschinen- bau , in the Encyclopadie der mathematischen , Vol. 4, von K arm angave the first presentation of the differential equations for large deflexions of plates. These equations, which bear his nam e, have been used extensively.

In this period he also collaborated with his friend M ax Born on work on

336 Biographical Memoirs



‘Theodore von K&rmanthe v ibrations o f crystal lattices and the theory o f specific heats. T he theory was very successful, in p a rticu la r in explaining certain anom alies in specific h ea t curves a t low tem peratures. In ano ther paper, w ith M ax Born & H . Bolza, rarefied-gas flows an d tem peratu re conditions in rarefied gases were studied. T he m athem atica l problem considered was th a t o f steady gas flow and steady hea t conduction th rough a fixed lattice o f very small solid spheres, thereby avoiding the need for hypotheses on m om entum and energy transfer a t a gas-solid interface. Essentially the d iam eter o f a sphere is taken as very sm all com pared w ith a m ean free p a th for gas-sphere collisions in the gas itself. T he m athem atical p rob lem is then practically iden tical w ith H . A. L oren tz’s form ulation of the problem of the m otion of electrons in metals, an d sim ilar m ethods of solution m ay be used. T he results ob tained agreed, sometimes w ith a suitable in te rp reta tion for the m ean free p a th for gas-sphere collisions, w ith the K nudsen and Knudsen-Sm oluchow ski results for tem peratures and flow in capillary tubes and hea t conduction betw een parallel p lates; they are in fact applicable to a fairly wide class of problem s of flow and tem peratu re distribution in highly rarefied gases. T he p aper concludes w ith an application o f the ideas to obtain a tem perature ju m p a t a solid surface proportional to the norm al tem perature gradient.

M ost im portan t o f all it was during this period th a t von K arm an ’s interest in fluid m echanics m atu red and becam e productive, and his interest in the science of flight and aerodynam ics was born and started to grow. In 1908 he was in Paris, and after a late party near the Sorbonne, instead o f going to sleep, he was taken— at five o ’clock in the m orning, it is said— to see his first aeroplane flight, the successful flight of H enri Farm an at Issy-les-M oulineaux. In addition he observed the m anufacture of lightw eight aero-engines, and sent a report on them for publication for his colleagues in H ungary. In G ottingen, Felix K lein, impressed by the possibility o f the resources of A m erican technology and the need for E uropean industrial efficiency if it was to rem ain com petitive, was insisting on the fullest use o f m athem atics and the basic sciences in engineering. M ore and m ore Ludwig P randtl, w ith the help of his co-workers, was bringing his genius for penetrating to the h ea rt o f a physical or engineering problem or situation to bear on fluid mechanics, which contained potential applications to the science o f heavier- than -a ir flight.

In 1911 von K arm an produced his famous work on the alternating double row of vortices behind a b luff body in a stream . In his book Aerodynamics: selected topics in the light o f their historical development he tells o f the period when every day he saw a doctoral student, K arl H iem enz, trying to produce a steady sym m etrical flow of w ater past a circular cylinder in order to check experim entally the position o f separation of the flow as calculated from boundary-layer theory. But the flow always oscillated, and every day when von K arm an politely asked if the flow was now steady, H iem enz sadly answ ered: T t always oscillates.’ T hereupon von K arm an com puted the stability of rows o f vortices and found instability except for the alternating,

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asym m etrical a rrangem en t w ith a definite ra tio o f the distance betw een the rows an d the distance betw een consecutive vortices. O thers h ad described the vortex p a tte rn before, b u t his was the first theoretical q u an tita tiv e analysis, n o t only o f the p a tte rn b u t also o f the m echanism o f d rag . T he p a tte rn has been know n ever since as a K arm an vortex street. M uch has been done since to refine the p a tte rn , and to find ou t the conditions under w hich it occurs, b u t von K arm an was the first to define a m odel from w hich calculations could be m ade and w hich was sufficiently nearly accurate to be m eaningful, an d to carry ou t the calculations.

T he periodic shedding o f vortices has m any physical an d engineering applications. I t provides the explanation , as R ayleigh showed, o f A eolian tones, and the exciting forces m ay be responsible for oscillations o f structures w hich can have serious consequences w hen resonance occurs. M any years la ter, in 1940, the collapse o f the bridge over the T acom a N arrow s was caused by resonance due to periodic vortices an d flutter. F la t plates had been used as side walls instead o f trusses. These gave rise to shedding vortices, and torsional oscillations o f the bridge developed. T he whole course of events was som ew hat com plicated. V on K arm an was called in as a consu ltan t to investigate the collapse o f the bridge. A erodynam ic tests were conducted a t a num ber o f laboratories, including some a t the California In stitu te o f Technology und er von K a rm an ’s guidance, an d in 1942 the results o f these tests were reported to the Federal W orks Agency. T he repo rt to W ashington S tate o f the com m ittee investigating the failure contained m any of von K a rm an ’s ideas. This was one o f the few times th a t von K arm an acted as a consultant to civil eng ineers; he said he found it a surprising experience in several ways, and quite different from aeronau tical engineering! Years la ter, in 1948, in an in teresting lecture to the civil engineers of F rance concerning aerodynam ic effects in engineering, von K arm an told the story o f the T acom a N arrow s bridge. T he lecture includes a list, going back to 1818, of suspension bridges wholly or p a rtly destroyed by w ind action, the destruction o f the Brighton C hain Pier in 1836 being given special m ention because an eye-witness’s account is ex tan t an d because Scott Russel, in a rep o rt to the R oyal Scottish Society o f Arts in 1839, h ad com pared the structure o f a suspension bridge to the string o f an A eolian harp .

V on K arm an excelled no t only in constructing models from w hich calculations could be m ade b u t in presenting clear physical ideas abou t the phenom ena to w hich the analysis should apply. H e was always interested in experim ental results and fam iliar w ith them , an d stim ulated m any students an d colleagues to conduct experim ents, b u t they all agree th a t he was no t equally skilful w ith the hand ling o f labora to ry equipm ent. M any assert th a t w hen he visited a labora to ry to inspect appara tus, w hich m ight well have been constructed to incorporate his own ideas, there was always a danger th a t there w ould be considerable repairs to be m ade after he left. So probab ly he supervised the conduct by technical assistants o f the experim ental researches described in his papers o f the G ottingen period.




Theodore von Karman 339In 1912 von K a rm an was inv ited to re tu rn to H ungary , to the chair o f

the T heo ry o f M achines a t the College o f M in ing E ngineering in Selmec- b an y a . H ow ever, in the sam e year he was also invited to organize an aerodynam ics in stitu te a t the T echn ical U niversity o f A achen. H e left H u n g ary to becom e Professor o f A erodynam ics an d M echanics and D irecto r o f the A eronau tica l In stitu te a t A achen. His tim e a t A achen was soon afterw ards in te rru p ted , du ring the F irst W orld W ar, b u t his association w ith A achen con tinued u n til 1930.

W hen the F irst W orld W ar started , von K arm an was called to active m ilita ry service w ith the ran k o f lieu tenan t, bu t, as a recognized au th o rity in ae ronau tica l engineering, he was soon m oved to the M ilita ry A ircraft F acto ry a t F ischam end in A ustria. A m ong o ther activities there he developed, w ith collaborators, the first stable hovering captive helicopter. (See N .A .C .A . Technical Note No. 47.) T h e design involved a substan tia l am oun t o f theore tica l work. L ater, in 1921, von K arm an published a well-know n p ap e r on ro tary -w ing aircraft. T his p ap e r was rep roduced in the Aachener Abhandlungen, 2 (1922), w ith some pictures o f the helicopter as an appendix .

3. Aachen, Pasadena, , ParisA fter the F irst W orld W ar von K arm an resum ed the task o f organizing,

bu ild ing-up , an d inspiring the aeronau tical institu te a t A achen. His ow n research interests a t this tim e were m ain ly in fluid m echanics and aerodynam ics. As his own rep u ta tio n grew, so th a t he rose to w orld-w ide em inence, so too d id th a t o f the In stitu te a t A achen. S tudents an d research w orkers cam e from m any countries, a ttrac ted by von K a rm an ’s rep u ta tio n an d by the descriptions they h ad h ea rd o f the social and in te llectual atm osphere a t the Institu te . For m any it was indeed a period o f free, in tellectual, in te rna tiona l, an d joyous creativity.

T o assist actively in in te rna tiona l scientific cooperation an d to prom ote personal friendships am ong scientists from all countries were strong and ab id ing urges in von K a rm a n ’s characte r an d thought. In the form ation o f this intense drive to assist in te rna tiona l co llaboration his fa th er’s influence seems to have p layed a p art. A fter the F irst W orld W ar there was an especial need to re-establish contacts an d friendships am ong scientists in various countries, and von K arm an d id w hat he could. H e ob ta ined the collaboration o f Levi-C ivita, Oseen, an d P rand tl, and invited the leading workers in aerodynam ics an d hydrodynam ics to a conference in Innsbruck in 1922. T he inv ita tion m et w ith an enthusiastic response, an d as a result von K arm an jo in ed w ith others, including in p articu la r J . M . Burgers o f Delft, in organizing m ore form al congresses o f a larger scope, to be held a t regular intervals. T he result was the series o f In te rn a tio n a l Congresses o f A pplied M echanics held a t Delft in 1924, Z urich in 1926, Stockholm in 1930, C am bridge, E ngland, in 1934, C am bridge, M assachusetts, in 1938, Paris in 1946, L ondon in 1948, Is tanbu l in 1952, Brussels in 1956, Stresa in 1960,



and M unich in 1964. V on K arm an was a m em ber o f the In te rn a tio n a l Congress C om m ittee from the beginning un til his death .

T he Innsbruck m eeting was inform al, an d von K a rm a n ’s sister Josephine, called P ipo by all her friends an d his, was m anager, hostess, an d secretary. T heir father h ad died in 1915, an d in 1923 she an d the ir m o ther cam e to live w ith T heodore (T o d o r) ; they m ade the ir hom e in H olland , ju s t across the border from A achen.

V on K arm an travelled w idely an d lectured in m any places. In 1926 cam e the visit to the U .S.A . w hich was shortly afterw ards to m ake a large difference in his life. T he nex t year he travelled rou n d the w orld studying aeronau tica l developm ents, an d lectured in Ind ia , J a p a n , an d C hina.

H e was a consultant, n o t only to Junkers A eroplane W orks (1912-1928) an d Luftschiffbau Zeppelin (1924-1928) in G erm any, b u t also to H andley- Page L td . (1926-1930) in E ngland. His visit to J a p a n in 1927 was a t the inv itation of the K aw anishi A ircraft C om pany, to w hom he acted as consu ltan t from 1927 to 1929. In 1927 he designed the K obe W ind T unnel, the first w ind tunnel bu ilt in Ja p a n . D uring his visit to C hina he was nam ed H onourab le C onsultan t to Tsing H u a U niversity, Peiping; as a result in due course (1936) F ran k W atten d o rf—who was one o f the students d raw n by von K arm an to A achen an d who la ter accom panied him to California— w ent to C hina for two years to teach , an d to design w ind tunnels.

V on K arm an , in the first D aniel an d Florence G uggenheim M em orial L ecture to the F irst In te rn a tio n a l Congress o f the A eronautical Sciences in 1958, has him self re la ted the story, as told to h im by R obert A. M illikan, o f the genesis o f his invitation to the U .S.A . in 1926. In th a t year the D aniel G uggenheim F und began to d istribute grants for the form ation of g raduate schools o f aeronautics, an d M illikan m ade a special trip to visit M r G uggenheim to persuade h im th a t a large g ran t should go to Southern California, w hich M illikan correctly foresaw w ould be an im p o rtan t centre o f A m erican aircraft production . M r G uggenheim prom ised to provide the funds ‘if M illikan w ould bring to Pasadena from Europe someone fam iliar w ith and active in aeronautical research, and especially fam iliar w ith the theoretical side of research’. So von K arm an , accom panied by his sister, w ent to A m erica in Septem ber 1926 on the invitation o f M illikan and G uggenheim . V on K arm an says th a t he was la ter told th a t M illikan had said th a t first he h ad aim ed a t P ran d tl— then he settled on K arm an .

O n his first visit to the U .S.A ., von K arm an ’s assignments were to act as adviser to the California Institu te o f Technology on the establishm ent of aeronautical courses an d consultant on the design o f the G uggenheim A eronautical Laboratories an d the 10-ft w ind tunnel. H e was also to lecture a t the o ther universities an d institutions a t w hich schools o f aeronautics were to be established. In addition to fulfilling these assignments, von K arm an visited W ashington and w ent to D ayton to see O rville W right. T h en he left the U .S.A ., b u t M illikan pressed him to re tu rn as director o f the aeronautical laboratories a t the California Institu te. F inally von K arm an




Theodore von Kdrmdnagreed to divide his tim e between A achen and Pasadena after his 1927 travels; in 1928 he form ally becam e a Research Associate of the California Institu te. G radually the balance swung in favour of Pasadena. To the positive persuasiveness of R obert M illikan and the opportunity of building a w ell-equipped and first-rate research institute provided by the Guggenheim F und was added the negative persuasion of the increasingly dark shadow of Nazism in G erm any. In 1930 he accepted M illikan’s offer of the directorship of the G uggenheim A eronautical L aboratory a t the California Institu te and m oved to the U.S.A . He also accepted the directorship of the G uggenheim A irstrip Institu te a t Akron, O hio, a post which he retained for five years.

V on K arm an again a ttracted students and research workers from m any places, even m ore widely than a t Aachen, and the institute a t Pasadena soon becam e one of the leading schools of the aeronautical sciences in the world.

His m other and sister m oved w ith him to Pasadena, where they established their home. M any of his students and jun io r colleagues have fond memories of visits to th a t home. O ne never knew whom one would m eet in the first p a rt of the evening— businessmen, actors, artists, musicians, writers, and scientists. V on K arm an ’s circle o f friends was wide. Sometimes there were visits to be m ade first, perhaps to private performances or discussions or to meetings w ith o ther friends and meals in restaurants. T hen, a t a late hour, von K arm an w ould work and discuss research questions w ith his jun io r co-workers.

In 1932, two years after he had settled in America, von K arm an was one of the founders of the Institu te of A eronautical Sciences. In 1936 he became a U nited States citizen, and the next year was chosen by the Royal Aeronautical Society, as a prom inent Am erican aeronautical scientist, to deliver the twenty-fifth W ilbur W right Lecture.

T he story of von K arm an ’s interest in rocket research, which led to the form ation of the J e t Propulsion Laboratory and the Aerojet General Corporation, has been told by Frank M alina, one of his students and later director for some time of the J e t Propulsion Laboratory. He tells how von K arm an had given a sym pathetic hearing in early times to estimates of the possibility of propelling a vehicle away from the earth , and had included in his lectures in Ja p a n in 1927 a discussion of the problem s tha t would have to be solved before space flight could become a reality. In 1936, at one of his weekly seminars, W illiam Bollay reviewed the experiments on rocket motors m ade by Eugen Sanger in V ienna, and a newspaper account brought to the laboratory two rocket enthusiasts, Jo h n W. Parsons and E. S. Form an, looking for someone with whom they m ight work. M alina formed a group to begin experiments on high-altitude sounding rockets, and von K arm an agreed th a t they should work a t the Guggenheim Laboratory in Pasadena. T hen in 1938 the N ational Academy of Sciences formed a committee for the study of several problem s of the U.S. Army Air Corps, one of which was to consider how to use some form of auxiliary power to assist the take-off of

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heavily loaded aircraft. V on K arm an was a m em ber o f the com m ittee, and offered his services and those o f the group to investigate the use of rockets. This was the beginning both o f von K arm an ’s own active work on rocket research and je t propulsion, and of the J e t Propulsion L aboratory , which grew into a large undertaking. M alina says the w ord ‘je t ’ was used and no t ‘rocket’ since the la tte r was no t a t th a t tim e a ‘respectable5 word. V on K arm an was director o f the J e t Propulsion L aboratory from 1938 to 1945.

T he efforts o f the group were directed towards developing long-duration solid-propellant rockets for assisted take-off, known as Ja to (jet assisted take-off) units. In 1939-1940 Parsons was m eeting failure after failure because o f excessive build-up of the pressure in the com bustion cham ber. Experts who were called in advised th a t it was not possible to make a pow der rocket w ith a burning tim e of more than two or three seconds; centuries of experience had shown this. V on K arm an a t once began to study the m atter theoretically. H e wrote down the equations, and he and M alina began to work out their im plications. As a result Parsons was told to keep trying. A few m onths la ter a rocket w ith a duration of abou t twelve seconds was ready for flight tests on a piloted aircraft. This was the beginning of the present A m erican large solid-propellant engines.

In 1941, established industry having failed to show interest in rocketry, von K arm an , on the advice o f M alina, took the initiative and in 1942, together w ith five associates, founded the Aerojet Engineering C orporation, la ter called the A erojet-G eneral Corporation. This was the first A m erican corporation specifically engaged in rocket developm ent. T he com pany, reputedly begun w ith an investm ent of $250 from each o f the founders, grew into a very large industrial enterprise.

In 1943 von K arm an organized the first postgraduate course on rocket propulsion a t the California Institu te of Technology. He later advised on the establishm ent o f sim ilar courses elsewhere.

A nother of von K arm an ’s interests— the dynam ics of w ind erosion— led to action a t the G uggenheim L aboratory in 1936. After the destructive dust storms in the M iddle W est of the U nited States, von K arm an offered his help to the Soil Conservation Service of the D epartm ent of Agriculture, and a program m e to study w ind erosion was started in the G uggenheim L aboratory under his guidance in 1936 and continued until A m erica’s entry into the Second W orld W ar in 1941.

His connexion w ith the m ilitary services of the U nited States until about 1938 lay largely in directing research work on their behalf a t Pasadena, bu t from then onwards he was more and more called in to advise on research activities elsewhere. In 1938 he becam e a consultant to the Ballistic Research L aboratory of the U .S. A rm y a t the A berdeen Proving G round, and supervised the design o f the first continuous-flow supersonic wind tunnel in America, which was erected there. He rem ained a consultant to the Ballistics Research Laboratory un til 1952.




!Theodore von KarmanFrom 1939 onw ards he was officially a consultant to the U .S. A rm y Air

Corps and, later, to the U .S. Air Force. O ne o f the first results of his efforts was a change in policy, w hereby the Air Corps was authorized to conduct its own research and developm ent, and he becam e the chief consultant on the design of the 40 000 horse-power 20-ft w ind tunnel a t W right Field, D ayton, O hio.

Feelings o f m utual respect and friendship developed between von K arm an an d G eneral H enry H . A rnold, C om m ander of the A rm y A ir Corps. T heir collaboration was to have a profound effect on the Air Force during and after the Second W orld W ar. Tow ards the end o f the war, in 1944, G eneral A rnold asked von K arm an to form and act as chairm an of a Scientific Advisory G roup to study the use o f science in w arfare in Europe and Ja p an , and to in terp ret the significance o f new technological developments for the future o f the U .S. A ir Force. T he G roup prepared two reports in 1945. T he first, ‘W here we stand5, reported on the results obtained overseas on the technical status in several fields. T he second, ‘Towards new horizons5, was concerned w ith the future. I t dealt w ith the im pact of new developments on air pow er and recom m ended, in effect, a future technical policy for the U .S. A ir Force. Both were massive studies. They served as models for similar reports in la ter years; scientific advisory groups were also later set up by the A rm y and Navy.

T he group was succeeded by the Scientific Advisory Board to the Chief of S taff o f the U .S. Air Force. V on K arm an was C hairm an until 1954 and C hairm an Em eritus un til his death.

His m other died in 1941. She was a very gracious, dignified, and wise lady, w ith a gentle and delightful sense o f hum our, and both Todor and Pipo were devoted to her. They continued to live in the same home in Pasadena after her death.

After the establishm ent o f the Scientific Advisory G roup in 1944 m uch of von K arm an ’s tim e was spent in W ashington on extended leaves of absence from Pasadena. M ore and more of his efforts were devoted to U.S. Governm ent activities, w ith some, as we shall see, directed towards the prom otion of in ternational scientific co-operation. In 1949 he resigned as Director of the Guggenheim Laboratory, w ith the status of a Professor Emeritus of the California Institu te o f Technology. He kept his home in Pasadena; in fact, he kept it right up to his death ; bu t his m ain headquarters were in W ashington, D .C ., until 1951.

A t the Sixth In ternational Congress o f Applied Mechanics in 1946 the form ation of a more formal organization was found desirable, to co-operate w ith the In ternational Council of Scientific Unions, organize colloquia on special subjects, etc. This led to the form ation of the International U nion of Theoretical and A pplied M echanics (Iu tam ), w ith von K arm an as H onorary President, a position he occupied until his death. However, on his advice, the organization of the In ternational Congresses rem ained with the more inform al In ternational Congress Committee, a self-perpetuating body with

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m em bers chosen on personal grounds, n o t com posed o f delegates from m em ber states.

In 1947, w hen the Econom ic an d Social C ouncil o f the U n ited N ations h ad u n d er consideration the possibility o f establish ing U n ited N ations R esearch L aborato ries, von K arm an , w ith the su p p o rt o f Iu tam , p roposed the estab lishm ent o f an in te rn a tio n a l research cen tre for fluid m echanics an d soil m echanics in the M idd le East. Such a research centre was n o t estab lished, b u t the proposal d id lead to the setting-up o f U nesco’s A rid Zone R esearch P rogram m e, w hich becam e one o f U nesco’s successful efforts.

D uring the years in the U .S .A ., an d p a rticu la rly w hile he was D irec to r o f the G uggenheim L aborato ries in P asadena, von K arm an , in ad d itio n to his activities for the F edera l G overnm ent, was engaged in m any consu lting activities for local an d sta te governm ents an d for industry . F rom 1933 to 1937 he was a m em ber o f the special com m ittee ap p o in ted by the U .S . N avy to investigate the A kron an d M acon D irig ib le disasters. H e was a consu ltan t on the developm ent o f p u m p in g eq u ip m en t for the M etro p o litan W ater D istric t o f S ou thern C aliforn ia (1934-1938), an d a consu ltan t on the G ran d Coulee D am P roject (1939-1942) an d the W ash ing ton P lan e ta riu m an d Space C en ter (1961). T h ere is a long list o f industria l concerns for w hich he was a c o n su lta n t: G enera l E lectric, a irc raft firms such as N o rth ro p , Boeing, an d the C onvair D ivision o f G enera l D ynam ics, T ool R esearch an d E ngineering , Allison D ivision o f G enera l M otors, S m ith -P u tn am W ind T u rb in e for D evelopm ent o f E lectricity , an d others. H is reports arising from consulting w ork frequen tly con ta ined im p o rta n t contribu tions.

V on K a rm a n conceived the idea o f a cooperative activ ity am ong the aero n au tica l scientists o f the m em ber nations o f the N o rth A tlan tic T rea ty O rgan iza tion , sim ilar in some w ay to th a t o f the Scientific A dvisory B oard o f the U .S . A ir Force b u t in te rn a tio n a l in characte r. In 1951 he ob ta in ed the coopera tion o f the U .S . A ir Force an d the M ilita ry S tand ing G roup o f N ato . A m eeting o f the directors o f ae ronau tica l research o f the N ato countries took place to discuss m eans o f streng then ing the com m on defence by exchange o f in fo rm ation ab o u t m odern developm ents in aeronau tics an d increasing the activ ity in ae ronau tica l research in the various countries. T he m eeting recom m ended the o rgan ization o f an A dvisory G roup for A eronau tica l R esearch an d D evelopm ent (A gard) to advise N ato on ae ro nau tica l questions. T h e recom m endation was approved by the N ato au thorities. V on K arm an was elected C h a irm an o f A gard , a position he held u n til his dea th . Paris was selected as the site o f A g ard ’s h ead q u arte rs . T he first G enera l A ssem bly o f A gard was held in Paris in 1952; subsequent m eetings have been held in R om e, L ondon, Scheveningen, O ttaw a , Brussels, W ashington , C openhagen, A achen, Is tan b u l, Oslo, Paris, an d A thens. Experts w ere ap po in ted by the m em ber states to form panels to w ork on special topics in research an d developm ent o f p a rticu la r im portance to N ato . Sym posia were held to discuss special technical topics an d the proceedings were published , as were o ther special reports an d m anuals, an d




!Theodore von Karmanconsulting service was provided. A gard did m uch to quicken and help the revival of aeronautical research in Europe, still suffering from the effects of the w ar, and the exchange of ideas and inform ation proved valuable. Sceptics were converted into enthusiasts.

V on K arm an ’s tim e and efforts were now largely devoted to Agard, and m uch of his tim e was spent in Paris. He personally partic ipated in the organization of the panels and symposia, and presented several papers h im self. F rom Paris he continued his frequent travels, to A gard meetings and E uropean capitals, and to W ashington and Pasadena. From tim e to tim e he lectured in E uropean countries and Israel, and in America.

His sister died in 1951. They were devoted to each other. She had constantly accom panied him on his travels, and had in her own way looked after h im and guarded him . N either bro ther nor sister had ever m arried. M any of his closest friends were fearful of the effect her death would have on him. H e was indeed w eary and sick for a long tim e, bu t he did recover so tha t he was almost, if no t quite, his old self. In 1953 he was well enough to give the Messenger Lectures on the Evolution of Civilization a t Cornell University. T he resulting publication, the book on Aerodynamics: selected topics in the light o f their historical development, is dedicated ‘To the m em ory of m y sister Josephine de K arm an whose devoted com panionship secured for me the peace of m ind necessary for scientific thinking.’ Pip6 had a doctorate in the H istory o f A rt from V ienna and was an adm irer of Spanish A rt in particular. T odor established in her m em ory a fund for the publication in M adrid of a large num ber of small volumes on Spanish art.

In 1956 von K arm an ’s efforts towards international co-operation, this tim e directly in the aeronautical sciences, resulted in the foundation of the In ternational Council of the A eronautical Sciences, w ith support from national scientific societies in 27 countries. V on K arm an was H onorary President until his death. The first In ternational Congress of the Aeronautical Sciences, sponsored by the Council, was held in M adrid in 1958. Two others followed during von K arm an’s life, in Zurich in 1960 and Stockholm in 1962. T he Council and Congresses have continued (under the leadership of M aurice Roy).

Among other services von K arm an contributed to international cooperation and aeronautics and positions he held until the time of his death was the chairm anship of the Board of D irection of the T raining Center for Experim ental Aerodynamics in Belgium.

Finally, a t the request of the International A stronautical Federation, von K arm an brought about the creation of the In ternational Academy of Astronautics. He was the Academ y’s President until his death. He also at the same tim e accepted the editorship-in-chief of the periodical Astronautica , which had already several years of publication. He had given support to the Federation when most of his scientific colleagues thought of astronautics as in the realm of science fiction; he knew it would not rem ain so, and said tha t he preferred enthusiasts with im agination to serious scientists w ithout

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im agination. For both the In ternational Council o f the A eronautical Sciences and the In ternational A cadem y o f A stronautics he obtained financial help from the D aniel and Florence G uggenheim Foundation. Activities in connexion w ith both these enterprises gave him m uch pleasure in his last years. This was especially true o f the First In ternational Symposium on the Basic Environm ental Problems of M an in Space, held in Paris in 1962. T he Symposium had the support o f Unesco, the In ternational Atomic Energy Agency, and the W orld H ealth O rganization, and for the first tim e scientists from the U.S.A . and the U .S .S .R . and o ther countries usefully exchanged inform ation on some of the difficult problem s of m anned space flight.

A nniversary volumes were published in von K arm an ’s honour on his 60th, 70th, 75th, and 80th birthdays. D uring the Second W orld W ar, in 1941, an anniversary volume for his 60th b irthday, containing papers by friends and colleagues in the U nited States and C anada, was arranged by a group from the California Institu te of Technology. O n his 70th b irthday several associates and students o f the A achen period organized the publication o f a volum e o f selected essays on engineering mechanics. His 75th b irthday saw the issue in his honour of special num bers of journals both by the Wissenschaftliche Gesellschaft fur Luftfahrt and the Institu te of the A eronautical Sciences. Again, on his 80th b irthday, a special volume was arranged by the editorial board of Astronautica , published by Springer, V ienna, and another volume sponsored by the U .S. A ir Force Office of Scientific Research, was published by the Institu te o f the A eronautical Sciences, New York.

For his 75th b irthday the M ay, 1956, issue of the Journal o f the Aeronautical Sciences, Vol. 23, was dedicated to von K arm an , and ‘in each of the technical areas in which D r von K arm an has m ade m ajor contributions, one of his form er students was invited to subm it a paper sum m arizing the present status of the field’. T he result was a num ber o f the jo u rn a l containing articles on the a rt and science of applied m athem atics, the behaviour o f nonlinear systems, the physics of solids and plastic flow, hydrodynam ics, boundary layers and transition to tu rbu len t flow, the problem of tu rbu len t m otion, flame theory and com bustion technology (aerotherm ochem istry, as von K&rman called it), applied aerodynam ics and flight mechanics, heat transfer, aerofoil theory, inelastic buckling, and supersonic wing theory.

A collection in four volumes of 111 papers th a t he had w ritten by 1951 was arranged in honour of his 70th b irthday, and was published under the title of Collected works in 1956 by Butterworths Scientific Publications o f London. Each paper in this collection is indicated in the bibliography a t the end of this m em oir by the initials C. W, and its num ber in the collection.

4. Aachen, Pasadena, Washingto, Paris: Publications(a) D uring the period a t Aachen, there were m ajor contributions to the

understanding o f the problem s of the frictional resistance of fluids,




Theodore von Kdrmdnboundary -layer calculations, an d tu rbu lence. T he m ost in teresting co n tribu tion to s tru c tu ra l analysis, a t any ra te for engineering practice , was a p ap e r on the effective w id th for a T -beam , the 1924 p ap e r w ith the expressive title ‘D ie m ittragende B reite5. T he effective w id th in this case o f a T -beam is the w id th o f the flange o f such a beam th a t effectively provides the sam e am oun t o f bend ing stiffness as th a t w hich m ust be taken in to account, together w ith the leg o f the T, to provide on a sim ple theory the correct bending stiffness o f the flange-stiffener com bination. V on K arm an clarified the concept an d gave it a nam e, showed how the effective w id th could be approx im ately calcu lated , an d h im self carried th rough the calculation for a sim ple case. In 1925 he published a p ap e r describing the m echanics o f the rolling process in p roducing sheet m etal. O th e r papers du ring this period included two w ith T refftz, the first on aeroplane stab ility an d the second on aerofoil sections, a short in teresting p ap e r on the in te rp re ta tion o f Reynolds num bers in term s o f the kinetic theory o f gases, a theory o f propellers (with Bienen), a well- know n p ap e r on the approx im ate calculation o f po ten tia l flow past bodies o f revolu tion (axial flow by a m ethod o f sources an d cross flow by using a d istribu tion o f doublets), an d one on im pact loads on seaplane floats during landing . N otew orthy also, in a different way, is the p aper von K arm an w rote for the eightieth b irth d ay o f Joseph Popper (‘Lynkeus’).

(b) In the p ap e r on aerofoil sections von K arm an an d Trefftz developed an idea o f K u tta ’s an d used conform ed transform ations to p roduce sections w ith finite angles a t the trailing edge instead o f the cusps o f the classical Joukow ski theory. T hey then briefly discussed tw o-dim ensional aerofoil theory.

In la ter years, during the Pasadena period, von K arm an w rote three papers on three-dim ensional aerofoil theory. T he purpose o f a p aper in 1935 was to develop m athem atically a theory of induced d rag by the m ethods used in discussing wave resistance; the p ap er concludes w ith an elegant application o f Fourier integrals to lifting-surface theory. T hree years la ter, together w ith W . R . Sears, he produced a theory o f the non-uniform m otion o f an aerofoil, the purpose being eto m ake the airfoil theory o f non-uniform m otion m ore accessible to engineers by showing the physical significance o f the various steps o f the m athem atical deductions, and to present the results o f the theory in a form suitable for im m ediate application to certain flutter an d gust problem s’. In 1945 he and H.-S. Tsien considered lifting-line theory for a wing in non-uniform flow. M eanw hile, in 1935, together w ith J . M . Burgers, he h ad co-authored a book on General aerodynamic theory—-perfect fluids. This was the second o f the volumes in the series on aerodynam ic theory edited by W . F. D urand .

(c) V on K arm an ’s 1921 paper on lam inar and tu rbu len t friction discussed m any m atters then curren tly under consideration, and m ade a t least two im portan t advances. I t contains the m om entum equation o f the boundary layer, which, by the use of reasonable assum ed velocity distributions across each section o f the layer, can be used to predict the grow th o f the boundary- layer thickness and the skin-friction, both for lam inar and for tu rbu len t

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flow. This is the basis of the rap id approxim ate ‘K& rm an-Pohlhausen’ m ethod of boundary-layer calculation. T he paper also contains the form of solution for the flow due to the ro tation o f a disk, the resulting ord inary differential equations (later solved num erically by C ochran), and their approxim ate solution by the ‘K arm an-Pohlhausen’ m ethod; tu rbu len t flow in the boundary layer of a ro tating disk was also considered. T he solution o f the rotating-disk problem in the lam inar-flow case is one of the few known solutions of the Navier-Stokes equations w ith non-linear terms. I t is o f some practical im portance in the tu rbu len t case in connexion w ith losses in turbom achinery, though developm ents beyond von K arm an ’s considerations are needed for p ractical use.

Later, in 1934, von K arm an , together w ith C. B. M illikan, retu rned to the question of the calculation of lam inar boundary layers, particu larly the calculation of flow in boundary layers under adverse pressure gradients and involving separation. T he question was the difficult one of finding a sufficiently accurate m ethod of calculation w ithout resorting to full num erical in tegration of the partia l differential equations or other very laborious methods. W hen the pressure grad ien t is adverse the graph of the velocity against distance from the wall has a poin t of inflexion, and von K arm an and M illikan suggested two different approxim ate solutions, one— the ‘ou te r’ solution— being applied in the region from the poin t of inflexion to the ou tside of the boundary layer, and the other— the ‘inner’ solution— in the region between the wall and the poin t of inflexion. In a later paper w ith M illikan the m ethod was used to discuss the variation w ith Reynolds num ber o f the m axim um lift coefficients o f aerofoils. Separation of a tu rbu len t boundary layer was neglected, and approxim ate calculations were m ade to determ ine the position of separation of a lam inar boundary layer a t each lift coefficient; this was then com pared w ith the probable position o f ‘transition’ to turbulence.

(d) T he 1930 papers on m echanical sim ilarity and turbulence contained the first appearance of the famous logarithm ic form ulae for the distribution of the velocity and for the resistance coefficient in tu rbu len t flow in a boundary layer along a flat plate and in straight pipes and channels. V on K arm an obtained the results by introducing the concept of sim ilarity of the tu rbu len t velocity fluctuations a t every po in t; it turns out th a t this is equivalent to keeping the idea of P ran d tl’s mixing length and assuming th a t it depends in the simplest way on the distribution of m ean velocity. T he logarithm ic form ulae were later derived in other, and ra ther simpler, ways, bu t von K arm an was the first to obtain and publish them .

O ne of these 1930 papers on turbulence was a lecture to the T h ird In te rnational Congress of A pplied M echanics a t Stockholm. Von K arm an had lectured on the stability of lam inar flow and turbulence a t the first congress in Delft in 1924. He returned to the subject of turbulence, and particu larly skin-friction in tu rbu len t flow, a t the ‘ship’ conference a t H am burg in 1932, and in a paper in the Journal oj Aeronautical Sciences in 1934. This last paper




Theodore von Karmancontains a discussion o f the influence o f surface roughness on the skin- friction. In 1934 he also lectu red again on the stab ility o f lam in ar flow and tu rbu lence, this tim e to the F o u rth In te rn a tio n a l Congress on A pplied M echanics a t C am bridge. In this lecture there is a significant section on hea t transfer in tu rb u len t shear flow. W hen the P ran d tl n u m b er is n o t unity , R eynolds’s analogy fails. T his was, o f course, know n an d extensions h ad been m ade (by P ran d tl an d T aylor) by considering a lam inar w all layer in w hich no tu rb u len t m ixing occurs as well as a tu rb u len t core in w hich R eynolds’s analogy applies. V on K a rm a n ’s con tribu tion was to consider also a tra n sition layer in w hich h ea t an d m om entum are transferred bo th by tu rb u len t m ixing an d by m olecular processes. T he resulting form ula was given in the lecture, b u t no t its derivation . T he derivation was given in Modern developments in flu id dynamics, b u t in 1939, in a p ap e r in the Transactions o f the American Society o f Mechanical Engineers, von K arm an himself, after a b rie f review o f the subject, published the details o f the calculation and a com parison o f theory an d experim ent.

T he 1934 lecture also includes a section on the transfer o f sedim ent in tu rb u len t flow, w hich contained the first theoretical evaluation o f the d istribu tion o f the various fractions o f sedim ent.

T h e year after the C am bridge Congress G. I. T ay lor published his p ap e r on the statistical theory o f isotropic tu rbu lence, an d w hen von K arm an re tu rn ed to the subject he dea lt m ain ly w ith a general system atic developm en t o f th a t theory in two papers published in 1937 an d one in 1938, the last w ith L. H ow arth . V on K arm an po in ted ou t th a t the correlations betw een two velocity com ponents a t any two points a t a distance r ap a rt are the com ponents o f a tensor, w hich is a function o f the vector distance betw een the points. In the case o f isotropy, the correlation divided by the m ean square velocity depends on ju s t two scalar functions o f the distance r and the tim e t. In an incom pressible fluid, the equation o f continuity yields a rela tion betw een these two scalar functions, so only one is involved. I f the trip le products o f com ponents o f velocities a t the two points are neglected, an equation can then be derived from the equations o f m otion for changes in this single scalar, w hich can be used to ob ta in inform ation abou t the ra te o f decay of the turbulence. T he trip le correlations were first neglected in this way, b u t this is incorrect, as G. I. T aylor po in ted out. V on K arm an in fact explicitly stated th a t if this is incorrect the vortex filam ents w ould have a perm anen t tendency to be stretched or com pressed along the axis o f vorticity, an d though t this was no t the case; T aylor poin ted ou t th a t the facts showed th a t it was, there being a tendency for the vortex filam ents to be stretched on the average. V on K arm an and H ow arth showed th a t the trip le correlation tensor also involves only one scalar function for the case of isotropy in an incom pressible fluid, and th a t the correlation between pressure and velocity is zero in this case. A p a rtia l differential equation connecting the double and triple correlation functions was then derived, and equations for the dissipation o f energy and vorticity deduced. For fu rther discussion

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35° Biographical Memoirscertain assumptions were m ade. These included a self-preserving hypothesis — th a t the graphs of the correlation functions preserve their shape and only the scale changes.

T he year 1937 also saw the publication of von K arm an ’s W ilbur W right Lecture to the Royal A eronautical Society— a general survey of turbulence. In 1938 he m ade some rem arks to the Fifth In ternational Congress o f A pplied M echanics, both on isotropic turbulence and on turbulence in shearing motion. In 1948 he in troduced his proposal for a relation between the transfer function and the spectral density in the spectral theory, together w ith an interpolation form ula for the spectral density; in 1949, together w ith C. C. Lin, he wrote an article on the concept of sim ilarity in the theory o f isotropic turbulence for the Reviews o f Modern ; and in 1951 he andC. C. Lin surveyed the various ideas th a t had been p u t forw ard concerning the types o f sim ilarity in the spectrum or the self-preservation o f the correlation functions, and m ade some definite suggestions.

(e) After his visit to the U nited States, in 1929 von K&rman gave the first calculations o f the im pact loads exerted on seaplane hulls during landing. He had learned o f m easurem ents of im pact pressure in A m erica, and he then obtained the relevant form ulae in his usual simple, approxim ate m anner, considering V-bottom floats w ith a small angle of dead rise, so the apparen t mass could be assumed to be th a t o f a flat plate a t each instant of penetration into the w ater surface. M ore refined calculations by others followed later.

(f) Two of von K arm an ’s earliest papers, one in G erm an and one in H ungarian , are on gasdynamics, bu t his abiding interest in the subject, in connexion w ith high-speed aerodynam ics, began early in his career in California. He was one o f a small group th a t laid the foundations of high-speed aerodynam ics long before the flight of aeroplanes a t the speed envisaged becam e possible. T he practical experience and em pirical knowledge th a t there were came from ballistics, bu t there is no doubt th a t von K arm an was looking forw ard to the piercing of the ‘sound b arrie r5 and flight a t supersonic speeds.

In 1932 von K arm an and N. R. M oore applied the linearized theory of supersonic flow to the m otion o f axially sym m etrical bodies, and introduced in the analysis the idea o f a distribution o f ‘supersonic sources5. By th a t tim e several analytical reports had appeared on two-dimensional supersonic flow, b u t the only published analysis of a three-dim ensional flow was a short note sum m arizing a lecture by Busemann on sym m etrical flow past a cone. G. I. Taylor was working on an exact solution for flow past a cone, and this was known to von Karm&n, b u t no num erical results were known a t the tim e.

By the tim e of the lecture a t the Fifth V olta Congress, in 1935, von K arm an had very m uch in m ind the possibility of the high-speed flight o f aircraft, especially a t large (stratospheric) heights, and said so explicitly. The report had the problem of resistance a t high subsonic and supersonic speeds as its general them e. I t contained som ething of a general review of frictional drag, potential flow and wake resistance a t high subsonic speeds, and wave



!Theodore von Kdrmanresistance a t supersonic speeds, m ore p a rticu la rly for slender bodies o f revolu tion . A discussion o f the lam in a r b o u n d ary layer in the high-speed flow o f a gas was included , m ention being m ade o f the w ork o f Busem ann an d o f F ran k l an d the discussion, p a rticu la rly o f h ea t transfer, by L. Grocco. A first a ttem p t was m ade to discuss the varia tion o f tu rb u len t skin-friction w ith M ach n u m b er a t h igh subsonic an d supersonic speeds. T h e K arm an ogive w ith m in im um w ave d rag for a given m ax im um cross-section was presen ted . F inally the theory o f flow a t very h igh M ach num bers— in w h at is now called the hypersonic range, an d w h at von K arm an called ‘u ltra supersonic5 speeds— an d its connexion w ith N ew ton’s conception o f air resistance w ere discussed.

V on K a rm a n re tu rn ed to the calcu lation o f lam in a r b o u n d ary layers in gases a t h igh speeds in 1938 in a p ap e r w ith H .-S . Tsien. T h e pressure g rad ien t along the surface was still taken as zero, the P ra n d tl n u m b er as un ity , an d the viscosity as p rop o rtio n a l to a pow er o f the absolute tem p era tu re (bu t n o t the square-roo t, as b e fo re ); an im proved m ethod o f calcu lation was used (successive app rox im ation in von M ises’s equation instead o f the m om en tum equation w ith a linear velocity d istribu tion).

T h e 1941 p ap e r in the Journal o f the Aeronautical Sciences opens w ith sta tem ents th a t illustrate von K a rm a n ’s outlook, work, an d influence on the connexion o f fu n d am en ta l an d m athem atica l theory an d engineering p rac tice. ‘A few decades ago the m ost enthusiastic ad m ire r o f m athem atica l analysis w ould n o t have expected th a t p rac tica l engineers engaged in the design o f a irc raft w ould have so m uch use for the m athem atica l theory o f fluid m otion as is the case in m odern ae ronau tica l engineering . . . T h e aero n au tica l engineer is pound ing h a rd on the closed door leading in to the field o f supersonic m otion . . . H e w onders w hether the m athem atica l theory could no t be used as a guide for avoiding a p rem atu re d rop o f aerodynam ic efficiency. T h e presen t p ap e r has the objective o f review ing the present status o f the theory o f com pressible fluids from the p rac tica l standpo in t o f its usefulness for in te rp re ta tio n o f experim ental research an d guidance in design.5 A m ong m any o ther m atters the p ap e r included the K arm an-T sien m ethod o f app rox im ating the pressure varia tion along an aerofoil surface a t h igh subsonic M ach num bers, an d estim ating the critical M ach num ber, a t w hich the local relative a ir velocity a t some p o in t o f the surface reaches the local velocity o f sound. In the sam e year von K arm an gave a presen tation o f the fundam en ta l background an d problem s o f the flow o f com pressible fluids. By 1945 he was w riting (in Faster than sound) th a t ‘there is no evidence th a t the velocity o f sound should constitu te a “ stone w all55 o f despair. To be sure, it w ill be necessary to use all good advice w hich aerodynam ic science, chem istry o f com bustion, and therm odynam ics m ay con tribu te5.

L ater, in his fam ous W righ t Brothers L ecture (published in the Journal o f the Aeronautical Sciences in 1947), he said: ‘I believe we have now arrived a t the stage w here know ledge o f supersonic aerodynam ics should be considered by the aeronau tical engineer as a necessary prerequisite to his art. This

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b ran ch o f aerodynam ics should cease to be a collection o f m athem atica l form ulas an d half-digested, isolated, experim ental results. T he aeronau tical engineer should sta rt to get the sam e feeling for the facts o f supersonic flight as he acqu ired in the dom ain o f subsonic velocities by a long process o f theoretical study, experim ental research, an d flight experience.5 T he lecture itself con tribu ted greatly to the rea liza tion o f the aim so stated. I t contains a g raphic descrip tion o f the physical principles o f supersonic aerodynam ics an d a review o f the subject presented so as to b ring o u t the design im plications, an d concludes w ith a m ethod o f range p red iction for supersonic aeroplanes. T he difficulties associated w ith subsonic leading edges ofsw ept- back aeroplane wings were discussed. A ‘reverse-flow5 theorem — in this case the invariance o f d rag w ith respect to a reversal o f flight d irection in supersonic flight, according to a linearized theory, was arrived at. T he transonic p rob lem was discussed, an d the transonic sim ilarity rules for lift an d d rag were announced . In the sam e year (1947) a separate p ap e r on the analy tical derivation o f the sim ilarity law for transonic flow was published. Also in the sam e year von K arm an lectured on stability an d control a t h igh speeds to a jo in t conference o f the R oyal A eronautical Society an d the In stitu te o f the A eronautical Sciences. In 1950 he published , w ith J . F abri, a short analy tical p ap e r on transsonic flow along a w avy wall. H e was C hairm an o f the Board o f Editors o f the P rinceton U niversity series o f volum es on High speed aerodynamics and je t propulsion, and w rote a section on the general theory o f high speed aerodynam ics in volum e 6, w hich appeared in 1954, an d included a short interesting section on ‘superaerodynam ics5, slip flow, and the valid ity o f con tinuum theory.

In 1955 von K arm an presented a review p ap er— ‘Solved and unsolved problem s o f high speed aerodynam ics5— to the conference on h igh speed aeronautics a t Brooklyn Polytechnic Institu te . In 1958, in the first D aniel an d Florence G uggenheim M em orial L ecture a t the F irst In te rn a tio n a l Congress o f the A eronautical Sciences— ‘Some significant developm ents in aerodynam ics since 19465— he said th a t he had the im pression th a t the goal stated nearly twelve years earlier in his W righ t Brothers lecture had generally been achieved. ‘H ow ever5, he w rote, ‘the honeym oon was short. Soon new problem s were facing the aeronau tical engineer, w ho now adays is pleased to call him self a missile engineer or even a space technologist.5 H e w ent on to give a review o f certain topics in high-speed aerodynam ics and tu rbu lence.

(g) D uring the Pasadena period von K a rm an ’s in terest in struc tu ra l analysis, particu larly in questions o f ‘effective w id th 5 and in buckling problem s, led to the pub lication of im p o rtan t papers. E arly in the period he considered a different kind of effective w id th from th a t for the bending stiffness o f a T -beam , showing how sim ple approxim ate m ethods o f calculation could be found for the effective fraction of the w id th o f a p late betw een stiffeners th a t rem ains effective in carry ing axial loads after the p la te buckles betw een stiffeners. A lternative assum ptions are possible as bases for




Theodore von Karmanthe ap p ro x im ate calculations, an d the resu lt is a k ind o f sem i-em pirical theory , leav ing one co n stan t (a function o f the Poisson ra tio o f the m aterial) to be d e te rm in ed experim entally . H e also gave a short b u t valuab le review o f the analysis o f several typ ical th in -w alled structures.

In 1939 the im p o rta n t p ap e r (w ith H .-S . Tsien) ap p eared w hich, together w ith tw o la te r papers (one w ith L. G. D u n n an d T sien, the o ther w ith Tsien) constitu ted a m ajor b reak th ro u g h in the un d erstan d in g o f the buckling o f th in shells. L in ear theories o f buckling o f spherical shells under ex ternal pressure an d cylinders u n d er ax ial com pression give notoriously un co n servative results. T h e co n tribu tion o f von K arm an an d his co-workers was to show th a t the post-buckling behav iou r o f ideal spheres an d cylinders involved sharp reductions o f ex ternal load, an d th a t only th rough the use o f ‘large-deflexion’ theory could progress be m ade in the analysis o f the buckling streng th o f real shells.

L a te r there were jo in t papers w ith ju n io r au thors on the com puta tion o f the stress d istribu tion in th in -w alled colum ns subjected to variab le tw ist u n d er torsion. F inally the last two published papers th a t I have found re tu rn to the buckling p rob lem for curved shells.

(h) In 1938 von K a rm an published for the benefit o f hyd rau lic engineers the theory o f the supercritical flow o f w ater in open channels as analogous to the theory o f the supersonic flow o f gases in two dim ensions. H e gave general relations for supercritical tw o-dim ensional flow for sm all transverse d istu rbances an d negligible energy dissipation, b u t in such a form th a t the w ork becam e a sp ringboard for la te r w ork on a m ore general theory an d its app lica tion to a n u m b er o f hydrau lic problem s. T h e w ork h ad its origin in a very p rac tica l question, the high-velocity flows in the flood channels o f Los Angeles C ounty after heavy rains, an d subsequent experim ents by K n ap p an d Ip p en , a t the C alifornia In stitu te o f Technology, o f supercritical flow in curved sections o f open channels.

(i) V on K a rm a n ’s theoretical study (together w ith F. J . M alina) o f solid p ro p ellan t rocket m otors was published in 1940, and showed for the first tim e th a t the com bustion process o f a long-dura tion restric ted -burn ing solid p ro p ellan t m otor is no t an unstab le one. T he p ap e r appeared as a Report o f the Jet Propulsion Laboratory o f the California Institute o f Technology. Sum m aries o f two o ther publications o f the J e t Propulsion L abora to ry w hich ap p ear in the Collected works show the kind o f activ ity th a t was requ ired in je t propulsion a t th a t tim e (1943). O ne was on the perform ance an d design o f long-range rocket projectiles; the o ther was a com parison o f various proposed types of je t p ropulsion systems for applications in w hich a large propulsive effort is req u ired for a short or m oderate du ra tion .

(j) In 1950, together w ith G uiseppi G abrielli, aeroplane designer an d d irec to r o f the F ia t aeroplane factories, von K arm an m ade a factual study o f existing vehicles, m oving on the earth , on or in the w ater, an d in the air (including w alking an d ru n n in g m en, cyclists, an d horses, b u t neither fish nor fowl nor sw im m ing m en), com paring the powers required per un it gross

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w eight as a function of m axim um speed. L ater von K arm an said the study was m ade ‘partly for fun, partly from scientific in terest’. Two results were th a t the racehorse has the same specific power as a good battleship a t about the same m axim um speed, and th a t from the po in t o f view described the most prom ising innovation for substantial increases in speed would be the availability for use of new m aterials w ith increased stress/density ratios; titan ium alloys were held to be prom ising a t th a t tim e.

(k) T he 1950 paper, w ith Pol Duwez, on plastic wave propagation had its origin in 1941, bu t publication was delayed by w artim e security requirem ents. I t contains an account of the analysis and of certain experim ents arising from the idea of a plastic wave travelling w ith a velocity equal to the square- root of the slope of the stress-strain curve divided by the density. Duwez has reproduced (in J.Ae. Sci. 23 (1956), 436) the letter sent to him by von K arm an containing the original analysis and w ritten while travelling on the New York C entral R ailroad. T he letter gives, very briefly, the theory of the propagation of plastic deform ation in solids and an explanation of the experim ental results of Duwez and Clark. Duwez mentions unpublished extensions by von K arm an and others of the work published in 1950.

(l) Early in the 1950’s von K arm an began to lecture and write on aerotherm ochem istry, the nam e he gave to the com bined study in all its aspects — gasdynam ical, therm al, and chem ical— of com bustion processes and flame theory. T he aerotherm odynam ic and chem ical-kinetic fundam entals m ust not only be com bined, bu t the relation of all these fundam entals to the com plex processes o f com bustion technology must be clarified. This, w ith com plete m athem atical form ulations and the in troduction of p ractical and sufficiently accurate approxim ations, was the task von K arm an set himself. There were rem arks—for exam ple, on enthalpy transport due to diffusion— in his lectures a t the Sorbonne in Paris in 1951-1952, papers w ith M illan in 1953, w ith Penner in 1953 and 1954, and w ith both in 1956, and a review and clarification of the basic processes to the second A gard Com bustion Colloquium in 1955 (published 1956).

T he first paper w ith M illan discusses in detail the problem of the ‘cold boundary ’ in the theory o f lam inar flames— the result th a t no non-zero solution of the resulting equations exists if the usual principles of chem ical kinetics apply over the whole tem perature range. An ignition tem perature was assumed, below which no chem ical reaction takes place, and im portan t techniques for approxim ate integration of the equations were then used w ith results th a t showed th a t the calculated flame speed was insensitive to the value assumed for the ignition tem perature over quite a wide range. This paper dealt w ith a single global reaction w ithout diffusion of chemical species. The extension to more complex systems of chem ical kinetics and to the inclusion of diffusion was m ade by von K arm an and Penner for two cases in which the knowledge of the kinetic details was reasonably assured.

T he second paper w ith M illan contains the first a ttem pt a t analysis from first principles of the problem of flam e-quenching near a cool surface. O f




considerable interest is von K arm an ’s m ethod of dealing w ith the transition betw een a lim iting state near the surface and th a t sufficiently far away.

(m) D uring the second h a lf of his life von K arm an spoke and wrote from tim e to tim e on the three related m atters of the education of engineers, m athem atics in engineering science, and the im portance of, and correct approach to, applied m athem atics.

H e lectured to m athem aticians on m athem atical problem s arising from engineering, the best-known lecture being the 15th Josiah W illard Gibbs Lecture, ‘T he engineer grapples w ith nonlinear problem s’, published in the Bulletin of the A m erican M athem atical Society in 1940. Together w ith M . A. Biot he wrote a well-known text-book Mathematical methods in ,also published in 1940, which has been translated into m any languages. (A second volume was p lanned b u t never com pleted.)

T hree papers in particu lar describe von K arm an ’s ideas about applied m athem atics, particu larly in relation to engineering science: ‘M athem atik und technische W issenschaften’ (1930); ‘Some rem arks on m athem atics from the engineer’s view point’ (1940); and ‘Tooling up m athem atics for engineering’ (1943). In the first of these there is a specially interesting passage on the influence of Felix K lein in G ottingen on the position of applied m athem atics, and the influence on K lein of his visit to the U.S.A. in 1893. I t was not, von K arm an pointed out, th a t K lein im ported applied m athematics into Europe from Am erica—far from it— there was no serious applied m athem atics in the U.S.A. I t was because K lein believed th a t w ithout applied m athem atics (and physics and chemistry) European industry could no t achieve the efficiency th a t would be required to compete w ith the future enormous resources of A m erican technology. In the second of the three papers m entioned von K arm an specially chose the names of two men ‘closely connected w ith the renaissance of applied m athem atics’— Lord Rayleigh and Felix Klein.

In his later years von K arm an returned frequently to the subject of the education of engineers. For example, the article ‘Atomic engineering?’ (1945) is m uch more concerned w ith education than anything else. He was, for one thing, fighting the still-continuing cam paign against teaching mainly for im m ediate applications.

(n) There are occasional papers of von K arm an’s of interest to historians of aerodynam ics, such as the articles on Isaac Newton and aerodynamics and on Lanchester’s contributions to the theory of flight. (The first Lanchester M em orial Lecture, delivered M ay 1957.)

In 1953, two years after his sister’s death, W. R. Sears invited him to give a series of lectures on the historical development of the science of aerodynamics a t the G raduate School o f A eronautical Engineering of Cornell University. The lectures were published in 1954, dedicated to the memory of his sister, under the title ‘A erodynam ics: selected topics in the light o f their historical developm ent’. To a large extent the volume deals with m atters tha t had occurred during von Karm&n’s lifetime, in m any of which

Theodore von Karmdn 355



he took part, and the result is a lively discussion, frequently lightened by anecdotes. T he lectures were the 1953 M essenger Lectures on the Evolution of Civilization, given under a foundation w hich provided annually for ‘a course or courses of lectures on the evolution of civilization, for the special purpose o f raising the m oral standard of our political, business, and social life’.

(o) Late in his life von K arm an ’s interests extended to m agnetofluid- mechanics, and several papers in the last five years of his life were on this subject. In 1961 he gave three invited public lectures a t the Institu te for Fluid Dynamics and Applied M athem atics a t the U niversity of M aryland, which form ed the basis for a small book, From low speed aerodynamics to astronautics, published in 1963, and the last lecture was on M agnetofluid- dynam ics. T he o ther two are entitled Aero therm odynam ics and Aero- therm ochem istry, and there is a short (four-page) section in the book on rarefied gasdynamics. V on K arm an wrote in the in troduction th a t ‘this booklet consists really of a series of review articles, w ritten for non-specialisls bu t trying to satisfy the specialists to o \ T he book was published posthumously, b u t von K arm an com pleted his work on it four m onths before his death.

(p) V on K arm an ’s published papers do not describe all his original scientific contributions. M any arose from his activities as a consultant for industrial firms and rem ain buried in the files of the firms. T here is no record of these industrial reports. As w ith all good teachers m any of his ideas were developed by his students and ju n io r colleagues— but von K arm an was a very good and inspiring teacher and inspired m any, m any students and ju n io r colleagues. O ther contributions m ay still be hidden in his correspondence. T here is no t even a sure list of all his com pleted m anuscripts, even ap a rt from industrial and restricted reports. 5

5. ConclusionV on K arm an received m any honours and awards during his lifetime. He

was a C om m ander of the Legion d ’H onneur of France, a G rand Officer of the O rder ‘al M erito della R epubblica’ of the Ita lian Governm ent, and G rand Officer o f the O rder of O range-N assau of the N etherlands. From France he also received the G rand Cross of the ‘O rdre du M erite pour la Recherche et 1’Invention’, from Spain the G rand Cross of M erit for Aeronautics and the G rand Cross of the Civil O rder of Alfonso X el Sabio, from W est G erm any the Federal G rand Cross for M erit w ith Star, and from the K ing of Greece the Cross of C om m ander of the O rder of George the First. In the U.S.A . he was aw arded the M edal for M erit in 1946, the M edal of Freedom in 1956, and the first N ational M edal of Science in 1963.

In addition he received a t least thirty-tw o medals and other awards, twelve of them described as gold medals. These included, for example, the Franklin Gold M edal, the Lord Kelvin Gold M edal, the Gold M edal of the Royal A eronautical Society, the Daniel Guggenheim Gold M edal, the




Theodore von KdrmdnC hristopher Colum bus Gold M edal, and the Jam es W att In ternational Gold M edal.

By the end of 1961 he had received honorary degrees from a t least twenty- nine universities in Belgium, France, G erm any, Greece, H ungary, Israel, Ita ly , the N etherlands, Spain, Switzerland, Turkey, and the U.S.A.

He was elected a Foreign M em ber of the Royal Society in 1946. In addition to being a m em ber of the N ational Academ y of Sciences o f the U .S.A . and the A m erican Philosophical Society, he was also a m em ber o f the national academ ies of France, Italy , and Spain, and of the Pontifical A cadem y of Sciences, and was an H onorary Fellow, H onorary M em ber, or Foreign M em ber of over tw enty other academies and professional societies in Belgium, C anada, France, Ind ia , Israel, Italy , Spain, Switzerland, and the U .S.A . He was a m em ber of m any other A m erican societies and clubs, since he had wide interests and a gregarious nature.

W hen the V on K arm an Gas Dynamics Facility a t the Arnold Air Force Station in Tennessee was renam ed in his honour on 30 O ctober 1959, he said in his acceptance speech: ‘There is no doubt in my m ind th a t this is the greatest honor th a t I have ever experienced. But I would have thought th a t the Air Force would have w aited a few years un til I had the occasion to look back a t this from beyond infinity. I th ink th a t , th a t is really the tradition, th a t the scientist or statesm an— if something gets dedicated in his nam e— should already be dead. But I am glad the Air Force m anaged an exception and gave me the opportunity to accept this honour . . . *

O n 8 February 1963 he received from President K ennedy a t the W hite House in W ashington the first N ational M edal of Science, aw arded in recognition of his ‘incom parable contributions not only in the field o f applied mechanics, aerodynam ics and astronautics, and to education in general, b u t also to industrial, national, international, and hum an affairs in their broadest sense5. ‘I know of no m an,5 said President K ennedy, ‘who so completely represents all the areas involved in this m edal—science, engineering, and education.5

Von K arm an then returned to Paris to continue his guidance of the work of the Advisory G roup for A eronautical Research and Developm ent of N ato and of the In ternational Academy of Astronautics. His plans included visits to and attendance at scientific meetings in Paris, W arsaw, Oslo, Athens, W ashington, and Pasadena during the four months Ju n e to Septem ber 1963. All the urgings of his assistants and jun io r colleagues th a t he should curtail his activities had been of no avail, bu t he did agree to go in A pril to Aachen to rest. From the beginning of M ay he weakened rapidly. O n 3 M ay he had a heart attack; this led to pneum onia, and he died on the m orning of 7 M ay. Late on the previous day word reached his closest friends in America tha t the end was near. The next m orning his death was announced a t official meetings in W ashington. His body was flown by the U.S. Air Force to California to be buried, a t his own wish, near his m other and sister.

357



T he record of his writings and activities does m uch to show his creative scientific ability, his success in finding the essential physical elements in com plicated engineering situations, and the clarity, sim plicity, and elegance w ith w hich he in itiated and continued the closing of gaps between theory and engineering practice. T aken together w ith the history of events and procedures during his long lifetime, they show m uch of his influence in reducing the gulf between sterile theory on the one hand and pure em piricism on the other, and also on the fostering of in ternational scientific co-operation.

Nevertheless, the factual record falls short by a wide m argin of giving a correct picture not only of the m an b u t also of his influence and contributions. M ention has already been m ade o f the scientific contributions not to be found in his published writings. In his scientific conversations he bubbled w ith ideas, b rought a feeling of excitem ent and adventure in scientific exploration, discussed w ith clarity, and yet spoke w ith his students and juniors as if they were all his equals.

He was noted as a w it and raconteur. His sayings and stories were passed round the world am ong his large circle of friends, and given wide oral circulation. Several have been published. I f a more complete collection is ever published, it would give pleasure both to m any who do no t know them and to those who cherish them w ith fond memories.

His friends were m any and their interests and activities varied. He was deeply interested in every creative hum an activity. He was interested in and friendly w ith people from all walks of life. H e had an am azing m em ory for people. W henever he m et them , he w anted to know about the welfare not only of all his past acquaintances and all his form er students, b u t also of every m em ber of their families as well. He never offended the dignity of any hum an being. His genius for friendship and his wide travels probably did as m uch for in ternational collaboration as the organizations he created or helped to create.

M any notices and memoirs about von K arm an have appeared from which I have derived m uch help, especially the notices p repared by D r H ugh D ry den for the N ational Academ y of Sciences and for Applied Mechanics Reviews, and by D r W attendorf and D r M alina for Astronautica Acta. I am

m uch indebted to these authors for showing me these articles and for permission to use the m aterial in them . W ith so m uch w ritten about von K arm an repetition has been unavoidable.

O n special points I gratefully acknowledge the assistance received orally or by correspondence from Professors Budiansky, Pai, Penner, R annie, Scholem, Sears, and Singer, and Drs M alina and W attendorf. S.


S. G o ld st ein



Theodore von Karmdn 359

B IB L IO G R A P H Y

T h e in itia ls C .W . fo llow ed by a n u m e ra l in d ica te p u b lica tio n in th e Collected Works p u b lish e d b y B u tte rw o rth s in 1956.

1902. G om bo lyu vegevel vizszintes la p ra tam aszkodo sulyos p a lcza m ozgasa . (T h e m o tion o f a h eav y ro d su p p o rte d on its ro u n d e d en d by a h o rizo n ta l p la te .) M ath. Phys. Lapok, 11, 34-41; 69-78; 134-140. £ Jou o f Mathematics and , Mathematical

Society, Budapest.] (C .W . 1)1906. A k iha jlas e lm ele te es hosszu ru d a k o n vegze tt nyom as-kiserletek . (T h e th eo ry o f

bu ck lin g a n d com pression tests on long slender co lum ns.) Magyar M em .K ozl. 40 , 329-334. [ , Journal o f the Society o f Hungarian Engineers and Architects.]

(C .W . 2)1907. O b e r s ta tio n a re W ellen in G asstrah len . Phys. Z - 8, 209-211. (C .W . 3)1908. A gozok es gazok a ram lasi je lensegeire vonatkozo u ja b b v izsgalatok. (R ecen t in

vestigations reg a rd in g th e flow p h en o m en a o f v apou rs a n d gases.) Magyar M em . tp it.-E g y l. K ozl. 42 , 103-110. (C .W . 4)

1908. Ig e n k o n n y u m o to rok ro l. (V ery ligh t-w eigh t engines.) Magyar M em . Epit.-Egyl.Heti-Ertes. 27 , 248-251. [ Bulletin o f the Society o f Hungarian Engineers and Architects.]

(C .W . 5)1908. D ie K nickfestigkeit g e rad e r S tabe . Phys. 9, 136-140. (C .W . 6)1909. (W ith A . H a a r .) Z u r T h eo rie d e r S p an n u n g szu stan d e in p lastischen u n d san d artig en

M ed ien . Gott. Nachr. 204-218. (C .W . 7)1909. H u llam os tuzcsovek szilardsaga. (T h e s tren g th o f co rru g a ted fire tubes.) Magyar

M em . Epit.-Egyl. Heti-Ertes. 28 , 302-305. (C .W . 8)1910. M ito l ffigg az an y ag igenybevete le ? (W h a t determ ines th e stress-strain b eh av io u r o f

m a tte r? ) Magyar Mern. Epit-Egyl. K ozl. 44 , 212-226.1910. U n te rsu ch u n g en u b e r d ie B edingungen des B ruches u n d d e r p lastischen D efor

m a tio n , insbesondere bei quasi-iso tropen K o rp e rn . H ab ilita tio n sch rift, G o ttingen .1910. U n te rsu ch u n g en fiber K nickfestigkeit. M itt. Ver. dt. Ing. 81 . (C .W . 9)1910. F estigkeitsprob lem e im M asch in en b au . Encyk. M ath. Wiss. 4 , 311-385. (C .W . 10)1911. Festigkeitsversuche u n te r allseitigem D ruck . Z- dt. Ing. 55, 1749-1757. (C .W . 12)1911. U b e r d ie F o rm an d e ru n g d fin nw and iger R o h re , insbesondere fed ern d er Aus-

g le ich rohre . Ver. dt. Ing. 55, 1889-1894. (C .W . 13)1911. U b e r d ie T u rb u le n z re ib u n g versch iedener Flfissigkeiten. Phys. Z- 12, 283-284.

(C .W . 14)1911. U b e r d en M echan ism us des W iderstandes, d en ein bew egter K o rp e r in einer

Flfissigkeit e rfah rt. 1 T eil. Gott. Nachr. p p . 509-517. (C .W . 15)1912. U b e r d en M echanism us des W iderstandes, den ein bew eg ter K o rp e r in einer

Flfissigkeit e rfah rt. 2 T eil. Gott.Nachr. p p . 547-556. (C .W . 16)1912. (W ith H . R u b a c h .) U b e r d en M echanism us des Flfissigkeits- u n d L uftw iderstandes.

Phys. Z• 13, 49-59. (C. W . 17)1912. (W ith M . B o r n .) U b e r Schw ingungen in R au m g itte rn . Phys. Z ' 13, 297-309.

(C .W . 18)1913. (W ith L . F o p p l .) Physikalische G ru n d lag en d e r Festigkeitslehre. Encyk. M ath. Wiss.

4 , 695-770. (C .W . 11)1913. (W ith M . B o r n .) Z u r T h eo rie d e r spezifischen W arm e. Phys. • 14, 15-19. (C .W . 19)1913. (W ith M . B o r n .) U b e r d ie V erte ilu n g d e r E igenschw ingungen von P u n k tg itte rn .

Phys.Z- I 4 , 65-71. (C .W . 20)1913. N aherungslosungen von P rob lem en d e r E lastiz ita tstheorie . Phys. Z- 14, 253-254.

(C .W . 21)



q6o Biographical Memoirs1913. (W ith H . B o l za & M . B o r n .) M oleku larstrom ung u n d T em p era tu rsp ru n g . Gott.

Nachr. pp . 221-235. {C.W. 22)1913. E lastiz ita t. Handbuch Naturw. 3, 165-193. 23)1913. Festigkeit. Handbuch Naturw. 3 , 1014-1030. ( . 24)1913. G leichgew icht. Handbuch Naturw. 4 , 245-261. {C.W . 25)1913. H a rte u n d H artep ru fu n g . Hamdbuch Naturw. 5, 198-202. {C.W . 26)1914. (W ith E . T r e f f t z .) f ib e r L angsstab ilita t u n d L angsschw ingungen von F lugzeugen.

J b . wiss. Ges. Luftf. 3, 116-138. {C.W . 27)1915. V izsgalatok a rugalm assagi h a td r es a tores felteteleirol. (R esearch on the conditions

of elastic lim it a n d ru p tu re .) M at. es ertesito.1916. D as G edach tn is d e r M aterie . Naturwissenschaften4, 489-494. {C.W . 28) 1918. (W ith E . T r e f f t z .) P o ten tia lstrom ung u m gegebene T rag flachenquerschn itte .

Z.Flugt. Motorl. 9 , 111-116. {C.W . 29)1918. Lynkeus als In g en ieu r u n d N aturw issenschaftler. Naturwissenschaften, 6 , 457-463.

{C.W . 30)1921. D ie B edeutung der M echan ik fu r das S tud ium der technischen Physiker. techn.

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{C.W.45)1927. Selected problem s in aeronau tics (in Jap an ese ). Aero. Res. Inst. Tokyo, 37, 353-410.1927. Ideale Flussigkeiten. In Die Differential- und Integralgleichungen der Mechanik und

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1928. D ie Schleppversuche m it langen V ersuchsflachen u n d das A hnlichkeitsgesetz d erO berflachenreibung . Werft, Reederei, Hafen, 9 . {C.W. 46)

1928. M athem atische P roblem e der m odernen A erodynam ik. A tti Congr. Internaz. Matem.Bologna, pp . 1-14. {C.W . 47)

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Theodore von 3611929. (W ith K . F r ie d r ic h s .) Z u r B erechnung fre itragender Fliigel. Z- angew. M ath. Mech.

9 , 261-269. (C .W . 49)1929. (W ith F . L . W a t t e n d o r f .) T h e im p ac t on seaplane floats d u rin g land ing .

Tech. Note 321. {C.W . 50)1930. M ath em atik u n d technische W issenschaften. Naturwissenschaften, 18, 12-16. {C.W . 51)1930. M echanische A hnlichkeit u n d T u rb u len z . Gott. Nachr. pp . 58-76. {C.W. 52)1930. M echanische A hn lichkeit u n d T u rb u len z . Proc. Third Intern. Congr. Appl. Mech.

Stockholm, 1,85-93. {C.W . 53)1931. D ie Seitenw ege d e r L u ftfah rt. Z- Flugt. Motorl. 22, 481-487. {C.W. 54)1932. (W ith E. E . Se c h l e r & L. H . D o n n e l l .) T h e streng th of th in p lates in com pression.

Trans. Amer. Soc. Mech. Engrs, A ppl. M ech. Section, 54, 53-57. {C.W . 55)1932. (W ith N . B. M o o r e .) R esistance o f slender bodies m oving w ith supersonic velocities,

w ith special reference to projectiles. Trans. Amer. Soc. Mech. Engrs, 54, 303-310. {C .W . 56)

1932. T heo rie des R eibungsw iderstandes. Konferenz iiber hydromechanische Probleme des Schiffsantriebs, Hamburg, pp . 50-73. {C.W . 57)

1932. Q uelques problem es actuels de l ’aerodynam ique. Journees Techniques Internationales deVAeronautique,Paris, pp . 1-26. {C.W . 58)

1933. Som e aerodynam ic problem s of airships. The Daniel Guggenheim Airship InstitutePublication N o. 1, p p . 45-52. {C.W. 59)

1933. Analysis o f some typical th in-w alled structures. Amer. Soc. Mech. Engrs Aero. Engrg, 5,155-158. {C.W . 60)

1934. T u rb u len ce a n d skin f r ic t io n .^ . Aero. Sci. 1, 1-20. {C.W. 61)1934. T u rb o len za e a ttr ito superficiale. L* Aerotecnic 14, 1156-1184.1934. (W ith G. B. M il l ik a n .) T h e use of the w ind tunne l in connection w ith aircraft-

design problem s. Trans. Amer. Soc. Mech. Engrs, 56, 151-166. {C.W. 62)1934. (W ith G. B. M il l ik a n .) O n the theory o f lam in ar b o u ndary layers involving sep ara

tion. N .A .C .A . Report N o. 504. (C .W . 63)1934. Som e aspects of the tu rbu lence problem . Proc. Fourth Intern. Congr. App. Mech. Cam

bridge, pp . 54-91. [See also the rep o rt o f the Berkeley m eeting o f the A eronautics a n d H ydrau lics D ivision o f the A m er. Soc. M ech. Engrs (1934).] (C .W . 64)

1935. (W ith J . M . B u r g er s .) G eneral aerodynam ic theory : perfect fluids. V ol. 2 o fAerodynamic theory (Ed. by W . F . D u ran d ). B erlin: Springer.

1935. (W ith G. B. M il l ik a n .) A theoretical investigation o f the m axim um -lift coefficient. J.Appl. Mech. 2, 21-27. (C .W . 65)

1935. N eue D arstellung der T ragflugel theorie. Z- angew. M ath. Mech. 15, 56-61. (C .W . 66) 1935. T h e p rob lem of resistance in com pressible fluids. Fifth Volta Congr. Rome. (C .W . 67)1937. O n the statistical theory of tu rbu lence. Proc. Nat. Acad. Sci. Wash. 23, 98-105.

(C .W . 68)1937. T h e fundam entals of the statistical theory o f tu rbu lence. J . Aero. Sci. 4, 131-138.

(I ta lian version in UAerotecnica, 17.) (C .W . 69)1937. T urbu lence. J.Roy. Aero. Soc. 41, 1108-1141. (C .W . 70)1938. [W ith L. H o w a r t h .) O n the statistical theory o f isotropic tu rbu lence. Proc. Roy. Soc.

A, 164, 192-215. (C .W . 71)1938. E ine praktische A nw endung der A nalogic zwischen U berschallstrom ung in G asen

u n d iiberkritischer S trdm ung in offenen G erinnen. Z- angew. M ath. Mech. 18, 49-56. (C .W . 72)

1938. (W ith H .-S . T sien .) B oundary layer in com pressible fluids. J . Aero. Sci. 5, 227-232. (C .W . 73)

1938. (W ith W . R . Se a r s .) Airfoil theory for non-uniform m otion. Aero. Sci. 5, 379-390. (C .W . 74)

1938. Som e rem arks on the statistical theory of turbulence. Proc. Fifth Intern. Congr. Appl. Mech. Cambridge, Mass. pp. 347-351. (C.W . 75)



1938. (W ith F. J . M a l in a .) A series o f lectures on aeronau tics. Popular N atio n a lA lliance, Inc . N ew Y ork. N o. 2, 93-96 a n d elsew here, w ith various titles.

1939. T h e analogy betw een fluid friction a n d h e a t transfer. Trans. Amer. Soc. Mech. Engrs,61, 705-710. (C .W . 76)

1939. (W ith H .-S . T s ie n .) T h e buckling o f spherical shells by ex terna l pressure. J . Aero. Sci. 7, 43-50. {C.W . 77)

1939. U se o f o rthogonal functions in s tru c tu ra l problem s. Contributions to the Mechanics o fSolids, Timoshenko 60th Anniversary Vol. pp . 114-124. {C.W . 78)

1940. (W ith M , A. B io t .) Mathematical methods in engineering. N ew Y ork : M cG raw H ill.1940. Som e rem arks on m athem atics from the eng ineer’s v iew point. Mech. Eng. 62, 308-310.

{C.W . 79)1940. (W ith L. G . D u n n & H .-S . T s ie n .) T h e influence o f cu rv a tu re on th e buckling

characteristics o f structu res. J . Aero. Sci. 7, 276-289. {C.W . 80)1940. T h e engineer grapples w ith non linear problem s. Bull. Amer. M ath. Soc. 46, 615-683.

{C.W . 81)1940. (W ith F. J . M a l in a .) C haracteristics o f the ideal solid p ro p e llan t rocket m otor. J e t

Propulsion Laboratory, Calif. Inst. Techn., Rept. N o. 1-4. {C.W. 82)1941. (W ith H .-S . T sien .) T h e buckling o f th in cy lindrical shells u n d e r ax ial com pression.

J . Aero. Sci. 8 , 303-312. {C.W . 83)1941. C om pressibility effects in aerodynam ics. J . Aero. Sci. 8 , 337-356. {C.W. 84)1941. Problem s o f flow in com pressible fluids. U . of P enna. B icentenn. Conf. Fluid mechanics

and statistical methods in engineering, p p . 15-39. {C.W. 85)1942. Isaac N ew ton a n d aerodynam ics. J . Aero. Sci. 9 , 521, 522, 548. {C.W . 86)1943. T ooling u p m athem atics for engineering. Quart. Appl. M ath. 1, 2-6. {C.W . 87)1943. T h e role o f fluid m echanics in m odern w arfare. Proc. Second Hydraulics Conf. Iowa,

pp . 15-30. {C.W. 88)1943. (W ith H .-S . T sien & F. J . M a l in a .) T h e possibilities o f long-range rocket projectiles.

J .P .L . {Jet Propulsion Laboratory, Calif. Inst. Techn.) Memo. N o. 1. (S um m ary in Collected Works, 4 , 206-208.) {C.W. 89)

1944. (W ith F. J . M a l in a , M . S u m m e r f ie l d , & H .-S . T s ie n .) C om para tive study o f je tpropulsion systems as app lied to missiles an d transonic aircraft. J .P .L . Memo. N o. 2 (S um m ary in Collected Works, 4 , 209-211.) {C.W . 90)

1944. (W ith N . B. C h r is t e n s e n .) M ethods o f analysis for torsion w ith variab le tw ist.J.Aero. Sci. 11, 110-124. {C.W . 91)

1945. (W ith H .-S . T s ie n .) Lifting-line theory for a w ing in non-un iform flow. Quart. Appl.M ath. 3 , 1-11. {C.W.92)

1945. A tom ic eng ineering? Mech. Eng. 67, 672, 679. {C.W. 93)1945. F aster th a n sound. J.Wash. Acad. Sci. 35, 144-155.1946. (W ith W .-Z . C h ie n .) T orsion w ith v ariab le tw ist. Aero. Sci. 13, 503-510. {C.W . 94)1946. O n lam in ar an d tu rb u len t friction. N .A .C .A . Tech. Memo. 1092.1947. Supersonic aerodynam ics— principles an d applications. J . Aero. Sci. 14, 373-409.

{C.W . 95)1947. T h e sim ilarity law of transonic flow. J . M ath. Phys. 26, 182-190. {C.W. 96)1947. T heo re tica l considerations on stab ility an d contro l a t h igh speeds. Proc. Joint Aero.

Conf. Roy. Aero. Soc. and hist. Ae. Sci. London, pp . 19-36. {C.W. 97)1947. S and ripples in the desert. Technion Yearbook, pp . 52-54. {C.W. 98)1947. Aerothermodynamics. C olum bia U niversity Lectures. (Lectures in the D ept, o f Physics

p rep a red by W illiam Perl.)1948. S ur la theorie statistique de la tu rbu lence. C.R. Acad. Sci. Paris, 226, 2108-2111.

{C.W. 99)1948. (W ith J . V a l e n s i.) A pplication de la theorie de la couche lim ite au probldm e des

oscillations d ’u n fluide visqueux et pesan t dans u n tu b e en U . C.R. Acad. Sci. Paris, 227, 105-106. {C.W. 100)




Theodore von Kdrmdn1948. Progress in th e s ta tis tica l th eo ry o f tu rb u len ce . Acad. Sci. Wash. 34 , 530-

539. (See also J . Marine Res. 7, 252-264; 116 (1951), 43-52.)(< C.W. 101)

1948. L ’ae ro d y n a m iq u e d ans P a r t de l ’ingen ieu r. M em . Soc. Ing. Civ. , p p . 155-178.{C .W . 102)

1948. Progress in av ia tio n . J . Franklin Inst. 246, 451-452. (C .W . 103)1949. (W ith C . G. L in .) O n th e co n cep t o f s im ilarity in th e th eo ry o f iso trop ic tu rb u len ce .

Rev. M od. Phys. 21 , 516-519.1949. O n th e th eo ry o f th ru s t au g m en ta tio n . Reissner Anniversary Vol. Contributions to Applied

Mechanics, p p . 461-468. A n n A rb o r, M ic h .: E dw ards.1949. A ccelera ted flow o f a n incom pressib le flu id w ith w ake fo rm ation . Ann. matem. pur. e.

appl. 29 , 247-249. (C .W . 104)1950. (W ith G . G a b r ie l l i .) W h a t p rice speed? Specific pow er re q u ire d for p ropu ls ion o f

vehicles. Mech. Eng. 72, 775-781. (See also , 118 (1952), 43-60.)(C .W . 105)

1950 (W ith P . D u w e z .) T h e p ro p a g a tio n o f p lastic defo rm atio n in solids. J . Appl. Phys. 21, 987-994. (C .W . 106)

1950. (W ith J . F a b r i .) fico u lem en t transson ique a d eu x d im ensions le long d ’u n e p a ro iondulee . C.R. Acad. Sci. Paris, 231, 1271-1274. (C .W . 107)

1951. C onsidera tions aero d y n am iq u es su r la fo rm atio n des o ndu la tions d u sable. Coll.Intern, du Centre N at. de la Rech. Sci. X X X V , Actions eoliennes. Phenomtnes

d*evaporationet d ’hydrologie superficielle dans les regions , A lgiers, p p . 103-108.1951. (W ith G. G. L in .) O n th e sta tistica l th eo ry of iso tropic tu rb u len ce . Advanced Appl.

Mech. 2 , 1-19. (A cadem ic Press, N ew Y ork.) (C .W . 108)1951. In tro d u c to ry rem arks on tu rb u len ce . Problems o f cosmical , p p . 129-148.

(C en tra l A ir D ocum ents Office, D ay ton , O h io .) (C .W . 109)1951. T h e th eo ry o f shock w aves a n d th e second law o f therm odynam ics. 31,

82-83. (C .W . 110)1952. (W ith F . B. F a r q u a h a r s o n & L. G . D u n n .) A erodynam ic s tab ility o f suspension

bridges, P a r t IV . T h e investigation o f m odels o f the o rig inal T ac o m a N arrow s B ridge u n d e r th e ac tio n o f th e w ind . Bull. Univ. Wash. Eng. Exp. Sta. 116, p a r t I I I .

1952. Je t-ass is ted take-off. Inter avia, 7, 376-379.1952. O n th e fo u n d a tio n o f h ig h speed aerodynam ics. Proc. First U.S. N at. Congr. Appl.

Mech. p p . 673-685.1953. (W ith G . M il l a n .) T h e th e rm a l theo ry o f con stan t pressure deflagration . C. B .

Biezeno Anniversary Vol. on Applied Mechanics, p p . 59-69. (N .V . D e T echnische U itgeverij H . S tam , H aarlem .) (C .W . I l l )

1953. A ero therm odynam ics a n d com bustion theory . L . Aerotecnica, 33 , 80-86.1953. F o u n d a tio n s o f o p era tio n a l research . Proc. 3rd Agard Gen. Ass. A G 6/P3, 82-84.1953. (W ith G . M il l a n .) T h e rm a l theo ry o f a la m in a r flam e fro n t n ea r a cold w all. Proc.

Fourth Intern. Symposium on Combustion, p p . 173-177.1953. (W ith S. S. P e n n e r .) T h e th e rm al theory o f co n stan t pressure deflag ration for first-

o rd e r g lobal reactions. Scientific Papers presented to M ax Born, pp . 49-58. E d in b u rg h a n d L o n d o n : O liver & Boyd.

1953. A few com m ents on rocketry . Interavia, 8 , 628-629.1954. Aerodynamics. Selected topics in the light o f their historical C ornell U niversity

Press.1954. O n th e fo undation o f h igh speed aerodynam ics. F rom High speed aerodynamics and je t

propulsion, V ol. 6, General theory o f high speed aerodynamics (Ed. by W . R . Sears), pp . 3-30. P rince ton U n iv . Press.

1954. (W ith S. S. P e n n e r .) F u n d am e n ta l ap p ro ac h to la m in a r flam e p ro p ag a tio n . A gard P ubl. Selected combustion problems: P t. 1, Fundamentals and aeronautical applications, p p . 5-41. L o n d o n : B utterw orths Scientific Publications.

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1955. Solved a n d unsolved problem s o f h igh speed aerodynam ics. High Speed Aero.Brooklyn, p p . 11-39.

1955. M odels in therm ogasdynam ics. I Modelli Nella Tecnica, A tti del Convegno di Venezia, 1, 643-651.

1955. G u id ed missiles in w ar a n d peace. Aero. Digest, 71, 21.1955. T h e nex t fifty years. Interavia, 10, 20.1955. (W ith G. G. L in .) T heore tica l com m ents on th e p a p e r of M r E . N . Fales. Franklin

Inst. 259, 517-518.1956. F u n d am e n ta l equations in aero therm ochem istry . A gard P ub l. Selected combustion

problems, Pt. 2, 167-184, 245-247. L ondon : B utterw orths Scientific Publications.1956. A erodynam ic h ea ting— the te m p era tu re b a rr ie r in aeronau tics. Proc. H igh-Tem

perature Symposium, Stanford Res. Inst, and Univ. o f Calif. Berkeley, Calif, p p . 140-142: also in Technion Yearbook, 1958.

1956. D im ensionslose G rossen in G renzgeb ieten der A erodynam ik . Flugwiss. 4 , 3-5.1956. Faster, h igher, ho tte r. Interavia, 11, 407.1956. T h e theory o f one-dim ensional la m in a r flam e p ro p ag a tio n for hydrogen-brom ine

m ixtures. P a r t I . (W ith S. S. P e n n e r .) D issociation neglected. P a r t I I . (W ith G . M il l a n .) D issociation included . Guggenheim J e t Propulsion Center, Calif. Inst. Tech. Tech. Rep. N o. 16.

1956. Som e rem arks on com bustion instab ility in rockets. A gard p aper.1956. (W ith B. J . D risc o ll .) Pathw ays for cooperation in N A T O research a n d develop

m en t. A gard pub lica tion .1957. A erodynam ische E rw arm u n g — die H itzeschw elle in der L uftfah rt. Flugwelt, 9 ,

163-164.1957. A lgunas reflexiones sobre el estado ac tu a l de la astronau tica . Ingenieria Aeronautica,

9 , 4-9.1957. Som e observations on gu ided missiles. Interavia, 12, 777-778.1957. M ore or less seriously. Interavia, 12, 1227-1228.1958. L an ch este r’s con tribu tions to the theory o f flight a n d opera tional research. J . Roy.

Aero. Soc. 62, 80-91.1958. A ero therm odynam ic problem s o f com bustion. G eneral aspects o f the com bustion

p rob lem . High speed aerodynamics and je t propulsion, V ol. 3, Fundamentals o f gas dynamics, (Ed. by H . W . E m m ons), pp . 574-584. P rinceton U niv . Press.

1958. M agnetoflu idm echanics. Proc. Ninth Intern. Astronautical Congr. Amsterdam, 2, 644-651.1958. M agnetoflu idom ecanica. Ingenieria Aeronautica, 10, 50-55.1959. Som e significant developm ents in aerodynam ics since 1946. First Intern. Congr.

Aeronautical Sciences, Madrid. P rin ted in Aero I Space Sci. 26, 129-144.1959. A pplications of m agnetoflu idm echanics. Astronautics, 4 , 30, 86.1959. Som e com m ents on applications of m agnetoflu idm echanics. In tro d u c tio n . Proc.

Third Biennial Gas Dynamics Symposium, IX -X I . N orthw estern U niv . Press (1960).1959. M agnetoflu idm echanics. Some com m ents in m em ory o f D. Banki. Acta Technica

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N o. 6, 22-23.1961. H ow to im prove scientific cooperation in N A T O . N A T O Journal, 1, 42-43.1961. From low-speed aerodynamics to astronautics. L ondon : Pergam on Press.1961. Space-age education . Astronautics, 6, 44-45.1961. E ngineering education in ou r age. J . Engrg Education, 52, 29-31.1961. (W ith C. C. L in .) O n the existence of an exact solution of the equations of N avier-

Stokes. Comm. Pure Appl. M ath. 14, 645-655.1962. In tro d u c to ry rem arks on space propulsion problem s. In Advances in astronautical

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Theodore von Karman 3651962. T h e develop ing ro le o f n u c lea r energy in aerospace technology. Inst. Radio Engrs

Trans, on Nuclear Sci. (N .S .), 9 , 49-51.1962. N o n -lin ea r buck ling o f th in shells. Fluid dynamics and applied mathematics (E d. by J . B.

D iaz a n d S. I . P a i), p p . 1-19. N ew Y ork a n d L o n d o n : G ord o n a n d B reach. 1965. (W ith A . D . K e r r .) In s tab ility o f spherica l shells sub jected to ex terna l pressure.

Topics in applied mechanics {Schwerin Memorial Volume) (E d. by D . A bir, F . O llendorff, a n d M . R e in e r), p p . 1-22. A m sterdam , L ondon , a n d N ew Y ork : E lsevier P ub lish ing C om pany .



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Theodore Von Kármán, 1881-1963 - Royal Societyrsbm.royalsocietypublishing.org/content/roybiogmem/12/334.full.pdf · Theodore von Karman received his secondary education at the