Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 17
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The water in various masses was observed gathering in a heap of a welldefined form around the centre of the length of the vessel. This accumulated mass,raising at last to a pointed crest, began to rush forward with considerable velocitytowards the prow of the boat, and then passed away before it altogether, andretaining its form, appeared to roll forward alone along the surface of the quiescentfluid, a large, solitary, progressive wave. I i=ediately left the vessel, and attemptedto follow this wave on foot, but finding its motion too rapid, I got instantly onhorseback and overtook it in a few minutes, when I found it pursuing its solitary pathwith a uniform velocity along the surface of the fluid.
After having followed it formore than a mile, I found it subside gradually, until at length it was lost among thewindings ofthe channel. This phenomenon I observed again and again as often as thevessel, after having been put in rapid motion, was suddenly stopped; and the accompanying circumstances of the phenomenon were so uniform, and some consequencesof its existence so obvious and important, that I was induced to make The Wave thesubject of numerous experiments.Russell soon suspected a connection between the existence of solitary waves andHouston's resistance paradox.
A few trials confirmed that 'the velocity of the motion ofthe solitary wave had a peculiar relation to a certain well-defmed point of transition in theresistance of the fluid.' Russell performed the necessary experiments 'during the leisure oftwo summers', 1 834 and 1 835, with the support of canal, naval, and academic authorities,and with the help of 'two scientific friends' and 'a dozen hired assistance'. Four differentvessels were towed in canals of various depths at a velocity ranging between 3 and 1 5 milesper hour. Horses provided the towing force, directly in 1 834, and through a suspendedweight regulator in 1835 (see Fig.
2.5). A dynamometer measured the resistance. Russellfound it to increase regularly until a certain critical velocity depending on the depth wasreached, then to suddenly diminish, and finally to increase again (see Fig. 2.6). The criticalvelocity turned out to be identical to the velocity of the solitary wave for the given depth h.With a gun-shooting friend and a chronometer, Russell measured the time that this wavetook to travel between two distant points. This gave him Lagrange's velocity formula yg!i,or more precisely .Jg(h + O"), where O" is the height of the wave crest above the undisturbedwater surface.41Russell also described how the shape of the water surface around the moving vesselevolved with the velocity (see Fig. 2.
7). Fo.r velocities inferior to the critical value, thewater level is raised around the prow, thus forming 'the great primary wave of displacement'. The resulting inclination of the vessel, Russell reasoned, increases its effectivetransverse section of immersion and the corresponding resistance. When the velocity ofthe vessel reaches the critical value, this wave has the velocity of a solitary wave. The pushfrom the vessel is no longer necessary for its progression. If the velocity is further41 Russell [1 835a], [1837b], [1839] pp.
61 (quote), 47 (friends), 49-50.- ��- - Fig. 2.5.Russell's towing mechanics of 1835 (plate of Russell [1839], redrawn in Thomson [1887f]).f1AFig. 2.6.Resistance as a function of towing velocity according to Russell [1839] p. 49.WATER WAVES51�=::::::;;r_===-==·--=---(b)Fig. 2.7.__ _Positions of a canal boat towed at a velocity inferior to the critical velocity (a), superior to the criticalvelocity (b) (Russell[1839] p. 70).increased, the vessel catches up with its own wave, so as to be 'poised on its sununit'. Theeffective transverse section is much smaller, and so is the resistance.42For subcritical velocities, Russell also noted the 'posterior wave of displacement',namely, the depression of the water surface at the stern that necessarily accompanies itsrise at the prow.
As water rushes into this depression from both sides, Russell reasoned, itinduces a series of oscillations of the water behind the vessel (see Fig. 2.7(a)). The violenceof these oscillations increases until the critical velocity is reached. They subside beyondthis velocity, because the posterior wave no longer exists.432.2.3Wave-lined vesselsWhatever the value of this intuitive reasoning, it convinced Russell that the accumulationof water at the prow of a vessel was a major obstacle to its progression.44 In canals of smalldepth this obstacle could be overcome by exceeding the critical velocity. For maritimenavigation, this cause of resistance necessarily grew with increased velocity. It couldcompromise the high-speed, steam-powered navigation in which the city of Glasgow hadthe highest stakes.
Russell soon suggested a remedy, namely, to shape the prow of thevessel according to hollow lines, so that it could enter the water without ruffling its surface.Specifically, he recommended lines made of two parabolic arcs, for this shape wouldinduce a uniformly-accelerated motion of the water along the lines. As he later put it,'There is a way of setting about the removal of the water from the place the ship wants toenter, which is pleasant and profitable to both.' Russell noted that hollow lines had longbeen used by pirates, to whom speed was essential.
They occur spontaneously in a mostprimitive mode of ship construction: binding the extremities oftwo planks, and separatingtheir middle part through a transverse beam. Russell only claimed to be first in showingtheir theoretical superiority.4542Russell[1835a], [1839] p. 40.43Russell[1 839] pp. 65-7.44According to the modern understanding of ship resistance, the wave component derives from the waves thatpropagate away from the ship, not from a direct action on the prow.45Russell [1 835b], [1837a], [1839] p. 51, [1865] vol.
1, p. 161 (quote and pirates). Cf. Wright [1983], pp.71-80.52WORLDS OF FLOWTHE 1f:d7TE.:=-�� ���SIFig. 2.8.Russell's first hollow-line model: "The Wave" ([1 839] plate).In 1 835, Russell built The Wave, a model with a 75-foot keel and a 6-foot beam, to testthis new principle of ship construction (see Fig.
2.8). The following year he continued witha series of more important wave-lined vessels: the Storm, the Skiff, . . . and the ScottRussell. In the early 1 840s, while steering a British Association committee 'on the form ofvessels', he performed some twenty thousand observations with models and full-scalevessels, ranging from 30 inches to 1300 tons. The wave profile came out best, thoughwith some modification.
Russell found the hollowness of parabolic lines to be excessive,and ultimately adopted sine-shaped lines for their analogy with the harmonic waves ofLagrange's theory. For the rest of his career, he pressed for the systematic use of the 'waveprofile' and repeatedly denounced British conservatism in matters of ship design.
In themid-1850s he applied his wisdom to the Great Eastern, a monster metal vessel built for theEastern Navigation Company.462.2.4Taming water wavesRussell's investigation of the best form of ships went along with further studies of waterwaves. At the Bristol meeting of 1 836, the British Association appointed a 'Committee onWaves' directed by Russell and John Robison. Russell gave a first report of this research atthe Liverpool meeting of 1 837.
A section of this report was devoted to an attempt atexplaining tides in terms of solitary waves, which will be discussed shortly. In most of hisreport, Russell described experiments he made in canals and in a 20-foot long and 1-footbroad experimental reservoir. Through a clever optical method, he established theJg(h + CT) velocity formula for solitary waves.
He found that these waves had a quasicycloidal form which was independent of the way they were produced. He described theinduced motion of the fluid particles: 'By the transit of the wave the particles of the fluid46Russell[1 835b], [1841], [1842a], [1843b], [1865] vol. I, pp. 210--1 1 (sine lines), [1852], [1865] vol. 1, p. XXX[1 854], [1857] (Great Eastern). Cf. Emmerson [1977], Wright [1983] p. 80, who claims that Russellapplied far less hollow lines to the Great Eastern than required by his theory, despite a lot of propaganda. In the(denouncing),later conceptions of ship resistance developed by Rankine and Froude, wave formation still played a role, thoughwith mechanisms different from Russell's and in competition with two other forms of resistance, namely skinfriction and eddy formation.
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