Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 13
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Applying d'Alembert's principle of dynamics to thefluid particles, and neglecting the vertical acceleration of the water as well as any quantityof second order with respect to the fluid velocity, he obtained the fundamental equations oftidal motion. As will appear in a moment, the former approximation requires the depth ofthe water to be small compared to the length over which the tidal elevation varies sensibly;the latter approximation requires the tidal elevation to be much smaller than the depth.43Saint-Venant [1 888] provides the most competent and thorough history ofthe water-wave problem to date.
Seealso Craik [2004] for French and British contributions before 1 850, and Craik [2005] for Stokes's contributions.4Cf. Cartwright [1 999] chap. 6.33WATER WAVESFor a modem reader, it is obvious that Laplace's equations are those for the propagation of small waves in shallow water, with an additional term corresponding to the Coriolisforce and an external force density corresponding to the lunar and solar perturbations.La place could not state so much, since at that time the theory of water waves remained tobe developed. He did realize, however, that his derivation of the tidal equations opened theroad to the simpler problem of the propagation of small disturbances in a large pond ofuniform depth.
Laplace knew Isaac Newton's analogy between water waves and theoscillations of a fluid in aU-shaped tube,which gave a propagation velocity proportionalto the square root of the length of the wave, but he judged this argument to be 'very5uncertain'. His own theory rested on well-established mechanical principles.Laplace focused on free propagation, which only occurs if the cause of the wave islocalized in space and tinie. The obvious example is a stone thrown into a pond. In order toease calculation, Laplace considered a narrow canal instead of a pond, and the emersion of6a solid body instead of its impact:The simplest manner to conceive the formation of waves is to imagine an arbitrarycurve, dipped'.into the fluid to a very small depth and held in this state until all the'fluid is in equilibrium; when this curve is thereafter withdrawn from the canal, it isclear that the fluid will tend to retrieve its equilibrium state by forming successivewaves.La place then used the so-called Lagrangian picture, in which the fluid motion is describedby giving the position (X + g, Y +position(X, Y)rt) of a particle of the fluid at time t as a function of itsat the origin of time (the moment when the curve is withdrawn).
To firstorder in g and Tf, the incompressibility of water implies the continuity equation(2. 1)According t o d 'Alembert's principle o f dynamics, the work o f the sum of inertial, gravitational, and pressure forces during a virtual displacement d(X + g, Y + rt) of the positionof a fluid particle at any given time must vanish.
Taking the ordinate axis to be vertical anddirected upwards, this givesazg&rtd(X + g) + !iT d( Y + rt) +;;zututwheredP= 0,g d ( Y + rt) + p(2.2)g is the acceleration of gravity, p the density of water, and P the pressure. To firstorder, this equation makes(82gjot2) dX + (82rt/8P)d Y an exact differential, so that(2.3)As the expression in parenthesis and its first time.derivative vanish identically for t =must vanish at any time. Together with the continuity equation, this gives5Laplace [1776l6/bid. p. 302.0, it34WORLDS OF FLOW(2.4)Laplace first took TJ to be a function of Y and t only, multiplied by coskX. Then thedifferential equation (2.4) and the boundary condition TJ = 0 at the bottom Y = 0 of thecanal further restrict TJ to the formTJ = a(t) sinh k Y coskX.(2.5)The function a(t) is determined through the condition that, for a virtual displacementalong the water surface, the pressure does not vary.
Using eqn (2.2), assuming the formY = h + s coskX for the water surface at t = 0, and retaining only terms of first order ing, TJ, and s, this implies that.&g + g OTJ = sgksmkX(2.6)012oXfor Y = h. The derivation of this equation with respect to X and the continuity equation(2.1) yieldg82 OTJa2TJ- 8t2 aY + 8X2 = sg� coskX(2.7)for Y = h. Substituting the form (2.5) for TJ then leads to the equationd2ak cosh kh + ag� sinh kh = -sg�.dt2(2.8)The only solution of this equation that agrees with the vanishing of a and da/dt for t = 0 is(2.9)withw2 = gktanh kh.(2.10)The corresponding elevation of the water surface above its original height h is, at the sameorder of approximation,u(X,t) = s cos kX + TJ(X,h;t) = s cos kX cos wt.(2.1 1)Laplace thus obtained what we would now eaU a standing wave, as a consequence of hisseeking a factored solution.
The modem reader may wonder why he did not also find asolution of the form sin kX sin wt and superpose it with the former solution to get theprogressive form cos (kX - wt). The reason is that the initial condition of zero velocityimposes the cosine form of the time dependence. Hence Laplace did not reach theprogressive sine solution for the free propagation of small disturbances on water of finitedepth, although he came very close to it from a formal point of view.The rest ofLaplace's analysis was unfortunately flawed.
To proceed from a sine-shapeddisturbance to a disturbance caused by local emersion, Laplace could not rely on FourierWATER WAVES35synthesis, which was unknown at that time. Instead, he truncated the sine functionby taking u(X,O) = s(coskX - cos ka) for IXIsa, and u(X,O) = 0 for IXI;;:a. In whathe called 'a delicate application of the calculus of partial differentials', he then rewrotethe product coskXcos wt in the expression (2.1 1) for cr(X,t) ascos (kX - wt)+cos (kX + wt)], and replaced the latter cosines by their truncated values. This gives apropagation of the depression toward the two extremities of the X-axis, and withoutdeformation.
The propagation velocity w/k only depends on the depth of water and onthe spatial period of the truncated cosine (roughly determined by the curvature of theoriginally immersed solid). As the calculus of partial differentials was still in its infancy,Laplace did not realize that the truncated wave no longer satisfied his differentialequations.
7H2. 1 .2Lagrangian foundationsIn his memoir of 1781, loagrange addressed the problem of water waves in a most elegantmanner, with no mention of Laplace's earlier analysis. As already mentioned in theprevious chapter, his purpose was to apply the methods of analytical mechanics tohydrodynamics, and thus solve a large class of useful problems, including the traditionalefflux from a vase and the less-explored water-wave problem. 8In these two problems, the fluid is (nearly) incompressible and the gravity is a constantg. Lagrange based his analysis on eqn (1.37):P ("Vrp)2arp- = g · r - - - -+C2atP(2.12)for the velocity potential <p, which he knew to exist whenever the motion was started fromrest by the sole effect of gravity and external pressures. He then assumed that the fluidmass never left the space between two mutually-close parallel planes, so that a powerdevelopment of the potential with respect to the perpendicular coordinate could be used.This condition is met in Bernoulli's problem of efflux from a narrow vase, as well as in thepropagation of surface disturbances in shallow water.
In the latter case, Lagrange'smethod is simply illustrated by assuming two dimensions only, a flat horizontal bottom,and velocity and surface disturbances so small that terms involving their second powerscan be neglected.9At the lowest non-trivial order, the expansion of the potential has the formxrp( ,y, t)ox= 'P ( ,t) + Y'PI (x,t) + i'P2(x,t),(2.1 3)where x is the horizontal coordinate and y is the vertical one. The incompressibility ofwater gives(2.14)7Ibid.
p. 307.8Lagrange [1781]. Lagrange did not mention Euler's memoirs, although they were probably a major source ofinspiration. Cf. Grattan-Guinness [1990] pp. 664-5.9Ibid. pp. 728-48.36WORLDS OF FLOWso that cpg + 2cp2 = 0 (primes denote derivation with respect to x). The vanishing of thevertical velocity at the bottom y = 0 implies that cp 1 = 0.
To summarize, the potential musthave the form(2.15)The equation of the surface is obtained by making P a constant and neglecting secondorder terms in eqn (2.12):8cp+ g(y - h) = 0.8t(2. 16)The condition that a particle of the surface should remain on the surface 'yields(2. 1 7)where (.X, y) is the velocity of the fluid particle. To first order, this gives(2.18)Combining this condition with eqn (2.15), we obtain(2.
1 9)The general integral of this equation, which d'Alembert had given in his theory ofvibrating strings, is'Po (x,t) = f(x - et) + g(x + et),(2.20)wheref and g are two arbitrary (differentiable) functions, andC=Vifz.(2.21)According to eqn (2.16), the elevation of the water surface has the same form. The twocomponents represent the distortionless propagation of any (small) perturbation with thevelocities +c and -c.Lagrange concluded his analysis with a speculative extension to waves on deep water.He argued that the 'tenacity and the mutual adherence' of the particles of water confinedthe agitation to a superficial layer of water, the thickness of which would depend on thepropagation velocity through formula (2.21). 1 0Like Laplace, Lagrange selected physics problems according to the possibilities ofmathematical analysis.
Both mathematicians came to the water-wave problem after realizing that mathematical procedures they had designed in other contexts, namely tides and"·1 0Lagrange [1781]. Lagrange did not mention Euler's memoirs, although they were probably a major sourceof inspiration. Cf.
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