Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 24
Текст из файла (страница 24)
His basic strategy was to develop the velocity componentspowers of the vertical distancey from theu and v inbottom of the channel, and to determine thecoefficients of this development through the boundary conditions. Lagrange had alreadytried this route and written the resulting series of differential equations, but had foundtheir integration to exceed the possibilities of contemporary analysis unless nonlinearterms were dropped.
A century later, Boussinesq managed to include these terms. 100To second order in y, Lagrange's expression(2. 1 5) of the velocity potential implies theformu = a - 2I aIfy2,of the velocity components, whereav=-aIy{2.68)is a function of x only and the primes denotederivation with respect to x.
Denote byu the elevation of the surface above its originallt+cr u y = cu.Jheight. The conservation of flux in a reference system bound to the wave implies0dThe resulting constraint on the unknown function(2.69)a is1a(h + u) - 6 a"(h + ui = cu.(2.70)Boussinesq solved this equation perturbatively. At the lowest order of approximation, thecubic term is dropped on the left-hand side, andu is neglected with respect to h, so thata = cjh.
At the next order of approximation, the latter value of a is substituted into thecubic term, andu is neglected with respect to h in this term only. This givesau1- = -- + -u11hc h+u 6(2.71)andu-c=u1 u'' 2+(h - 3y2) ,h+u 6 h--- -(2.72)Boussinesq then obtained the equation of the surface by substituting these expressions into1the boundary condition 0 199Bazin [1865]; Boussinesq [1872a].1 00Boussinesq [1 871a]; Lagrange [1781].1 01The other boundary condition, that a particle of the surface should remain on the surface, is a consequenceof eqn. (2.69).78WORLDS OF FLOWzil + if - 2�� + 2g(yh) = 0 for y = h + cr.(2.73)As the potential cp is a function of x - et only, 8cpl8t is the same as -cu.In order to clarify subsequent approximations, it is convenient to introduce the dimensionless variables e = crlh, e' = er, and e" = her'. Boussinesq assumed the wave to besmall and gently sloped, and therefore treated e, e'Ie, and e"Ie as small quantities.
Hethus obtained the equation of the surface(3 er 1 h2er'2 h 3 (T)2 = gh 1 + - - + - -'(2.74)where terms in e2, e12Ie, and e" and all smaller terms are neglected.102 This equation maybe rewritten as(2.75)A first integration yields(2.76)The maximum e' = 0 of the corresponding curve is reached when e = K. Consequently,the velocity of the wave is related to the height CTM of its summit through(2.77)which is Russell's formula. Boussinesq then integrated a second time to reacherh12K+ cosh [VJK(x - ct)lh] ·(2.78)His plot of this curve is presented in Fig. 2.23.A couple ofmonths later, Boussinesq submitted to the French Academy a more generaltheory that gave the deformation of a small, gently-sloped, but otherwise arbitrary waveduring its progression in a channel of constant depth.103 His calcnlation was still based onLagrange's development of the velocity potential in powers of y.
To fourth order, thisdevelopment has the form104(2.79)102Boussinesq kept the s'2/s terms, but neglected them when he integrated the equations.103Boussinesq [1871c]; [1872b]. For a brief but accurate discussion of this memoir, cf. Miles [1981]. Miles notesthat thememoir implicitly contains the Korteweg-<le Vries (KdV) equation, but does not mention that Boussinesq[1877] explicity contains it (see later on pp. 83-4).1<>+r'he reader may wonder why Boussinesq now includes the fourth-order term, which he seems to haveneglected in his earlier determination of the solitary profile. The reason is that the use of the differential condition(2.81) instead of the integral condition (2.69) requires a higher approximation of the potential.WATER WAVES79Profil d'une onde solitaire....j..·��\I=\\\\r-J,IIii':'\\Fig.
2.23.1IIII0.795R;-ijIIIf'i<-0.655R->tID\-i.1.3i7BjAThe profile of a solitary wave (curved solid line) according to Boussinesq ([1872b] p. 90).The vanishing of the pressure at the free surface givesocp 1g£T + + z- ('Vcp)2 = 0 for y = £T(x,t).(2.80)otThe condition that a particle originally on the surface should remain on the surface givesocp o£T ocp o£T- = - + - - if y = <l>(x,t).(2.81)oy ot ox oxAt the lowest order of approximation, using dots for time derivatives and primes forderivatives with respect to x, these two conditions yield (in the reverse order)(r = -f3"h, /3 = -glT.(2.82)The elimination of {3 gives Lagrange's wave equation6" = gh£T" .(2.83)Consequently, at this order cp is the sum of a function of x - c0t and a function of x + c0t,with eo = ,fi!i. Boussinesq retained only the first component, which represents a perturbation traveling at the constant speed eo in the direction of increasing x.At the next order of approximation, the two conditions give(r�= -{3"h - {3"£T - {3' £T' + {3"" h\/3 = -glT + /3"h2 - {3'2 '��(2.84)where h + <l> has been replaced by h in terms that have a derivative of third order or higherin factor.
In order to eliminate {3 , Boussinesq derived the first equation with respect to timeand the second equation twice with respect to x. This givesWORLDS OF FLOW80(2.85)In the terms that follow the first, dominant term in each of these equations, & and can bereplaced by their first approximationand the operatorsandareinterchangeable. This gives(2.82),8l8t{3-co8l8xer = -{3"h + g(c?)" -�gh3a'11 , {3" = -gcr - �gh2a'1 1 - �gh-1 (cr2)".(2.86)Hence follows Boussinesq's equation for the evolution of the perturbation: 105<J= ghcr" + �g(c?)'l + lgh3a'm.(2.87)In order to ease the integration of this equation, Boussinesq imagined a series of fictitiousvertical planes moving in such a manner that the volume ofliquid between two consecutiveplanes remains constant. The velocity of these planes is easily seen to depend on theirabscissa in such a way thatwx&= -(crw)'.(2.88)With the notation(2.89)eqn(2.87) leads to8crw + cocr + = 0.(2.90)'Yx = cr(w - co) - 2eo '(2.9 1)�2 18t_1yIn terms of the auxiliary quantitythis equation can be rewritten as.x=8 atIcox ,-c08 ox(2.92)provided that the operator I can be replaced byI when applied to the smallonly.
As it is also a combination ofquantity y. This means that x is a function ofquantities that are functions ofwhich vanish at infinity, it must vanish. Thisimpliesx - cotx + cot'Y-,w = co + -2cocr1 05Boussinesq [187lc], [1872b] p. 74.(2.93)81WATER WAVESand, approximately,(3 CT2hBoussinesq then substituted his expression forvariation of the height of the fluid slices as&+1wcr =-eo(3CT-)1 h2 cr''w2 = gh 1 + - - + 3wCT.into eqn(2.94)(2.88)to get the convective1+ 1 h2cr") .2 h 3 --;;:-(2.95)He also verified that the volume, momentum, and energy of a swell evolving according tothis equation were invariable. Most importantly, he identified a fourth invariant of themotion, namely the 'moment of instability'M=J (era _ 3�n dx.+oo(2.96)He probably came to suspect its existence while studying the condition ofpermanent shapeas follows.106Remembering that w is the velocity of constant-volume slices of the swell, the shape of aswell is permanent if and only if w is a constant c which represents the celerity of the wave:This condition is identical to that reached earlier by Boussinesq using a more directmethod (see eqn (2.74)).
For anyone familiar with the calculus of variations, this equationobviously derives from the condition that the integral M should be a minimum for a fixed+oovalue of the integral J pgu2dx that approximately gives the energy of the wave. Thisremarkable property oTthe quantity M presumably prompted Boussinesq to examine itstime evolution for arbitrary swells. He found it to be a constant of motion. From the latterproperty, he inferred that M measured the departure of a swell from a solitary wave, or thespeed at which its shape varied in time.
This remark justified the name 'moment d'instabilite' . It also explained the ease with which Russell and Bazin had produced solitarywaves:107If the moment of instability of a wave slightly exceeds the minimum value, the shapeof the swell will oscillate about that of a solitary wave with the same energy, withoutever differing much from the latter wave: indeed a notable difference would imply anincrease of the moment of instability, which is impossible, since this moment does not106/bid. pp.
76 (slices), 78 (eqn. 2.94), 79 (eqn. 2.95), 87 (moment). Equations (2.88) and (2.93) giveer = -(o-w)' = -coa' - y'f2co (KdV). The latter equation, time derivation under the integral sign of eqn (2.96),andeqn. (2.89) give M = -6 J -ycrdx = 6co J -ya'dx + (3/co) J -yy'dx = 0, since the integral ofthe derivative ofanyfunction that vanishes at infinity is zero.107Ibid. p.I 00.82WORLDS OF FLOWvary in time; or, rather, a solitary wave will soon be formed; because frictional forces,which we have neglected so far, damp the oscillations of the effective form of the swellabout its limiting form .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
