Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 12

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Despite its name, it was not invented by Lagrange. 66 Euler introduced it inhis theory of sound of 1759, and Laplace used it in his theory of tides of 1776. In thesecases the neglection of second-order terms with respect tomore manageable ones.r- R turns the equations intoAn interesting question is why the Eulerian picture historically preceded the Lagrangianone. There may be no simple answer, however. Daniel Bernoulli naturally focused on thefluid's velocity, since he based his analysis on the principle of live forces. His father, whodid not rely on this principle, still focused on velocity, presumably because it was the mainquantity of interest in the efflux problem.

The same could be said for d'Alembert's treatiseon fluids, with its classical emphasis on efflux. In his memoirs on winds and on fluidresistance, velocity was again the most relevant quantity, the more so because the flow wassteady (or uniformly rotating in the wind case). Perhaps orie should instead wonder whyEuler introduced the Lagrangian picture in an acoustic context. The answer may be that64Lagrange [1788] p.

442.66Ibid.65Ibid. p. 283.p. 280. Cf. Truesdell [1954] pp. CXIX--CXXIII. Lagrange's earliest discussion of this picture is inLagrange [1761] pp. 448-52.WORLDS OF FLOW30earlier solutions of problems of elastic motion, such as d'Alembert's for vibrating stringsor Lagrange's for sound propagation, were formulated in terms of the displacements of theparticles of the system that determined the elastic response.The vanity of seeking the true founder of hydrodynamics should now be clear. AlthoughEuler's name is legitimately attached to the equations of motion of inviscid fluids, hiscontribution should only be regarded as one step toward the end of a long formativeprocess.

Particular cases of his equations or closely-related statements already appeared inthe works of the Bernoullis. D'Alembert invented a general method through which theequations of any problem of fluid motion could be formulated, and obtained the firstpartial differential equations of fluid mechanics. Euler brilliantly capitalized on theseearlier achievements. Lagrange offered alternative foundations, and· powerful methodsfor solving the equations.An essential element of this evolution was the recurrent analogy between the efflux froma vase and the fall of a compound pendulum.

Any dynamic principle that solved the latterproblem also solved the former. Dauiel Bernoulli appealed to the conservation of liveforces, Johann Bernoulli to Newton's second law together with the idiosyncratic conceptoftranslatio, and d'Alembert to his own dynamic principle of the equilibrium of destroyedmotions. With this more general principle and his taste for partial differentials, d'Alembertleapt from parallel-slice flows to higher problems that involved two-dimensional anticipa­tions of Euler's equations. His method implicitly contained a completely general deriv­ation of these equations, as Lagrange later showed.

Another important element was theconcept of internal pressure. So to say, the door on the way to general fluid mechanicsopened with two different keys, namely, d'Alembert's principle, or the concept of internalpressure. D'Alembert and Lagrange used the first key, and introduced internal pressureonly as a derivative concept. Euler used the second key, and ignored d' Alembert'sprinciple. As Euler guessed (and as d'Alembert suggesteden passant), Newton's old secondlaw applies to the volume elements of the fluid, if only the pressure of fluid on fluid is takeninto account. Euler's equations derive from this deceptively simple consideration.2WATER WAVESOf all the beautiful forms of water waves that of Ship Waves is perhaps mostbeautiful, if you can compare the beauty of such beautiful things.

The subject ofship waves is certainly one of the most interesting(William Thomson, August 1887)in mathematical science. 1As d'Alembert and Euler admitted, one could well know how to write the basic equationsof hydrodynamics without knowing how to apply them to concrete problems.

In the case offluid resistance, this gap could only be filled in the twentieth century. Yet there is one kindof problem that earlier fluid theorists could solve to their satisfaction, namely, the motionof waves on the free surface of water. In1 78 1 , Lagrange wrote the basic equations ofwaterwaves, and solved them in the simplest case of small waves on shallow water. Hisnineteenth-century followers determined the celerity of small, plane, monochromaticwaves on water of constant depth, the pattern of waves created by a local action on thewater surface, the shape of oscillatory or solitary waves of fmite size, and the effect of2friction, wind, and a variable bottom on the size and shape of the waves.There is, however, a puzzling contrast between the conciseness and ease of the modemtreatment of these topics, and the long, difficult struggles of nineteenth-century physicistswith them.

For example, a modem reader of Poisson's old memoir on waves fmds abewildering accumulation of complex calculations where he would expect some ratherelementary analysis. The reason for this difference is not any weakness of early nineteenth­century mathematicians, but our overestimation of the physico-mathematical tools thatwere available in their times. It would seem, for instance, that all that Poisson needed tosolve his particular wave problem was Fourier analysis, which Joseph Fourier had intro­duced a few years earlier.

In reality, Poisson only knew a raw, algebraic version of Fourieranalysis, whereas modem physicists have unconsciously assimilated a physically 'dressed'Fourier analysis, replete with metaphors and intuitions borrowed from the concrete wavephenomena of optics, acoustics, and hydrodynamics. In our mind, a Fourier component isno longer a mere coefficient in an algebraic development, it is a periodic wave that mayinterfere with other :waves in a manner we can easily imagine.The transition from a dry mathematical analysis to a genuinely physico-mathematicalanalysis occurred gradually in the nineteenth century, through reversible analogies betweendifferent domains of physics.

It concerned not only Fourier analysis, but also the theory of1Thomson [1887fl p. 410.2Nineteenth-century wave theorists did not understand the random, statistical character of ocean waves, nor. the mechanisms responsible for their formation. Progress on these difficult questions only occurred in the 1950s,cf. Kinsman [1965].WORLDS OF FLOW32ordinary differential equations, potential theory, perturbative methods, Cauchy's methodof residues, etc.

The modern recourse to such mathematical techniques involves a great dealof implicit knowledge that only becomes apparent in comparisons with older usage.The motivation for the introduction of more powerful tools of analysis was mainlyexperimental. Most water-wave phenomena were known well before they could beexplained.

In most cases, they were discovered in connection with navigation problems.Not surprisingly, the wave theorists after Poisson and Cauchy shared an interest in therational development of navigation. Waves were relevant to several aspects of this science,namely: tide prediction, ship rolling, ship resistance, harbor safety, the wearing of canals,etc. British natural philosophers such as Airy, Stokes, Thomson, Rayleigh, and Lambwere evidently more concerned with these questions than their continental counterparts.They did most to bring the theory of water waves to the service of sea and canal travel,although there were a few French contributions in Saint-Venant's wake.3Section 2.1 is devoted to the theories of waves developed between 1775 and 1 825 by thefour French mathematicians Lap1ace, Lagrange, Poisson, and Cauchy, mostly for the sakeof mathematics, on the basis of the new hydrodynamics.

Section 2.2 is devoted to ScottRussell's many instructive experiments on waves of various kinds, including his nowfamous and then infamous solitary wave, in the context of British Association sponsoredresearch on ship design. Section 2.3 presents Airy's wave theory of tides and his criticalanalysis of Russell's results. Section 2.4 deals with the problem of finite waves of perman­ent shape, as studied by Stokes, Boussinesq, and Rayleigh. It also includes Boussinesq'streatment of the evolution of an arbitrary swell, through which he arrived (in 1877) at theequation which is now attributed to Korteweg and de Vries [1895]. Section 2.5, the lastsection in this chapter, concerns the application of optical or acoustic ideas of interferenceto the explanation of water-wave phenomena.

Due to such innovations, Stokes, Reynolds,and Rayleigh forged the concept of group velocity, Rayleigh solved the problem of wavescreated by a drifting fishing line, and Kelvin computed the pattern of ship waves, therebyinventing the celebrated method of stationary phase.2.12.1 . 1French mathematiciansLaplace 's attemptIn 1 775/76, Pierre-Simon de Laplace published his celebrated theory of tides, based on thehydrodynamics of Jean le Rond d'Alembert. Laplace represented the oceans as a layer ofperfect liquid of variable depth on a uniformly-rotating spheroid, subjected to tj:J.e variableattraction of the Moon and the Snn.

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