bined with earlier work by Jeffreys [14] and Dressier [7. 8, 9] shows that these flows with 

 C > V -f- c are unstable and tend to break up into the so-called "roll-waves," which 

 can be regarded as alternations of kinematic shock waves* and bores (Fig. 13). Good 

 two-dimensional roll-waves are observed only with a fairly flat bottom of regular slope, 

 but a similar instability involving, if necessary, three-dimensional disturbances occurs 

 in any sufficiently rapid torrent. 



On a small scale, one can easily observe roll waves on a thin film of water 

 flowing down a vertical glass surface; it would be interesting to try to develop the theory 

 of these capillary roll waves, taking surface tension instead of gravity as the restoring 

 force and laminar instead of turbulent friction. 



27. Conclusion 



In conclusion, I hope that the steeplechase on which I have led you over, and 

 round, the difficulties of this extensive subject has at least suggested to you that inter- 

 esting questions abound in the study of river waves. There may be some in the 

 audience who like myself have devoted much of their working life to the quite different 

 question of trying to unravel the mysteries revealed by the beautiful schlieren and 

 interferometric photographs which are taken in supersonic wind tunnels and shock 

 tubes. To them I would suggest that, fascinating though these subjects are which relate 

 to the behaviour of gases under very extreme conditions, there is sometimes an even 

 greater delight to be derived from fluid motions in which one can become immersed 

 not only metaphorically but also literally. 



REFERENCES 



1. Abbott, M. R., Proc. Camb. Phil. Soc. 52, 344 (1956). 



2. Benjamin, T. B. & Lighthill, M. J., Proc. Roy. Soc. A, 224, 448 (1954). 



3. Boussinesq, J., "Essai sur la theorie des eaux courantes", Mem. pres. Acad. Sci. (Paris, 



1877). 



4. Burns, J. C, Proc. Camb. Phil. Soc. 49, 695 (1953); with an appendix by M. J. Lighthill. 



5. Cole, J. D., Q. Appl. Math. 9. 225 (1953). 



6. De, S. C, Proc. Camb. Phil. Soc. 51, 713 (1955). 



7. Dressier, R. F., Comm. Pure & Appl. Math. 2, 149 (1949). 



8. Dressier, R. F., Nat. Bur. Stand. Circ. 521, 237 (1952). 



9. Dressier, R. F. & Pohle, F. V., Comm. Pure & Appl. Math. 6, 93 (1953). 



10. Favre, H., "Etude theorique et experimentale des ondes de translation dans les canaux 



decouverts" (Dunod, Paris, 1935). 



11. Forchheimer, P., "Hydraulik" (Teubner, Leipzig, 1930). 



12. Green, G., Trans. Camb. Phil. Soc. 6, 457 (1837). 



13. Hopf, E., Comm. Pure & Appl. Math. 3, 201 (1950). 



14. Jeffreys, H.. Phil. Mag. (Ser. 6) 49, 793 (1925). 



15. Keulegan, G. H. & Patterson, G. W., J. Res. Nat. Bur. Stand. 24, 47 (1940). 



16. Kishi, T.. Proc. 4th Jap. Nat. Congr. Appl. Mech. p. 241 (1954). 



17. Korteweg, D. J. & De Vries, G., Phil. Mag. (Ser. 5) 39, 422 (1895). 



18. Kynch, G. J., Trans. Faraday Soc. 48, 166 (1952). 



19. Lamb, H., "Hydrodynamics,'* 6th Edition (Cambridge 1932). 



20. Lemoine, R., "Sur les ondes positives de translation dans les canaux et sur le ressaut 



ondule de faible amplitude." La Houille Blanche, No. 2 (Grenoble. 1948). 



* More strictly, kinematic shock waves in reverse; these become possible at super- 

 critical speeds. 



39 



