Celebratio Mathematica

Cathleen Morawetz

including mathematical problems,
from my early years

by Cathleen S. Morawetz

It seems im­possible to re­call when I first met Olga Taussky-Todd. She was an old Göttin­gen friend of both Cour­ant and Friedrichs, and she spent a year with John at the fledgling Cour­ant In­sti­tute in the early 1950s, but I feel sure I met her be­fore then. I also heard about her from my par­ents, who had known Olga’s Ir­ish­man, John Todd, for a long time.

I re­call two con­ver­sa­tions with Olga. In the first one, very long ago, I asked Olga about her war work on flut­ter prob­lems (flut­ter is caused by the res­on­ance in­ter­ac­tion between an air­foil and the gas flow­ing past it). But she was no longer in­ter­ested in the sub­ject, and my in­terest was also mostly gone. By then, al­gebra was her main con­cern, and her great arena.

Our later talk took place around 1968. I had just been ap­poin­ted to the AMS Com­mit­tee on Wo­men. I used a trip west to ask all the wo­men math­em­aticians I knew who were roughly my age about their ca­reers and po­s­i­tions. It was a sober­ing mis­sion as none of the wo­men, ex­cept the stat­ist­i­cian Betty Scott, had the stand­ard aca­dem­ic job I had.

Olga in­sisted our talk was off the re­cord, and so it shall es­sen­tially re­main. However, it was an op­por­tun­ity for her to put away her won­der­ful smile and air her com­plaints. Her greatest dif­fi­culties had come from be­ing both Jew­ish and a wo­man. Her early year in Bryn Mawr had been dif­fi­cult, and not hav­ing a reg­u­lar po­s­i­tion at Cal­tech rankled with­in her. But her be­loved work in math­em­at­ics saved her.

I shall talk now about my own early years in math­em­at­ics. I worked on my first sci­entif­ic prob­lem dur­ing 1943–1944. I was meas­ur­ing the muzzle ve­lo­city of shells for the In­spec­tion Board of the United King­dom and Canada. I was there be­cause I had wanted to be a radar of­ficer in the Navy, as my male fel­low stu­dents be­came, but I was told I would have to start with boot-train­ing, scrub­bing floors, etc. (That made me angry!)

I found an er­ror in a graph that was used whenev­er the weath­er was bad. Fix­ing it up was great fun. I will not bore you with the de­tails, but it showed me that a little bit of un­der­gradu­ate mech­an­ics could get you something use­ful. Well, use­ful, al­though maybe the old graph had prag­mat­ic in­form­a­tion that I had not used. I owe it to my boss, Mal­colm Macphail, a very able phys­i­cist, for mak­ing me write it up.

At the In­spec­tion Board, I had my first ex­pos­ure to vis­ible pre­ju­dice — against my French–Ca­na­dian cowork­ers. So I re­tained a lifelong sym­pathy for their na­tion­al­ist re­ac­tion as I did for my par­ents’ Ir­ish na­tion­al­ism, that is, na­tion­al­ism tempered by un­der­stand­ing and good­will.

My second prob­lem in math­em­at­ics was a mas­ter’s thes­is at MIT in elasti­city. It was in fact a rather un­man­age­able prob­lem in flut­ter the­ory. I did not really un­der­stand it or find it in­ter­est­ing. I am glad to see that mas­ter’s theses in math are mostly re­placed today by learn­ing more math­em­at­ics.

The at­mo­sphere at MIT, though es­sen­tially male, was not at all hos­tile to wo­men. I was there be­cause Cal­tech took no wo­men and Har­vard only put them thor­ough via Rad­cliffe, which I found of­fens­ive. However, I did find at MIT, as I have found al­most every­where, men whose at­ti­tude to­ward pro­fes­sion­al wo­men is best told through a story of today:

A few weeks ago I was trav­el­ing in a re­mote part in the west of Ire­land, where I met a coun­try man. He asked me if it was true that I “was in the high­er edu­ca­tion.” I replied that I was a pro­fess­or of math­em­at­ics, to which he re­spon­ded, “It would not do a man to make some kind of fail­ure and you around.”

On to my third math­em­at­ic­al prob­lem. I mar­ried and moved to New York in 1946 to the group to the work­ing with Cour­ant, Friedrichs, and Stoker. They had worked to­geth­er on mil­it­ary prob­lems dur­ing the war. For bet­ter or worse, since the time of Archimedes’s cata­pult, the mil­it­ary has been the source of very in­ter­est­ing prob­lems in mech­an­ics, and I was en­gaged to work for and be sup­por­ted by an ONR con­tract. The job, shades of Olga’s past at Göttin­gen, was to fix the Eng­lish and not tamper with the math­em­at­ics of Cour­ant and Friedrichs’s great book, Su­per­son­ic Flow and Shock Waves,1 which was be­ing made out of a pre­vi­ously clas­si­fied set of notes, writ­ten dur­ing the war. This had been, I now think, the greatest war­time con­tri­bu­tion of that group. Ex­cept for an­oth­er year at MIT, I spent the rest of my ca­reer at NYU as Cour­ant’s group evolved.

Friedrichs gave me, from time to time, spin-off prob­lems, but mostly I ed­ited. I learned flu­id dy­nam­ics, at the same time com­plain­ing about the ob­scur­ity of Friedrich’s writ­ing and the short­cuts Cour­ant in­tro­duced to make the book more read­able.

My first “pub­lished” con­tri­bu­tion was an ap­pendix to a pa­per us­ing sta­tion­ary phase. I learned sta­tion­ary phase that way. For me, do­ing, not read­ing, has been the best way to learn. The pa­per was by Friedrichs and Hans Lewy on flow near a dock [e2]. My job was to find the second term in an asymp­tot­ic ex­pan­sion. The first was of course found by Friedrichs and Lewy, and, I might add, no es­tim­ate of the er­ror was made by me or any­one else.

I found that I had some skill and got great pleas­ure out of the ma­nip­u­la­tion of the ne­ces­sary for­mu­las. Friedrichs urged me to be a coau­thor but out of some per­verse van­ity I de­clined. This was 1948, and the pa­per was pub­lished in one of the first is­sues of Com­mu­nic­a­tions.2 I was duly cred­ited in the text.

By then I had one child. Cour­ant had a big ONR grant. He gen­er­ously al­lowed me to work part-time. No timetable was men­tioned ex­cept that I should take my or­als as soon as pos­sible, which I did. I star­ted to work on a thes­is in quantum mech­an­ics, Friedrichs’s lifelong love, but by that time I had one-and-a-half chil­dren, and Friedrichs and Cour­ant per­suaded me to con­vert one of my Navy prob­lems in­to a thes­is.

The thes­is was a study of the sta­bil­ity of a spher­ic­al im­plo­sion un­der a per­turb­a­tion of the ini­tial con­di­tions, while pre­serving spher­ic­al sym­metry. To me, the ap­plic­a­tion was to col­lapsing stars, and in fact the in­terest today would again be in the stars. The mil­it­ary ap­plic­a­tion to ig­nit­ing the first nuc­le­ar bombs was un­known to Friedrichs and to me. It was not made pub­lic un­til the spy tri­al of the Rosen­bergs in 1952.

I nev­er pub­lished my thes­is be­cause it was dis­ap­point­ingly in­com­plete. I know now that al­most every thes­is has un­fin­ished busi­ness, and one should think twice be­fore not pub­lish­ing one’s thes­is. The un­der­ly­ing prob­lem is a per­turb­a­tion of a par­tic­u­lar solu­tion of Euler’s equa­tions that is spher­ic­ally sym­met­ric. Thus the equa­tions are for \( u \) ra­di­al ve­lo­city, \( \rho \) dens­ity, and \( p \) pres­sure, re­spect­ively. The ra­di­al dis­tance is \( r \). the equa­tions are \begin{equation*} \begin{aligned} \rho_t+u\rho_r+\rho\Big(u_r+\frac{2u}{r}\Big) &=0 &&\mathrm{Mass}\\ u_t+uu_r+\frac{1}{\rho}p_r &=0 &&\mathrm{Momentum}\\ (p\rho^{-\gamma})_t+u(p\rho^{-\gamma})_r &=0 &&\mathrm{Entropy} \end{aligned} \end{equation*}

The ex­po­nent \( \gamma \) is the ra­tio of the spe­cif­ic heats. The spe­cial solu­tions are of the form \( u=r/tU(\eta) \) etc. with \( \eta=r^{-\lambda}t \). The pic­tures are in Fig­ure 1.

Figure 1: \( t < 0 \) (left); \( t > 0 \) (right).

The prob­lem re­duces to solv­ing a single autonom­ous or­din­ary dif­fer­en­tial equa­tion: \begin{equation*} \frac{dC^2}{dU}=\frac{B(U,C)}{A(U,C)} \end{equation*}

\( C \) is es­sen­tially the sound speed. There are lots of lovely sin­gu­lar points where \( A=B=0 \). The para­met­er \( \lambda \) has to be chosen so that a solu­tion ex­ists which rep­res­ents an im­plo­sion.

This solu­tion was found dur­ing World War II by v. Neu­mann and oth­ers at Los Alam­os, G. I. Taylor in Bri­tain, and, of all things, by G. Guder­ley in Ger­many, where it was pub­lished dur­ing the war in the open lit­er­at­ure [e1]. This lit­er­at­ure was combed by sci­ent­ists after the war, and the res­ult was de­scribed in Cour­ant and Friedrichs’s book.

To look at sta­bil­ity we per­turb around this solu­tion and look for com­plex solu­tions of the res­ult­ing lin­ear equa­tion that grow ex­po­nen­tially in time, like \( \exp(k,t) \) with \( \operatorname{Re} k > 0 \). I suc­ceeded in re­du­cing the prob­lem to study­ing a second-or­der or­din­ary dif­fer­en­tial equa­tion (o.d.e.) with real coef­fi­cients de­pend­ing on \( \eta \); i.e., \( r^{-\lambda}t \) and the com­plex para­met­er \( k \). It was sin­gu­lar where \( \eta=0 \); i.e., \( t=0 \), the mo­ment of re­flec­tion. I had to prove there were no ei­gen­val­ues with \( \operatorname{Re}k > 0 \). But \( k \) ap­peared in an odd non-stand­ard way and, even worse, the equa­tion was sin­gu­lar at \( \eta=0 \) in a man­ner not con­sidered in the then stand­ard works of Langer on o.d.e. Langer’s res­ults worked for “something” less than zero. I needed equals zero. Tor­ture! I tried hand com­pu­ta­tion on a Marchand cal­cu­lat­or — too time-con­sum­ing and in­ac­cur­ate. Later asymp­tot­ic the­ory for such equa­tions was worked out, but too late for me. I nev­er looked back. What was really wrong? (1) The prob­lem had too many cliff­hangers for a thes­is. (2) I had too many du­ties as a moth­er of two. But I got my de­gree or, as Cour­ant liked to say, my uni­on card.

The next prob­lems were sim­il­ar: sin­gu­lar o.d.e. with a para­met­er. But this time they were solv­able. I was at MIT work­ing for C. C. Lin and paid for part-time from NASA grant. I wrote my first two pa­pers, and they were pub­lished [1], [2].

Even be­fore this, Cour­ant in­vited me to come back to his in­sti­tute, again to be sup­por­ted by ONR. For the next six years, I worked on an as­sort­ment of prob­lems, mostly on tran­son­ic flow. Everything worked well, more or less. And I had two more chil­dren.

Figure 2

Cour­ant, around 1956 I think, pro­posed that I join Har­old Grad’s group in mag­neto hy­dro­dynam­ics (MHD) and plasma phys­ics. He en­ticed me with the pro­spect of help­ing to solve the world’s en­ergy prob­lem through ther­mo­nuc­lear fu­sion. Still an open ques­tion to say the least! But it was a golden lode for prob­lems, from non­lin­ear hy­per­bol­ic par­tial dif­fer­en­tial equa­tions in MHD to stat­ist­ic­al mech­an­ics without col­li­sions. I nibbled away for twelve years, but the re­wards were all mod­est. As an ex­ample, here in Fig­ure 2 is a plasma prob­lem, con­nec­ted ac­tu­ally to the sol­ar wind as well as ther­mo­nuc­lear fu­sion. An ion­ized gas or plasma, the sol­ar wind flows to­ward earth. The mag­net­ic field of the earth re­pels the wind. Be­cause of the high speed of the wind, a shock, like the bow shock in front of a su­per­son­ic air­plane, is formed between the earth and the sun. Ques­tion: What is the mech­an­ism for the shock?

Figure 3

In gas dy­nam­ics you ar­gue that the shock (jump in flow quant­it­ies) is smoothed out by adding vis­cous terms to the ques­tions. In Fig­ure 3, a one-di­men­sion­al mod­el, the width of the trans­ition goes to in­fin­ity with the vis­cos­ity or mean free path. I do not know what the mech­an­ism is for the sol­ar wind, but for the con­trolled ther­mo­nuc­lear fu­sion setup for mak­ing en­ergy it is enorm­ous. If there is go­ing to be a real­ist­ic shock lim­it, there has to be a bet­ter mech­an­ism. I did cre­ate a crude mod­el, but this is still today an in­ter­est­ing un­solved prob­lem in­volving asymp­tot­ics and stat­ist­ic­al mech­an­ics and with lots of room for new ideas.

Aside from un­treat­able math­em­at­ics prob­lems, and that’s the way most prob­lems from phys­ics are, the prob­lem of those years was that I was mov­ing for­ward pro­fes­sion­ally only very slowly. I had suc­cess­fully avoided teach­ing. I con­sidered that I could not raise four chil­dren, do re­search, and teach, so I settled for the first two. I hoped to get a reg­u­lar po­s­i­tion when my young­est child went to school. In 1957, six years after my Ph.D., Cour­ant offered me an as­sist­ant pro­fess­or­ship in the In­sti­tute. I ac­cep­ted know­ing that I might not have a second chance. So in 1957 I stopped be­ing a hy­brid, part-time re­search work­er cum postdoc and be­came a reg­u­lar fac­ulty mem­ber.

Those six early years gave me a very spe­cial start. I wish something like those op­por­tun­it­ies were more avail­able to wo­men — as I like to call them, those vali­ant wo­men — who have to choose between hav­ing chil­dren and risk­ing ten­ure, or get­ting ten­ure and risk­ing not hav­ing chil­dren.


[1] C. S. Mor­awetz: “The ei­gen­val­ues of some sta­bil­ity prob­lems in­volving vis­cos­ity,” J. Ra­tion­al Mech. Anal. 1 (1952), pp. 579–​603. MR 51648 Zbl 0048.​19205 article

[2] C. S. Mor­awetz: “Asymp­tot­ic solu­tions of the sta­bil­ity equa­tions of a com­press­ible flu­id,” J. Math. Phys­ics, Mass. Inst. Techn. 33 : 1–​4 (April 1954), pp. 1–​26. MR 60664 Zbl 0059.​20101 article