REFERENCES 



1. Comprehensive surveys can be found in G. Birkhoff and E. Zarantonello, Jets, Wakes, 



and Cavities, New York, Academic Press 1957, and in D. Gilbarg, Jets and Cavities, 

 Encyclopedia of Physics, vol. 9 (to appear). 



2. A more refined theory of the wake, which employs the experimentally observed base 



pressure, provides better agreement with drag observations; see A. Roshko, A new 

 hodograph for free streamline theory, NACA Tech. Note 3168 (1954), and R. Eppler, 

 Beitrage zu Theorie und Anwendung der unstetigen Stromungen, J. Rational Mech. 

 Anal. 3, 591-644 (1954). 



dq 



3. This follows immediately from the irrotationality condition in the form — -j- k q = O, 



on 

 where n denotes the streamline normal and k its curvature. 



4. The essentials of a proof in the axially symmetric case have been given by P. Garabedian, 



Calculation of axially symmetric cavities and jets, Pac. J. Math. 6, 611-684 (1956); 

 the same proof can be carried over to plane flows. 



5. See Garabedian, preceding footnote. 



6. N. Levinson, On the asymptotic shape of the cavity behind an axially symmetric nose 



moving through an ideal fluid, Ann. of Math. 47, 704-730 (1946); M. Gurevich, Flow 

 past an axi-symmetrical semi-body of finite drag, Prikl. Mat. Mekh. 11, 97-104 (1947); 

 (in Russian with English summary). 



7. J. Leray, Sur la validite des solutions du probleme de la proue, Volume du Jubile de 



M. Brillouin, Paris, Gauthier-Villars 1935, 246-257. 



8. See R. N. Cox and J. W. Maccoll, Recent contributions to basic hydro-ballistics, in this 



volume, pp. 215-233. 



9. E. Trefftz, Uber die Kontraktion kreisformiger Fliissigkeitsstrahlen, Z. Math. Phys. 64, 



34-61 (1916). 



10. R. Southwell and G. Vaisey, Fluid motions characterized by 'free' streamlines. Philo. 



Trans. Roy. Soc. (A) 240, 117-161 (1948); H. Rouse and A. Abul-Fetouh, Charac- 

 teristics of irrotational flow through axially symmetric orifices, J. Appl. Mech. 17, 

 421-426 (1950). 



11. P. Garabedian, Calculation of axially symmetric cavities and jets. Pac. J. Math. 6, 611- 



684 (1956). 



12. D. Young, L. Gates, R. Arms, D. Eliezer, The computation of an axially symmetric free 



boundary problem on NORC, Naval Proving Ground Rep. No. 1413 (1955). 



13. Cf. R. N. Cox and J. W. Maccoll, Recent contributions to basic hydroballistics, in this 



volume, pp. 215-233. 



14. P. Garabedian, The mathematical theory of three-dimensional cavities and jets. Bull. 



Amer. Math. Soc. 62, 219-245 (1956). 



15. For references and a survey of the comparison method, see D. Gilbarg, Jets and Cavities, 



Encyclopedia of Physics, vol. 9 (to appear). 



DISCUSSION 



J. D. Nicolaides 



I wonder if the speaker would care to comment on the efforts that I mentioned 

 yesterday, of attempting to determine the transient development of the cavity, using 

 the NORC computer? 



D. Gilbarg 



No. I think — well, I will say something. Because of the difficulties around the 

 singular point, I would say the finite difference method would give a useful answer 

 only if you are not too concerned with accuracy. 



However, if they are pressed to their limits of capability, these machines may 



292 



