Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 23
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The modem,statistical theory of ocean waves contradicts this view: cf.·Kinsman [1965].90Gerstner [1802], [1804]. This motion has the same form as the large-depth limit of Airy's equations (2.51) forinfinitesimal oscillatory waves. The only difference is that for Airy the surface of the water could only correspondto a large negative value of Y, whereas for Gerstner any negative value would do.9 1 Cf. Stokes [1880a] Lamb [1932] pp. 421-3; Weber and Weber [1825] p. 368 (for Moebius's remark). InGerstner's original reasoning [1802], steady waves are investigated first, and a uniform translation is superposedonto these waves to yield progressive waves.92Weber and Weber [1825] pp. 338-72, 368 (quote); Russell [1845] p.
368n.�. .. r�1 · Ay.I.N, ,n�s:iJJ,Al:"Ila'D----------------------------1-i\'•A.Fig. 2.22.U<---1--I "-•Gerstner's waves ([1802] plate). The lines A'B1C1 • • • represent possible wave profiles; the circles represent the orbits of fluid particles; the remaining linesrepresent the successive forms of a line of particles that is vertical when passed by a crest or a through.WATER WAVES75In the 1860s, British and French interest in ship rolling led to three rediscoveries ofGerstner's waves, by the Edinburgh engineering professor William Rankine, by the navalengineer William Froude, and by the Director of the Ecole du Genie Maritime FerdinandReech. When, in the early 1 870s, the French leader in applied mechanics Adhemar Barrede Saint-Venant and his disciple Joseph Boussinesq became aware of Gerstner's theory,they fully endorsed it.
As they noted, Gerstner's waves imply a rotational motion of thewater and therefore cannot be regarded as being generated by pressures acting on a perfectliquid originally at rest. In their eyes, this fact did not preclude the application to seawaves, for the latter usually have a long history in which the imperfect fluidity of waterplausibly plays a role. Stokes judged differently: in his view, only irrotational waves couldbe produced by natural causes. Consequently, these waves were worth analytical efforts,despite the much greater simplicity of Gerstner's waves.932.4.4From wedge-shaped waves to solitary wavesNext to his dismissal of Gerstner's waves, Stokes inserted a supremely elegant proof that,if the crest of an irrotational wave has a sharp edge, then this edge necessarily makes anangle of 120°.
As a fluid particle travels along the surface, its velocity (in a reference systemin which the wave is stationary) must vanish at the angular points. At a short distance rfrom such a point, the velocity must vary as .fi according to Bemoulli's law. Theirrotational character of the wave implies the existence of a velocity potential. As thispotential is harmonic, it is the real part of a function of the complex variable x + iy thatcan be developed in whole powers of this variable.
Taking the origin of coordinates at theangular point, the potential behaves as the real part of a power of x + iy. In polarcoordinates, this gives the form <p ex T' cos ne. On the vertex, the normal velocity o<pjoemust vanish, and the tangential velocity o<pjor must be proportional to .fi. The lattercondition implies n = 3/2. The former then requires that the angle of the vertex shouldbe 120°.94By 1 880, Stokes believed that the highest possible wave (for a given wavelength) had this120° cusped shape. Yet his correspondence with Thomson shows that a few months earlierhe still hesitated. It also shows that he sought opportunities to verify this prediction:I have in mind when I have occasion to go to London to take a run down to Brightonif a rough sea should be telegraphed, that I may study the forms of waves about tobreak. I have a sort of imperfect memory that swells breaking on a sandy beachbecame at one phase very approximately wedge-shapes.During the next summer, Thomson invited him 'to see and feel the waves' on his yacht.
Inthe fall, Stokes wrote to his friend:You ask if I have done anything more ab6ut the greatest possible wave. I cannot saythat I have, at least anything to mention mathematically. For it is not a verymathematical process taking off my shoes and stockings, tucking up my trousers as93Rankinc [1862]; Froude [1862]; Reech [1869]; Saint-Venant [1871b] Boussinesq [1877]: pp.
345-6; Stokes[1880a]. In principle, wind could exert a shear stress on the water surface and thus induce vorticity of the water. Inreality1 however, the normal pressures are more important and observed waves are very nearly irrotational, asStokes expected; cf. Kinsman94Stokes[1965].[1880b]. Twentieth-century experiments have confirmed the 120" cusps, cf. Kinsman [1965].76WORLDS OF FLOWhigh as I could, and wading out into the sea to get in line with the crest of some smallwaves that were breaking on a sandy beach.These adventurous observations seemed to confirm the 120° edge for the highest possiblewaves.95From a theoretical point of view, what convinced Stokes of the existence of wedgeshaped waves was a new perturbation method that enabled him, in the fall of 1 879, to pushthe calculation of finite oscillatory (and irrotational) waves to third order for fmite depthand to fifth order for infinite depth.
The trick was to simplify the expression of theboundary conditions by using the potential 'P and Lagrange's stream function 1/J (theharmonic conjugate of <p) as independent variables instead of the coordinates x and y.The calculations indicated that, for large amplitudes, the tip of the waves came closer tothe 120o -cusp shape when the order of perturbation increased.
For the exact oscillatorysolutions, Stokes expected the cusp shape and divergent series to occur for a definite valueof the amplitude/wavelength ratio in the case of infinite depth, and for a definite value ofthe amplitude/depth ratio in the case of finite depth. In the latter case he realized that thewaves 'tend[ed] to assume the character of a series of disconnected solitary waves.'96In October 1 879, the latter finding prompted him to write to Thomson: 'Contrary to anopinion expressed in my [BA] report [of 1 846], I am now disposed to think there is such athing as a solitary wave that can be theoretically propagated without degradation.'Thomson disagreed: 'The more I think of it the more I am disposed to conclude thatthere is no such thing as a steady free periodic series of waves in water of any depth. I can'tbelieve in the solitary wave.' This divergence of opinion came from Thomson's suspicionthat Stokes's series for fmite waves never converged and only indicated approximatelysteady waves.
In the following years, there was indeed much controversy about theconvergence of these series. The story only ended in 1925, with Tullio Levi-Civita'srigorous proof of the existence of finite waves of permanent shape.972.4.5Boussinesq on solitary wavesUnknown to Stokes and Thomson, the mathematical existence of solitary waves hadalready been argued twice-in 1 871 by a remote French theorist, and in 1 876 by a risingstar of British natural philosophy.
The French investigator, Joseph Boussinesq, had beenworking on open-channel theory for some time. In the steps of his mentor Saint-Venant,he tried to subject every aspect of the motion of water in rivers and canals to mathematicalanalysis.98 He was aware of Russell's observations, and also of the more precise measurements of solitary waves performed by the French hydraulician Henry Bkin. He hadalready written a long memoir on water waves of small height on water of constantdepth. In addition to results that could be found in earlier memoirs by Green, Kelland,95Stokes to Thomson, 20 Sept.
1 879, 1 1 Oct. 1 879, 1 5 Sept. 1 880, ST; Thomson to Stokes, 14 July 1 880, ST.96Stokes [1 880c] pp. 320, 325. Stokes probably borrowed this method from Helmholtz [1868dj, discussed lateron pp. 1 64-5.97Stokes to Thomson, 6 Oct. 1 879, ST; Thomson to Stokes, 10 Oct. 1 879, ST; Levi-Civita [1925]. Cf. Lamb[1932] p. 420.98See Chapter 6, pp. 233-8.77WATER WAVESand Airy (of which he was unaware), he offered a few preliminary considerations on wavesof finite height that may have led him to reflect on Russell's wave.99In his first derivation of the solitary wave, published in1 87 1 in theComptes rendus,Boussinesq sought an approximate solution of Euler's equations that propagated at theconstant speed c without deformation in a rectangular channel. His success in this difficulttask depended on his special flair in estimating the relative importance of the various termsof his developments.
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