Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 55

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He found that the initial small waves could only have avery short wavelength (about10cm for a windof l O m / s). In order that higher and longerwaves could be formed, the wind had to keep blowing in the same direction and tocommunicate (by some unspecified mechanism) some of its energy and momentum tothe waves. Another cause of growth was the nonlinear superposition of different waves,as Helmholtz had already suggested in his previous paper.10810'1-he waves discussed in Helmholtz's paper are not the only possible steady waves, since in the no-wind casethey do not include the Rayleigh-Stokes solution (which can have any wavelength).

Helmholtz probably onlymeant to give the form offorced waves, although nowhere did he explain how free and forced solutions of theequations of motion should be discriminated. He soon became aware of the necessity of a broader class of waves,as is attested by a footnote in his collected paper (HWA 3, p. 325n); there he suggested to require the balance ofpressure only to second order, so that the ratio between wind velocity and wave velocity could be freely chosen.Wilhelm Wien realized this program in Wien [1900] pp. 1 69-99.105Helmholtz [1 890]. Ibid. p.

334, Hehnholtz noted a connection between the steady-wave problem and thetheory of polycyclic systems that he was developing to give a Lagrangian form to thermodynamics and electro­dynamics.107Ibid. pp. 340-4.106Ibid. pp. 335-40.108Ibid. pp. 349-55. Helmholtz's zero-momentum condition for the initial formationof waves excludes thewaves discussed in his previous paper, which never have zero momentum for a finite height. This contradictiondisappears for the more general waves considered later by Helmholtz (see footnote 1 04).VORTICES4.6.3181Dubious idealizationsIn March 1890, Helmholtz applied to the Staatsminister for a one-month journey to theFrench Riviera. He not only wished to bring back his family who had spent the winter there,but also 'to perform a few scientific observations on the behavior of sea waves .

. . in orderto test the truth of a few new theoretical propositions on the interaction between wind andwaves.' He spent a whole week of April at the tip of the Cap d'Antibes, measuring the windwith a portable anemometer and counting the number of approaching billows. His mainpurpose was to verify that these two quantities varied in inverse proportion, as resultedfrom the similitude argument of his first paper on wind and waves.

The results wereembarrassingly unconvincing. Helmholtz had to admit that the wave count on the shoremainly depended on the off-shore winds, with a delay corresponding to the propagationtime. He nevertheless reported his measurements to the Berlin Academy in July 1 890,perhaps to justify the official character of his time on the Riviera. 1 09Unlike his other hydrodynamic works, Helmholtz's papers on wind and waves had littlefollow-up. Their historical impact was limited to the concept of atmospheric waves, whichsoon became a basic meteorological reality.

Experts in hydrodynamics paid little attentionto Helmholtz's theory of wave formation, as can be judged from the very brief andfragmentary mention in Lamb's otherwise thorough treatise. Wilhelm Wien seems tohave been the only important physicist to pursue Helmholtz's line of thought, with nosignificant progress. Oceanographers only had a passing interest in it. The 1 9 1 1 edition ofOtto Kriimmel's Handbuch der Ozeanographie included a praiseful summary of Helm­holtz's findings.

Later treatises on water waves systematically ignored them.11 0The reasons for this neglect are not too difficult to guess. One may be that Helmholtzwrote his two papers on wind and waves in a hurry, neglected to provide intelligiblesummaries, and did not carefully read the proofs.

In the long run, a more fundamentalreason to ignore Helmholtz's conclusions was that they depended on a number of arbi­trary idealizations. He neglected capillarity forces, although they affect the energy andstability of short waves. He only considered steady waves, whereas more general wavescould have a different range of stability. He took the air and water flows to be irrotational,whereas the actual air flow is always turbulent. Until the mid-twentieth century, ocean­ographers could only complain that no theory properly took into account this complexityof wind waves.

Reasonable models later became available for the interaction between theturbulent air flow and the oscillatory water surface. Essential to their success was theconsideration of the random nature of ocean waves, which nineteenth-century theoristscompletely ignored. All of this explains why Helmholtz's ingenious memoirs on waves andwind have fallen into oblivionY 11 09Helmholtz to Botticher, 9 Mar.

1 890, i n Koenigsberger [1902] vol. 3 , p . 27; Helmholtz [1890] pp. 353-5.Helmholtz planned a third paper on this topic, see the manuscript fragment 'Forme Sationiirer Wogen' (HN,#684).110Lamb [1895] 409n, pp. 421-3; Wien [1900], and previous papers listed therein; Kriirnmel [19 1 1] vol. 2,pp. 61-4. Cf.

also Forchheimer [1905] pp. 429-32 for a summary ofHelmholtz's results, and Baschin [1899] p. 410for a generous assessment of Helmholtz's contribution: 'a theory . . . which in a single blow explains all thecircumstances of wave formation that are observed in nature.'1 1 1 0n older failures, cf. Russell and MacMillan [1952] pp. 61-2; on modern successes, cf. Kinsman [1965].182WORLDS OF FLOWFollowing Helmholtz's strange itinerary across worlds of fluid motion, we began with thepitch of organ pipes, spent a while on atmospheric motion, and ended with water waves.Helmholtz jumped from one domain to the next through an amazing series of conceptualinnovations and analogies. Most importantly, he identified the vortex filament as afundamental, invariant structure of inviscid incompressible flow, and inaugurated apowerful approach to hydrodynamics in which vortices and discontinuity surfaces con­trolled the flow.

With this new perspective, he elucidated basic processes of jet formation,shear instability, and mixing.Although nineteenth-century physicists and mathematicians recognized the depth ofHelmholtz's contributions to hydrodynamics, their practical importance only becameapparent in the twentieth century. In Helmholtz's times, the vortex theorems offeredmore to British theorists of ether and matter than they did to hydraulic engineers. As wewill see in the next chapter, Rayleigh's solution to d'Alembert's paradox in terms ofHelmholtz's discontinuity surfaces turned out to be quantitatively inadequate; Rayleighhimself did not believe in it, and Kelvin completely dismissed it. As we will see in the lastchapter, its connection to a practically useful treatment of fluid resistance was onlyunderstood in the early twentieth century.

Although Helmholtz offered many new insightsinto the motion of perfect liquids, he did not know precisely how to relate such motionswith those occurring in the slightly-viscous fluids of nature.5INSTABILITYThere is scarcely any question in dynamics more important for Natural Philoso­1phy than the stability of motion. (William Thomson and Peter Guthrie Tait,1 867)In the previous chapter, we encountered a special kind of instability, now called theKelvin-Helmholtz instability, which occurs when two fluid masses slide on each other,for instance along smoke jets or on a plane water surface under wind.

Helmholtz arrived atthis instability by reasoning on vortex sheets and used it to explain phenomena · thatseemed to elude Eu1er's equations. He was neither the first nor the last theorist toemphasize the role of instabilities in fluid mechanics. The nineteenth-century interest inthis question was for two reasons. Firstly, the discrepancy between actual fluid behaviorand known solutions of the hydrodynamic equations suggested the instability of thesesolutions. Secondly, the British endeavor to reduce all physics to the motion of a perfectliquid presupposed the stability of the forms of motion used to describe matter and ether.Instability in the former case and stability in the latter case needed to be proved.In nineteenth-century parlance, kinetic instability broadly meant a departure from anexpected regularity of motion.

In hydrodynamics alone, this notion included unsteadinessof motion, non-uniqueness of the solutions of the fundamental equations under givenboundary conditions, sensibility of these solutions to infinitesimal local perturbation,sensibility to infinitesimal harmonic perturbations, sensibility to finite perturbations,and sensibility to infinitely-small viscosity. Although this spectrum of meanings is muchwider than a modern treatise on hydrodynamic stability would tolerate, it must berespected in a historical study that does not artificially separate issues that nineteenth­century writers conceived as a whole.The first section of this chapter is devoted to Stokes's pioneering emphasis on hydro­dynamic instability as the probable cause of the failure of Eulerian flows to reproduce theessential characteristics of the observed motions of slightly-viscous fluids (air and water).Stokes believed instability to occur whenever the lines of flow diverged too strongly, ashappens in a suddenly-enlarged conduit or past a solid obstacle.

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