Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 15
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It must be recalled, however, that Poisson was discovering, or at least perfecting, much of the calculus he needed for his problem. Most importantly, he could not benefit from the physico-mathematical language later developed in thecontext of wave optics and acoustics. At that time, Fourier analysis and synthesis still18Ibid. pp. 1 1 3-14, 1 19-20. This means that, in the vicinity of a distant point, the progressive sine wave solutionwith wk = ../ilC approximately represents the traveling disturbance. Poisson, who did not have the modernpropensity to favor sine wave solutions, did not make this remark.19Jbid.
pp. 1 15-18.20Ibid. pp. 1 19-26. At a given distance x, only the first oscillations of the water surface are unaffected by thefinite width of the generating perturbation. After a time of order xj,fil, the modulation of these oscillationsbegins. Their amplitude, which originally grew linearly in time, now oscillates between limits that ultimatelydecrease as t-3.42WORLDS OF FLOWt = 15s0t = 20s0t = 25s001510xFig.2.1 .15(m)Computer drawing of Poisson's "dentate waves" caused by a local, parabolic disturbance of thewater surface (witb exaggerated vertical scale). The faster oscillations travel witb a constant acceleration,their slower modulation travel at constant velocity.were-despite Fourier's intentions-mostly formal operations.
They did nof·carry withthem the series of images and metaphors that later physicists learned together with them.Notions such as monochromatic wave and constructive/destructive interference werelacking. As we will see in a moment, these notions not only eased the expression ofPoisson's results, but they also suggested more expedient demonstrations. One author ofthis simplification, Horace Lamb, professed a 'deep admiration' for Poisson's memoir onwaves.2 121Larnb [1904] p.
372.WATER WAVES432.1 .4 Cauchy 's prize-winning memoirBeing himself an Academician and a member of the prize committee, Poisson could notcompete for the Academy's prize on waves. A young but already important mathematician, Augustin Cauchy, won the prize. 22 The original text of his memoir was publishedeleven years later in the Memoires des savants etrangers, with a few appendices taking intoaccount Poisson's contribution. The overlap between Poisson's and Cauchy's memoirs isconsiderable, even though they worked independently. They both used Lagrange's velocity potential and the relevant differential equations; they both considered a localperturbation of the fluid surface; and they both solved the equations through Fourieranalysis. This last point is the most remarkable because Cauchy, unlike Poisson, was notaware of Fourier's theory of heat when he submitted his memoir.
He simply reinvented thereciprocal relation between a function and its Fourier transform.23From a mathematical point of view, Cauchy was more systematic and more rigorousthan Poisson. In particular, he carefully attended to the existence conditions for variouskinds of solutions of his differential equations.
A major novelty of his memoir was arigorous proof of Lagrange's theorem regarding the existence of the velocity potential.24For this purpose, Cauchy used the Lagrangian form of the equations of motion.Denoting by X; the coordinates at timet of the fluid particle that has the coordinates X;at time zero, F; the components of the force density acting within the fluid, P the pressure,and p the density, these equations read (in anachronistic tensor notation):px;dx; = F; dx; - dP.(2.40)If the fluid is incompressible and if the force density F derives from a potential, x;be an exact differential. With respect to the coordinatesX;, this impliesdx; must(2.41)(Permutations of the partial derivatives then lead to)� avk axk avk a xk = 0'a t ax.
BJ0 - a10 a x.(2.42)or, by integrating from time zero to time t,22Cf. Belhoste [1991] pp. 87-91 , Grattan-Guinness [1990] pp. 674--8 1, Dahan [1989a]. In July 1815, a monthbefore Poisson submitted his first memoir on waves, Cauchy read a note containing the main results of his theory,namely, the constant acceleration of the waves, the decrease of the height of a wave during its propagation, and theincrease of the distance between two successive waves; cf. Academic des Sciences, Proces-verbaux 5 (1812-1815)p.
530, Cauchy [1827a] p. 188. Bruno Belhoste notes ([1991] pp. 297-8) that Cauchy also investigated theproduction of waves at the interface between a compressible and an incompressible fluid. This unpublishedmanuscript is inserted in the Cahier sur la theorie des ondes belonging to Madame de Pomyers.23Cauchy [1827a]. On Cauchy's ignorance ofFourier, cf. Cauchy [1818] and Cauchy [1827a] p. 291.24Cauchy [1827a] pp.
35-43. Cauchy's rigor was not flawless: although he was aware that eqn (2.1 8) only heldfor y = lz, he used reasoning that implicitly assumed its validity for any y and thus derived the equationl/' cpj8t4 + i'ff'cpjax'- = 0 (ibid. pp. 52-3). Fortunately, this assumption happens to be correct in the case ofinfinite depth, the only one treated in Cauchy's prize memoir.
Cauchy corrected this slip in an appendix to the finalpublication (ibid. pp. 173-4). Cf. Craik [2004] p. 6.WORLDS OF FLOW44a vk a xk avk axk avJ av?=axi aXj - 8Xj a xi axi - aXj ·(2.43)identity 8vk/8Xi = (8vk/8xi)(8xt/8Xi), the incompressibility condition(8xtf8Xj) = 0, and some algebra, Cauchy finally obtained the simple relation25Using thedetwj(t) = Wj(O)whereWJ= 8v2/ax3-axiaxJ- '(2.44)8v3/ ax2, and so forth.
Consequently, if a velocity potential existsat time zero, the condition for its existence is maintained at any later time. This isLagrange's theorem.Another mathematical difference between Poisson's and Cauchy's memoirs was thelatter's systematic recourse to dimensionless variables.
For example, Cauchy rewrote eqn(2.30) in terms of the variables JL = gtlk and K = gt2j2x to obtain(j=A7Tgt2J dJL COS 2JL COS JL112.+oo0(2.45)KUnder this form, it is immediately clear that the wave crests correspond to definite valuesofgt2j2x,so that their motion is uniformly accelerated. In general, Cauchy soughtuniversality beyond the specific physics problems he was studying.
He tried to extractformulas and structures that had intrinsic mathematical value and could eventually servein other physical situations.26Regarding the physical discussion of waves, the scope of Cauchy's differed fromPoisson's. Like Laplace, Poisson confmed his analysis to disturbances created by thesudden emersion of a solid body.
He briefly indicated how the case of an impulsivepressure applied on a portion of the fluid surface could be included in his general formulas,but he did not pursue the analysis of this case any further. In contrast, Cauchy showedhow the initial fluid velocity depended on the impulsive pressure, and thus reached aphysical interpretation of the velocity potential as the internal impulsive pressure resultingfrom the external impulsion (for unit density). He also proved that the motion of the fluidat any instant could be regarded as being created from rest by impulsive pressures appliedon its surface, a result important to later British hydrodynamicists. 27In other respects, Cauchy's physical discussion was less complete than, Poisson's.Cauchy only described waves independent of the shape of the original disturbance,28whereas Poisson regarded the effect of this shape as the most perspicuous aspect of25This is the Lagrangian expression of the fact, established by Helmholtz in 1858, that the convective derivativeof the vorticity vanishes in an incompressible, Eulerian fluid.
A much easier proof of the theorem (Lambp.[1932]17) is obtained by noting that v · dr = &(v · dr)/&t - d(if /2) in the Lagrangian picture, for which r denotes theevolving position of a given fluid particle. As v·dr is an exact differential at any time,at time zero then it must be so at any later time.26Cauchy [1 827a] p.28Cauchy88.27Poissonifv · dr is an exact differential[1816] p. 92; Cauchy [1827a] pp. 14-15.[1 827a] pp. 92-4 gave the validity condition for this.WATER WAVESFig. 2.2.45VDisturbed water surface with a convex profile, as imagined by Cauchy.wave motion. After reading Poisson, Cauchy investigated this question more thoroughlythan Poisson had done. He showed that, for any symmetric profile of the immersed body,the modulating envelope of the fast oscillations was the Fourier transform of the profile(eqns (2.38) and (2.39) ).
He confirmed Poisson's result for the parabolic profile andexpressed it with the slightly more apt metaphor of ondes sillonnees. He also showedthat, for convex profiles (as in Fig. 2.2), the modulating factor did not oscilJate. Inresponse to this nice theorem, Poisson argued that the only case of physical interest wasthe small-depth parabolic profile, because other profiles would not be consistent with thecontinuity of the fluid during the first instants of the motion.29The comparison between Cauchy's and Poisson's memoirs suggests that Poisson wasmore concerned with physical meaning, and Cauchy with mathematical meaning.
. Paisson's physics nonetheless remained idealized physics. As we wilJ see shortly, his andLaplace's emersion method for producing waves does not work in practice. Poisson didnot perform any experiment. He contented himself with calling, in the introduction to hismemoir, for an experimental confirmation of his theory.302.1.5Apparent confirmationsIn 1 820, the Turin-based hydraulician George Bidone claimed to have confirmed Poisson'smost striking prediction, namely, the uniformly-accelerated motion of the first wavescreated by a local perturbation of the water surface, as well as the numerical values ofthe accelerations of the two first waves (0.3253g and 0.
1 1 83g). Bidone operated with a 24inch wide and 24-inch deep canal. He did not say how he measured the velocity of thewaves, but he dwelt on the difficulty he encountered in applying the Laplace-Poissonemersion method for the production of waves. The immersed body did not instantly leavethe water surface upon withdrawal as the two mathematicians had imagined. On thecontrary, the water adhered to the body and followed it to a certain height until it violentlyfell down (see Fig. 2.3).
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