Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 16
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Bidone believed he could circumvent this difficulty by attending tothe two first waves only, which in his opinion were created before the fall of the raisedwater column. Apparently, he did not realize that Poisson's calculations did not apply tothis impulsive excitation either. It is not clear how he reached such 'a marvelous agreementbetween theory and experiment.'31In 1 825, the Leipzig professor Ernst Heinrich Weber and his brother Wilhelm publisheda very thorough Wellenlehre, which summarized all previous theories of waves and29Cauchy [1827a] note XVI, pp.
1 96, 220; Poisson [1 829b]. Fourier [1818] recommended the investigation of anon-parabolic profile.30Poisson [1816] p. 78.31Bidone [1820] p. 25. Poisson [1829b] p. 571 noted Bidone's confirmation ofthe accelerated waves. Strangely,he did not comment on the failure of the emersion method, even though his new memoir was about the permissibleprofiles of the initial water surface.46WORLDS OF FLOWFig. 2.3.The emersion of a parabolic solid according to Bidone ([1820] plate).provided many astute, quantitative experiments on this matter. Their motivation was therecent development of wave physics in acoustic and optical contexts, owing to the works ofEmst Chladni, Felix Savart, Thomas Young, and Augustin Fresnel.
They wanted toprovide the subject with a solid empirical basis, using water waves as an archetype ofwave motion. In their most extended series of experiments, they used two long, narrowwater tanks (see Fig. 2.4). They disturbed the water at one end of the tank, and obtained'self-drawn' wave profiles by suddenly withdrawing a vertically- and longitudinallyimmersed board. They also measured the time a wave took to travel along the tank, andvisualized the internal fluid motion through suspended dust particles.
32The Weber brothers became aware of Poisson's 'very important' theory of waves afterthey had performed their experiments, but before the final editing of their treatise. As theybelieved their observations to confirm some aspects of this theory, they included acommentary of Poisson's paper in French. They approved his general description of thewave pattern, with faint accelerated waves at the front, followed by constant-velocitywaves with a 'dentate' surface. They also confirmed the proportionality between theperiod of oscillation and the square root of the wavelength.33These conclusions would not have resisted a more accurate reading ofPoisson and moreadequate experiments.
As Scott Russell later commented, the Webers' tank was toonarrow, too shallow, and too short to approximate the ideal conditions of frictionlessdeep-water wave motion far from the source. In order to create their waves, the Webersdipped a glass tube vertically into water, drew up the water by suction, and let it fall back.This method differs widely from the static surface deformation imagined by Poisson. Mostfatally, the two brothers mistook Poisson's ondes dentelees to mean large waves with aruffied surface, whereas Poisson's formulas show that he meant what we would now callmodulated waves. What they actually observed was probably capillarity ripples superposed with gravity waves.
The lack of figures and concise summaries in Poissqn's. memoirfavored the confusion. As Thomson put it in 1 871:A great part of what they [Poisson and Cauchy] have to say would be much shortenedeven by the addition of graphic representations, and it would be much easier for anyone (the authors I believe included) to understand the whole character of thephenomena investigated, with illustration like this of the chief function on whichthe expression of these depends.32Weber and Weber [1 825] pp. V, 1 05-17 (self-drawn profiles), 166-99 (velocity), 1 1 7-55 (visualization).33Ibid.
pp. 377-434.WATER WAVES��t:1-··-·····. .······!!i': : : : : ::"'- - - -� -_J�····· · ····· · · · · ···-------''Q).rrrJ-- 'cl;...:::±1[!"'47I�j, .:1:f.::::-� -----------------·Fig. 2.4.likU,�IThe experimental tanks of the Weber brothers ([1825] plate).What had become a common practice in the 1 870s would, however, have seemed costlybad taste to a French mathematician of the early nineteenth century.342.2 Scott Russell, the naval engineer2.2.1Ahorse's discoveryIn 1 833, the Cambridge astronomer James Challis reviewed the present state ofhydrodynamics for the British Association. Although he praised Poisson's and Cauchy'stheories of waves and rejoiced over their verification by Bidone and the Webers, heconcluded on a pessimistic note, lamenting over the stagnation of the more pressingproblem of fluid resistance.
The hydrodynamics of d'Alembert and Eu1er completelyfailed on this matter, since it yielded a vanishing resistance. Newton's old theory ofresistance, based on individual impacts of the fluid molecules at the prow of the immersedbody, at least explained the usually observed proportionality of the resistance withthe square of the velocity. Yet even this simple law suffered exceptions. In particu1ar,Challis referred to a 'singu1ar fact' observed m canal navigation: for a speed of four or fivemiles per hour the hau1ed boat rose out of the water and the resistance was suddenlydiminished.
3534Russell [1845] p. 25n; Weber and Weber [1825] p. 106; Thomson to Stokes, 20 Nov. 1871, ST.35Challis [1833] p. 155. More will be said on Newton's theory in Chapter 7, pp. 264-265.48WORLDS OF FLOWJohn Scott Russell, a young Glasgow engineer who specialized in steam power and navalarchitecture, knew well of this striking anomaly. 3 6 He later described it in vivid terms:As far as I am able to learn, the isolated fact was discovered accidentally on theGlasgow and Ardrossan Canal of small dimensions. A spirited horse in the boat ofWilliam Houston, Esq., one of the proprietors of the works, took fright and ran off,dragging the boat with it, and it was then observed, to Mr. Houston's astonishment,that the foaming stem surge which used to devastate the banks had ceased, and thevessel was carried on through water comparatively smooth, with a resistance verygreatly diminished.
Mr. Houston had the tact to perceive the mercantile value of thisfact to the Canal Company with which he was connected, and devoted himself tointroducing on that canal vessels moving with this high velocity.There was indeed, in the 1 8 30s, a system of fly-boats carrying passengers on two Scottishcanals. A pair of horses drew each boat at a speed of about 1 0 miles per hour. 37Stimulated by Challis's interest in this paradox of fluid resistance, Scott Russell submitted his own simple solution at the Edinburgh meeting of the British Association in1 834.
The motion of a boat through water, he reasoned, raised the pressure of the water atthe bottom of the ship above its static value. This caused a partial emersion of the boat,and the observed decrease in resistance. Denoting by S and S' the transverse sections ofimmersion for velocities zero and v, respectively, Russell wrote the strange nondimensional equation S'v = S(v ..?j2g), and inserted the resulting value of S' in theNewtonian resistance formula R = S1..?pj2. Of this departure of the resistance law from aquadratic form, he said that he had found ample evidence in towing experirnents.38A fuller version of this argument displays Russell's crude misunderstanding of thelaws of mechanics.
There he derived the bottom pressure from the well-known frontpressure of the Newtonian theory of resistance, artistically combined with the isotropyof pressure. In the rest of his reasoning, he seems to have confused the Archirnedeandisplacement with the dynamic displacement pSv.392.2.2The great, solitary waveRussell was not a man to worry over such infractions of the laws of mechanics. He did,however, recognize that his consideration only gave a gradual correction to the Newtonianresistance, not the desired Houston jump.
In order to understand this stronger anomaly,he attended to the fluid motion induced by the boat. One day, 'the happiest of [his] life',something unexpected happened:4036For a biography, cf. Emmerson [1977]. On Russell and waves, cf. Bullough [1988]. On ship hydrodynamics inthe nineteenth century, cf. the excellent Wright [1983].37Russell [1839] p. 79. Cf. Thomson [1887./] pp. 41 8-20, with the lament: 'Is it possible not to regret the old flyboats between Glasgow and the Ardrossan and between Glasgow and Edinburgh, and their beautiful hydrodynamics, when, hurried along on the railway, we catch a glimpse of the Forth and Clyde Canal still used for slowgoods traffic; or of some swampy hollows, all that remains of the Ardrossan Canal on which the horse and Mr.Houston and Scott Russell made their discovery?'38Russell [1834].
Of course, Russell intended his formula to be used with fixed foot and pound units.39Russell [1839] p. 57.40Russell [1865], vol. I, p. 217, [1 839] p. 61.WATER WAVES49In directing my attention to the phenomena of the motion communicated to a fluidby the floating body, I early observed one very singular and beautiful phenomenon,which is so important, that I shall describe minutely the aspect under which it firstpresented itself. I happened to be engaged in observing the motion of a vessel at ahigh velocity, when it was suddenly stopped, and a violent and tumultuous agitationamong the little undulations which the vessel had formed around it, attracted mynotice.
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