Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 25

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. . . And we may even conceive, in the absence of any stableform about which a wave might oscillate, that any swell susceptible, by its positiveand moderate volume, to form a solitary wave with a height small enough not tobreak, should assume this form after a certain time. Thus is explained the ease withwhich solitary waves are produced.2.4.6Torrents and tidal boresLastly, Boussinesq used the expression (2.93) for the velocity w of constant-volume slicesto determine the evolution of an arbitrary swell. Wherever the cre u" is smallcompared to c?jh3, this velocity is given by Airy's formula w = g(h + � u) .

This applies,for instance, to the case of the flat horizontal part of a swell produced by the continuousinjection of fluid at one end of a canal. Boussinesq's interest in this case was concernedwith the distinction between river and torrents and with the theory of river tides. 108In 1 870, by elementary reasoning based on momentum conservation, Saint-Venant hadshown that a step-shaped swell propagated in a prismatic canal at the Lagrangian velocityVifi in a first approximation, and at the velocity g(h + �u) in a second approximation(u being the height of the step). On this occasion, he proposed to call the velocity of wavepropagation 'celerity' in order to distinguish it from the fluid velocity. Superposing auniform flow at the velocity -Vi/i onto this wave motion, he then synthesized a hydraulicjump (ressaut), that is, a sudden variation of the height of water on a constant stream.

In astream of velocity inferior to the critical value Vi/i, any such jump must drift in thedownstream direction; in a stream of velocity superior to this critical value, jumps recedein the upstream direction. Therefore, when the water encounters an obstacle in the bed ofthe stream, it tends to accumulate upstream from the obstacle in the subcritical case (theaccumulating water forms an upstream moving step); it tends to jump over the obstacle inthe supracritical case. The former case defines a river, and the latter a torrent according toSaint-Venant.

1 09A few months later, the Ponts et Chaussees engineer Henri Partiot gave a theory otrivertides based on Bazin's idea that the tidal flux entered the river through a succession ofsmall step-swells propagating at the Lagrangian velocity for the height of the water theyencountered during their progression.1 1 ° Following Bazin, Partiot explained the tidal boreor mascaret by the fact that successive step-swells encountered higher and higher levels ofwater, and therefore propagated at higher and higher velocities. In this process, the laterlaminas of water catch up with the earlier ones, so that the front of the tidal ;ave becomessteeper and steeper.

For strong tides or rapidly-narrowing beds, it can reach the verticalslope for which breaking occurs. 1 1 1j108Boussinesq[187lc], [1872b] pp. 1 00-3.109Saint-Venant [1870]. See also Chapter 6, pp.227-9. The connection between gravity waves and the torrent/[1865a] p. 34.river distinction is roughly expressed in Darcy and Bazin1 10Bazin was himselfinspired by Tbeodore Bremontier, who, in 1 809, analyzed river tides in terms of snccessivelaminas of water (though without recourse to Lagrange's formula).'"Partiot [1871]; Bazin [1865] pp.633-5.WATER WAVES83Mter reading Partiot, Saint-Venant showed that the same evolution ofthe level of wateralong the river resulted from the general equation of non-permanent, gradually-varyingflow that he had obtained by applying thee turn law. 112 For small step-swells, thisequation retrieves the celerity formula g(h + �u), in apparent contradiction with Rus­sell's and Bazin's vg(h + u) formula.ereas in his former communication Saint-Venantheld friction responsible for the discrepancy, he now understood that the formula ofRussell and Bazin applied to situations in which his approximation of gradually-varyingflow was not allowed.

For Russell, CT represented the height of a solitary wave. For Bazin,it represented the height of the surging head of a step-swell, which happened to be fifty percent higher than the step itself.When Boussinesq wrote on solitary waves, he made clear that Saint-Venant's formulaonly applied to a portion of a wave in which the curvature could be neglected. In thecurved part of the swell, convexity implies a decrease of the velocity w, and concavity anincrease.

Through this simple remark, Boussinesq managed to justify the oscillatory shapeof the front ofBazin's swell, as well as the oscillations behind Russell's negative waves. Inthe end, there was nothing in the multifarious wave phenomena observed by Bazin thatBoussinesq could not explain through his powerful analysis. Saint-Venant applauded: 1 13These numerous results of high analysis, founded on a detailed discussion and onjudicious comparisons of quantities of various orders of smallness, sometimes to bekept, sometimes to be neglected or abstracted, and their constant conformity with theresults obtained by the most careful experimenters and observers, appear mostremarkable to me.2.4.

7 Rayleigh on the solitary waveFive years after Boussinesq's note in the Comptes rendus, Lord Rayleigh independentlyreached the solitary wave equation and profile. With Lagrange and Boussinesq, he sharedthe idea of developing the fluid velocity in powers of the vertical coordinate y.

Hisimplementation of this idea was remarkably elegant, thanks to two subterfuges: heanalyzed the fluid motion in a reference system bound to the wave; and he conjointlyused Lagrange's potential <p and the stream function 1/J such that -v dx + u dy = di/J. Therequired power developments are(2.97)The stream line 1/J = 0 forms the bottom of the channel. In Rayleigh's reference system, themotion is stationary, and the condition that a particle of the fluid surface should remain onthis surface is replaced by the condition that this surface should be the stream line1/J(x, y) = -eh. The condition of uniform pressure at the free surface isz? + v2 = Cl - 2g(y - h).msaint-Venant [1871a]. Saint-Venant was apparently unaware of Airy's earlier theory.1 1 3Boussinesq [1 872b] pp.

103-8; Saint-Venant [1 873] p. XXI.(2.98)WORLDS OF FLOW84Rayleigh then inserted the power developments of <p and 1/J into these two conditions, andneglected terms that involved orders of derivation higher than two. This led him to thedifferential equation12112 (y (2.99)+= -y" y12g h)2y 3 y 3 y2 h2 c2h2for the function y(x) = u(x) + h.

The first integral of this equation is the same as Boussi­-- -- -----nesq's equation (2.76). Rayleigh discussed it and integrated it, and obtained resultsequivalent to those of Boussinesq.1 142.4.8The so-called KdV equationEaux courantes, Boussinesq remarked that his second-orderIn a note to his monumentalequation (see eqn (2.87))d-c5u" =y" ( withc� gh and y = �gu2 + �gh3u")=for the deformation of a swell during its propagation could be integrated without recourseto the constant slice-motion, by rewriting it as(at) (at� - eo !_ox))y .c (�at co !-.ox� + co !-. u = y" � __!__ox2 o_'(2. 1 00)A reasoning similar to that given for the vanishing of the quantity x of eqn (2.91) leads tothe first-order equation(2. 101)or(2.102)This is the so-called KdV equation, which Boussinesq wrote some twenty years before itsDutch rediscovery.

Rather than this equation, Boussinesq used the equivalent equations(2.93) for w and (2.95) for the convective variation of height, because they represented the"deformation of the swell in a more direct mannerY5In 1 895, the Dutch mathematician Diederik Johannes Korteweg and his doctoralstudent Gustav de Vries extended Rayleigh's method of 1 876 to include oscillatorywaves, arbitrary long waves of evolving shape, the effect of capillarity, and an investiga­tion of higher-order terms in the Lagrange-Rayleigh expansion. They thus rediscoveredthe 'very important equation' that now bears their name, apparently unaware of Boussi­nesq's relevant study.U61 14Rayleigh [1876a] pp.

256-61. Cf. Lamb [1932] pp. 424-6.l 1 5Boussinesq [1877] p. 360n.1 6Korteweg and de Vries [1 895] p. 428. These authors gave the evolution of the wave in a reference systemmoving together with the wave. Hence their equation involved an undetermined constant depending on the celerity185WATER WAVESKorteweg and de Vries also extended Rayleigh's derivation to periodic waves of per­manent shape, not knowing that Boussinesq had already solved this problem in hisIn this case, the condition of constant pressure at the surface involves anundetermined constant, since the disturbance no longer vanishes at infinity.

Consequently,the equation (2. 76) for the slope of the wave is replaced byEauxcourantes.tP. = 3( -s a)(s - b)(k - s),(2.103)a,where b, and k are three positive constants. The integral can be expressed in terms of theelliptic function 'en', which prompted Korteweg and de Vries to call these periodic waves'cnoidal'. As they showed, Stokes's finite oscillatory waves are large-depth approxima­tions of the cnoidal waves. Solitary waves correspond to the limit of infinite period. 1 17Korteweg and de Vries believed that the permanence of the shape of their cnoidal waveswas preserved at large orders, and gave a tentative proof of this long-debated fact. Theythus sided with Stokes, who had believed since 1 879 in the existence of waves of permanenttype, both solitary and oscillatory.

In 1891, Stokes identified the false step which hadearlier led to the widespread belief in the impossibility of permanent solitary waves,namely, the assumption that, for a given height, a solitary wave could be so long thatthe horizontal velocity was the same on a vertical line. This was indeed the starting-pointof Airy's theory of the nonlinear deformation of waves. The assumption is wrong, since thelength of a solitary wave is determined by its height. Stokes could have added that anargument of his own, according to which, for a given wavelength, the height of a solitarywave could be so small as to undergo finite-depth dispersion, similarly fails.

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