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On the development of the theory of the solitary wave. A historical essay. (English) Zbl 0732.76016
Summary: The mathematical evolution of the theory of a solitary wave is followed from its inception when analytical mechanics was formed. Very early already, it was recognized that the motion of a wave differs from that of the water particles, but the first exact solution of a wave theory, published by Gerstner in 1802, assumes that both motions coincide. With Russell’s experimental observations in 1838, and with his classification of waves into four orders, the motion of transmission, i.e. the motion of the wave, was solidly established as the transfer of momentum from one spatial point to its neighbour. His discovery of a solitary wave of permanent form, whose velocity depends on its height started a mathematical dispute. Several authors developed their own theory of waves. The most successful contribution were brought by Airy in 1845 with a nonlinear shallow water theory of waves of nonpermanent form, by Stokes in 1849 with a theory of deep water waves. Stokes concluded that his theory holds whenever the amplitude of the surface elevation relative to the water depth is smaller than the ratio of the waterdepth to the wavelength: \(\eta_{\max}/h<(h/\lambda)^ 2\), whereas Airy’s results imply \(\eta_{\max}/h>(h/\lambda)^ 2\) thus providing different physical phenomena. The first mathematical description of a wave of permanent form was obtained by Boussinesq in 1871; it was rediscovered by an other method by Lord Rayleigh in 1876. The next steps were to improve the stationary behaviour of the solitary wave. Rayleigh had remarked that in his solution the pressure distribution at the free surface is not constant, thus invalidating the stationarity of the solution. McCowan, 1894, Kortweg-de Vries, 1895 gave more accurate results for the wave form and the maximum possible waveheight. Later on, systematic expansions were introduced and allowed an entire hierarchy of waves to be developed, a process presently still being under way. In this context proofs of convergence and existence of the forementioned approximations to an exact solution of a wave in an irrotational perfect fluid are the focus of the 20th century research.

MSC:
76B25 Solitary waves for incompressible inviscid fluids
76-03 History of fluid mechanics
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
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[1] Abramowitz, M., Stegun, A.: Handbook of mathematical functions with formulas, graphs and mathematical tables, 3rd ed. United States Department of Commerce, National Bureau of Standards 1965. · Zbl 0171.38503
[2] Airy, G. B.: Tides and waves. Encyclopaedia Metropolitana, London5, Sect. 392, 241-392 (1845).
[3] Aiton, E. J.: The contribution of Newton, Bernoulli and Euler to the theory of the tides. Annals of Science11, 206-223 (1955).
[4] Boussinesq, J.: Théorie de l’itumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rendus Hebdomadaires des Séance de l’Académie des Sciences72, 755-759 (1871). · JFM 03.0486.01
[5] Boussinesq, J.: Théorie génénerale des mouvements qui sont progagés dans un canal rectangulaire horizontal. Comptes Rendus Hebdomadaires des Séance de l’Académie des Sciences73, 256-260 (1871). · JFM 03.0486.02
[6] Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Journal de Mathématiques Pures et Appliquées17, 55-108 (1872). · JFM 04.0493.04
[7] Colladon, D., Sturm, C.: Mémoires sur la compression des liquides. Mémoires Preséntés par Divers Savants a l’Academie Royale des Sciences de l’Institut de France5, 267-347 (1827), publ. (1938).
[8] Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Communication in Partial Differential Equations10 (8), 787-1003 (1985). · Zbl 0577.76030
[9] de Saint-Venant, J.-C.: Movements des molécules de l’onde dite solitaire, propagée à la surface de l’eau d’un canal. Comptes Rendus Hebdomadaires des Séances de l’Academie des Sciences, Paris101, 1101-1105, 1215-1218 (1885).
[10] Earnshaw, S.: The mathematical theory of the two great solitary waves of first order. Transactions of the Cambridge Philosophical Society8, 326-341 (1849).
[11] Euler, L.: Dècouverte d’un nouveau principe de mécanique. Mémoires de l’Academie des Sciences de Berlin6, 185-217 (1750), publ. (1752) Reprinted in: Leonhardi Euleri Opera Omnia, ed. Fleckenstein, J. O., auctoritate et impensis Societatis Scientiarum Naturalium Helveticae, (Schweizerische Naturforschende Gesellschaft), Lausanne, 1957, Ser. II, Vol. 5, 81-108, Orell Füssli, Zürich.
[12] Euler, L.: De la propagation du son. Mémoires de l’Academie des sciences de Berlin15, 185-209 (1759), publ. (1766) Reprinted in: Leonhardi Euleri Opera Omnia, eds. Bernoulli, E., Bernoulli, R., Rudio, F., Speiser, A., sub auspiciis Societatis Scientiarum Naturalium Helveticae (Schweizerische Naturforschende Gesellschaft), Lausanne, 1926, Ser. III, Vol. 1, 428-451, Teubner Verlag, Leipzig und Berlin.
[13] Fenton, J.: A ninth-oder solution for the solitary wave. J. Fluid Mechanics53 (2), 257-271 (1972). · Zbl 0236.76016
[14] Friedrichs, K. O.: On the derivation of the shallow water theory. Communications on Pure and Applied Mathematics and Mechanics1 (1), 81-87 (1948), Appendix to: The formations of breakers and bores, by J. J. Stoker.
[15] Friedrichs, K. O., Hyers, D. H.: The existence of solitary waves. Communications on Pure and Applied Mathematics and Mechanics7 (3), 517-551 (1954). · Zbl 0057.42204
[16] Gardner, C. S., Greene, S. M., Kruskal, M. P., Miura, R. M.: Method for solving the Kortewegde Vries equation. Physical Review Letters19, 1095-1097 (1967), publ. American Physical Society. · Zbl 1103.35360
[17] Gerstner, F.: Theorie der Wellen. Annalen der Physik32, 412-445 (1809), id. Annalen der Physik, Neue Folge,2. Reprinted from: Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Prag für das Jahr 1802.
[18] Green, A. E., Naghdi, P. M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mechanics78 (2), 237-246 (1976). · Zbl 0351.76014
[19] Grimshaw, R.: The solitary wave in water of variable depth. Part 2. J. Fluid Mechanics46 (3), 611-622 (1971). · Zbl 0222.76017
[20] Grimshaw, R.: Theory of solitary waves in shallow fluids. In: Encyclopedia of fluid mechanics (Cheremisinoff, N. P., ed.) pp. 3-25. Houston: Gulf Publishing Company 1986.
[21] Huygens, C.: Horologium oscillatorium sive de motu pendulorum ad horologio aptato, demonstrationes geometricae. Paris 1673.
[22] Keller, J. B.: The solitary wave and periodic waves in shallow water. Communications on Pure and Applied Mathematics and Mechanics1, 323-339 (1948). · Zbl 0031.33105
[23] Korteweg, D. J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on an new type of long stationary waves. London, Edingburgh and Dublin Philosophical Magazine and Journal of Science, Ser. 5,39, 422-443 (1895). · JFM 26.0881.02
[24] Lagrange, J. L.: Mécanique analytique. Paris 1788.
[25] Lagrange, J. L.: Recherches sur la nature, et la propagation du son. Miscellanea Taurinensia (Turiner Akademieberichte)1, 1-112 (1759).
[26] Laitone, E. V.: The second approximation to cnoidal and solitary waves. J. Fluid Mechanics9 (3), 430-444 (1960). · Zbl 0095.22302
[27] Lamb, H.: Hydrodynamics, 6th ed. New York: Dover Publications 1945. · Zbl 0828.01012
[28] Laplace, P. S.: Recherches sur quelques points du système du monde. Mémoires de l’Académie des Sciences, (1775), publ. (1778).
[29] Lavrent’ev, M. A. (Lavrentieff): On the theory of long waves. American Mathematical Society Translations102, 51-53 (1954).
[30] Longuet-Higgins, M. S., Fenton, J. D.: On the mass, momentum, energy and circulation of a solitary wave. II. Proceedings of the Royal Society of London. Ser. A, Mathematical and Physical Sciences340, 1-28 (1974). · Zbl 0292.76010
[31] Lord Rayleigh: On waves. London, Edingburgh and Dublin Philosophical Magazine and Journal of Science, Ser. 5,1 (4), 257-279 April (1876).
[32] McCowan, J.: On the solitary wave. London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Ser. 5,32, 45-58 July, (1891). · JFM 23.0964.01
[33] McCowan, J.: On the highest wave of permanent type. London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Ser. 5,38, 351-358 Oct. (1894). · JFM 25.1478.01
[34] Miles, J. W.: Solitary waves. Annual Review of Fluid Mechanics12, 11-43 (1980). · Zbl 0463.76026
[35] Miles, J. W.: The Korteweg-de Vries equation: a historical essay. J. Fluid Mechanics106, 131 to 147 (1981). · Zbl 0468.76003
[36] Miura, R. M.: The Korteweg-de Vries equation: a survey of results. SIAM Review18 (3), 412-459 (1976). · Zbl 0333.35021
[37] Newton, I.: Philosophiae naturalis principia mathematica. London 1687. · Zbl 0732.01044
[38] Poisson, S.-D.: Sur la vitess du son. Annales de Chimie et de Physique23, 5-16 (1823).
[39] Russell, J. S.: Report of the committee on waves. Report of the 7th Meeting of the British Association for the Advancement of Science, Liverpool, 417-496 (1838).
[40] Russell, J. S.: On waves. Report of the 14th Meeting of the British Association for the Advancement of Science, York, 311-390 (1845).
[41] Schwartz, L. W., Fenton, J. D.: Strongly nonlinear waves. Annual Review of Fluid Mechanics14, 39-60 (1982). · Zbl 0489.76025
[42] Shields, J. J., Webster, W. C.: On direct methods in water-wave theory. J. Fluid Mechanics197, 171-199 (1988). · Zbl 0658.76011
[43] Speiser, D.: Die Grundlagen der Mechanik in Huygens Oscillatorium und in Newtons Principia. In: Die Anfänge der Mechanik (Hutter, K., ed.), pp. 21-46. Berlin: Springer 1989.
[44] Stoker, J. J.: Water waves. New York: Interscience 1957. · Zbl 0078.40805
[45] Stokes, G. G.: On the theory of oscillatory waves. Transactions of the Cambridge Philosophical Society 8, 441-445 (1849).
[46] Stokes, G. G.: On the theory of oscillatory waves. In: Mathematical and physical papers1, pp. 197-229. Cambridge: University Press 1880. Appendix B.: Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form.
[47] Stokes, G. G.: The outskirts of the solitary wave. Mathematical and Physical Papers5, p. 163. Cambridge: University Press 1905. · JFM 36.0025.01
[48] Struik, D. J.: Determination rigoureuse des ondes irrotationnelles permanentes dans un canal à profondeur finie. Mathematische Annalen95, 595-634 (1926). · JFM 52.0876.04
[49] Thacker, W. C.: Some exact solutions to the nonlinear shallow water wave equations. J. Fluid Mechanics107, 499-508 (1981). · Zbl 0462.76023
[50] Truesdell, C. A., ed.: Leonhardi Euleri Opera Omnia. Ser. II, Vol. 12, auctoritate et impensis Societatis Scientiarum Naturalium Helveticae (Schweizerische Naturforschende Gesellschaft), Lausanne 1954. · Zbl 0215.31503
[51] Ursell, F.: The long wave paradox in the theory of gravity waves. Proceedings of the Cambridge Philosophical Society49, 685-694 (1953). · Zbl 0052.43107
[52] Weber, E. H., Weber, W.: Wellenlehre auf Experimente gegründet. Leipzig 1825.
[53] Wehausen, J. V., Laitone, E. V.: Surface waves. In: Handbuch der Physik (Flügge, S., ed.) Vol. 9, Strömungsmechanik III. Berlin: Springer 1960. · Zbl 0095.22302
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