Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 14
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Grattan-Guinness [1 990] p. 748.37WATER WAVESefflux, applied to this problem. They both found that their mathematics only gave limitedsolutions of the wave problem: standing sine waves for Laplace, and small-depth solutionsfor Lagrange. They both tried 'to overcome these limitations by speculative moves thatlater proved illegitimate.2.1.3Poisson 's thorny, but thorough analysisIn the following thirty years, mathematical analysis progressed so much that the flaws ofLaplace's and Lagrange's theories of waves became obvious. On27December1 8 13,anAcademic committee including Legendre, Poinsot, Laplace, Biot, and Poisson made 'thewaves at the surface of an indefinitely deep liquid' the subject of the Academy prize for theyear1 8 1 6.Laplace wrote the announcement: 1 1A ponderable fluid mass, primitively a t rest, and indefinitely deep, i s set into motionunder the effect of a given cause. It is asked to determine, after a given time, the formof the external surface of the flnid and the velocity of every of the molecules situatedon this surface.This was his old problem of1776,in a slightly more general form.Laplace's brilliant disciple Simeon Denis Poisson, who belonged to the prize committee,wrote the first memoir on this subject that reached the Academy.
He was one of the firstPolytechnicians, with an unusual capacity for labyrinthine mathematical analysis and adeep interest in fundamental physics.12In his memoir, Poisson first recalled the earlier contributions by Newton, Laplace, andLagrange. He judged Newton's siphon analogy to be 'insufficiently founded'. Laplace'ssolution of1776, he politely noted, only applied to an initial sine-shaped form of the watersurface, and could not be truncated to yield a solution of the local-perturbation problem.Lagrange's solution of1782was correct for small depth, but its extension to large depthwas illegitimate. In order to prove the latter point, Poisson appealed to 'the principle of thehomogeneity of quantities', probably borrowed from Fourier's theory of heat.
This earlydimensional argument went as follows.13Poisson, like Laplace, assumed that the waves were produced by the sudden withdrawalof a partially-immersed body. In infinitely-deep water, the only 'lines' of the problem arelthe breadth of the original depression of the water surface, and the producttgt2 , where g isthe acceleration of gravity and is the time of observation.
The distance traveled by a wavet must therefore be a homogenous function of l and grl. If this distance isl, then it must be proportional to gt2 and the wave is accelerated like a freefalling body. If the wave has constant velocity, this distance must be proportional to t-/ijl.summit at timeindependent ofTherefore, Lagrange's assumption of waves traveling at a constant velocity, independentof their mode of production, is impossible. Whether the waves produced by emersion11Cf.
the Proces-verbaux of the Academie des Sciences 5 (1812-1815) pp. 262, 292, 546, 556, 595, and thestatement in Cauchy [1 827a] p. 1 .12Poisson's memoir was read on 2 October 1 8 1 5, and a sequel on 1 8 December 1 8 1 5. It was published i n 1 8 1 8in a volume dated 1816. A summary o f the main conclusions appeared in the Annales de chimie e t de physique(Poisson [1817b]), Cf. Grattan-Guinness [1990] pp. 666--74, Dahan, [1989a]. For a modern treatment, cf. Lamb[1932] pp. 384-98. On Poisson's physics in general, cf. Arnold [1983].1 3Poisson [1816] pp. 71-5.38WORLDS OF FLOWtravel with constant velocity, with constant acceleration, or else with variable accelerationcan only be decided by calculation.Having thus dismissed Lagrange's approach to deep-water waves, Poisson adoptedLagrange's equations for the velocity potential rp.
In the two-dimensional case, and for asmall perturbation of the fluid surface, these equations are eqn& rpf)x2within the fluid mass, orpfoy+&rpf)y2(2.14), namely= 0,= 0 at the bottom y = 0, and eqn (2.1 8):& rpot2+gorp=0oyfory= h.Poisson, now imitating Laplace's procedure, sought factored solutions of the formcosh ky cos k(x - a) sin wt or cosh ky cos k(x - a) cos wt.Theboundarycondition(2.18) requires that eqn (2.10) holds, namely14w2= gk tanh kh.Poisson then obtained the most general solution by superposition of the factoredsolutions. Using Fourier's identity (without naming Fourier)f(x)and eqn=(2.16), namely� IIf(a) cos k(x - a) da dkorp(x, h;t)fii+g(y - h) =(2.22)0,for the fluid surface, he easily found that the superposition 1 5gI+ooIf'=-;I+oof(a) da-oo0dkcosh kycoshkhmet the initial conditions of zero velocity (rpcos k(x - a)sin wkt-;;;;-= 0) and surface shape y = hcorresponding elevation cr(x,t) of the water surface above the level h is�I+ooer=f(a) daI+oo0dk cos k(x - a) cos wkt.(2.23)+J(x).
The(2.24)Poisson then studied the behavior of these two double integrals in the case of largedepth, for which wk= .Jifk.He did this in a purely mathematical manner, by cleverlycombining changes of variables, integration by parts, and power series developments. To1 4Poisson [1816] p. 82.1 5Poisson [1816] p. 92.WATER WAVES39give a first idea of these 'rather thorny transformations' consider the first integral in theexpression (2.23) of the potential.
It is a linear combination of terms of the form1:�=J0-WkJ0+oo+oo-g-y sin Wk t.e kg dk = 2e-yw' sm wt dw,(2.25)where 'Y is a linear combination of x, y, and a with complex-number coefficients. Derivation with respect to time yieldst=Integration by parts then yields['J0+oo2we-yw' cos wt dw.J:oo -y-1 t J e-yw' sin wt dt,+oot = -y- l e-'Y"' coswtor(2.26)0(2.27)(2.28)The integral of this equation is(2.29)Poisson thus reached a familiar form, whose behavior for small and large times t heobtained through development in positive and negative powers, respectively, of t. Hethen computed the corresponding expression for the potential cp and the derived velocities,paying special attention to the case when the profile f (a) of the disturbance is very1narrow.
6Poisson's most detailed discussion of the wave pattern was based on the formula (2.24)for the surface disturbance. For a very narrow disturbance, the double integral in thisformula may be replaced by the simpler expressionu=A;J0J0( )+oo+oo2Aw2xdkcoskx cos t ,fik =:w dw cos g cos wt,7rg(2.30)where A is the area of a vertical section of the original disturbance. Poisson astutelyrewrote the last integral as u = (A j'TT'g)(h + L), where1 6Jbid. pp. 93-107; Poisson [1817a] p. 85 (thorny).40WORLDS OF FLOWh=w2xJ w dw cos (gw t) = J w dw cos [Xgt) 2- gP4x ) ]g (w 2x+oo+oo±±00·(2.31)For obvious symmetry reasons, it is sufficient to consider the case x > 0.
Putting(2.32)and(2.33)we obtain� J dw(w+ooh=±a2=J= a) cos (w - a2)= =Fg: J cos(w - a2) dw+oo(2.34)±aand(2.35)The modem reader may recognize the Fresnel integrals that appear in the theory ofdiffraction. Poisson, who had no such knowledge, developed these integrals in powers ofa and gave numerical estimates of the position of the first extrema of u. As he noted, theseextrema occur for well-defmed values of a = ..j l4x. Therefore, the crests of the wavesmove with the acceleration of gravity.17For large values of et, the two integrals in the last expression for a differ little from theirlimit ! Ffi. Hence the surface profile is approximately given bygt2u=� cos (a2 �x-(2.36)}The behavior of this function is mostly given by the fast oscillations of the cos,\ne, with anamplitude increasing linearly in time and decreasing with distance as x-312• Maximaapproximately correspond to a2 = -rrl4 + 2n-rr, where n is an integer.
The distance Abetween two consecutive crests at a given time, which Poisson calls wavelength, is givenby Aoa2 Iox = 2-rr, or A = 8-rrx2 I The period of the oscillations at a given place is suchgt2•+oo17Poisson [1816] pp. 108-14. Without any comment, Poisson ignored the indefinite contributionf wdwcos (W' - a2) = ! sin ( + oo) to the integral (2.34).
This indetermination results from the use of a singular±adistribution f(a) = AS(a) for the initial surface deformation. Convolutionindefinite, infinitely-oscillating terms.witha regular profile eliminates theWATER WAVESthatrfJa2 IEJt = 27T,41or r = 47Txlgt. As Poisson noted, the period is a function of thewavelength only, namely r =J2'TTAig.1 8Poisson also considered the more general case in which a is still large but the width ofthe original disturbance is no longer negligible.
He assumed a truncated parabolic profilea2(!(a) = h P z- )2.or"I a I :5 z,and performed the integration over a explicitly in the formula(2.37)(2.24) for cr. This led him,after painstaking consideration of the variation rates of the various factors in the remaining integral, to the formula-where-f(k) =J+oo-oo(gt )2= f 4x2erikaf(a)e-axfocos(a - 4'TT) ,24hZ4da = v( sinkl - klcos kl).(2.38)(2.39)This new factor involves the sine and the cosine of a2 lIx, which oscillate much slower thanthe cos (a2- 'TTI4) factor, as long as the distance x is much larger than the width l of theoriginal perturbation. 19As Poisson noted, the crests of the modulating envelope travel at a constant velocity,j occur for definite values of the dimensionless ratio g!P Ix2.
Poissonsince the maxima ofdescribed the resulting wave pattern asondes dente!ees (dentate waves, see Fig. 2.1). Thisexpression indicates that he regarded the envelope as physically more important than itsaccelerated corrugation. A dent, Poisson reasoned, corresponds to a fixed value of a andtherefore decreases like 1/x as it moves away from the origin; however, an anti-nodecorresponds to a fixed value of gt2ll4x2 and therefore decreases more slowly, asThis is why Poisson believed the anti-nodes to be more visible than the dents.201Ift.In the last sections of his memoir, Poisson obtained similar results in the more realistic,three-dimensional case. To a modern reader, much of his lengthy essay seems uselesslycomplicated and overly abstract.
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