Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 18
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Cf. Wright [1883] Chaps5-7, and Chapter 7.53WATER WAVESare raised from their places, transferred forwards in the direction of the motion of thewave, and permanently deposited at rest in a new place at a considerable distance fromtheir original position', in opposition to 'second-order', oscillatory waves in which theparticles oscillate around a fixed point. He found that two solitary waves 'cross[ed] eachother without change of any kind'. He observed that sea waves, originally of second order,evolved into solitary waves after breaking on a gently sloped shore.
He determined that thehighest possible wave had a relative heightuequal to the depthh. Lastly, he performed afew measurements on sea waves. Owing to unfavorable weather conditions, these gave47little more than the independence of the waves on the depth of the sea.In a later report, Russell confirmed the singular properties of solitary waves, extendedhis investigation to other sorts of waves, and compared his results with previous mathematical theories. As we wi!l see shortly, the Astronomer Royal, George Biddell Airy, hadalready denied the existence of solitary waves and downgraded Russell's observations to amere confirmation of Lagrange's shallow-water waves.
Russell, who saw Airy's text just48before sending his report to the printer, was naturally disappointed:This paper I have long expected with much anxiety, in the hope that it would furnisha final solution of this difficult problem [the discrepancy between wave theory andwave phenomena], a hope justified by the reputation and position of the author, aswell as by the clear views and elegant processes which characterize some of his formerpapers . . . It is deeply to be deplored that the methods of investigation employedwith so much knowledge, and applied with so much tact and dexterity, should nothave led to a better result.Russell insisted that his waves, unlike Lagrange's, had a definite shape for a given height,with a length about six times their height.
New experiments performed 'after the bestmethods employed in inductive philosophy' confirmed this point. The disturbance producedby the injection of additional water at one end of his tank soon evolved, while propagatingalong the channel, into the perfectly stable form of the solitary wave (see Fig. 2.9). When theinjection was irregular, a compound wave was produced which evolved into separate solitarywaves (see Fig. 2.12). Using Weber's self-drawing method, Russell showed that the shape ofthe solitary height was perfectly determined for a given height and tended to a cusped shapewhen the maximal height was reached (see Fig.formula for this wave,2.2.52.1 1).
Lastly, Russell confirmed his velocity.Jg(h + u), instead of Lagrange's or Airy's formulas.49The four ordersNo one, Russell argued, had predicted or observed his great solitary wave before him:Lagrange's waves were too small compared to the depth of water; the mode of productionof Poisson's and Cauchy's waves precluded solitary waves; and the Weber brothersbelieved that a positive wave never went without a correlative negative wave. In order toavoid confusion of his great wave with others' waves, Russell introduced the followingfour orders of waves (see Fig.2.10).5047Russell [1 837c] pp.
423 (reservoir, quote), 424 (cycloid), 425 (crossing waves), 426 (sea waves).48Russell [1 845] pp. 27, 30; Airy [1845].49Russell [1845] pp. 27 (quote), 33-4, 45-6.50Ibid. pp. 23-5 (priority), 9 (orders).54WORLDS OF FLOWPtr·lI---A.j;__t::;::;::;;:::::;::=:;;=::,:::;:;:;;::==::;=====::;:;:XA:X.A-1c=p��:;;:::::;::=;::;=;;;;;;:,.�;;;:;;;;;(a)(b)Fig.2.9.Two ways of producing a solitary wave: through the displacement of a wall (a); through theimmersion of a solid (b) (Russell [1 845] plate).�-System cif Water Waves.FouaTH,THIRD.SECOND.FrasT.ORDERS.Designation. Wave of translation . • . .
Oscillating waves. Capillary waves. Corpusenlar wave.1ICharacters ••. Solitary • . . . . • . . • • • . . . • . . • . Gregarious . • . . . . . . . Gregarious . . • • . . . Solitary.ciSpe es···yarietiesInstances{{{Positive . . . . • . . . . •. • • . . . • • . Stationa.rv•••••.••• Free.Negative . • • . . . . . . . •. • • • • . . Progressive •.••...
Forced.[1Free .... . . . .............. . Free.Forced •••. . . . • . • • . . . • . • . · Forced.e wave of resistauce. Stream ripple .... Dentate waves . . . Watet·�sound wave�The tide wave • . • • . • . • • • Wind waves....... Zepbyral waves.The aerial sound wave. Ocean swell. ..... .Fig. 2.10.Russell's wave orders ([1845] p.
9).WATER WAVES55(i) Waves of translation. They involve mass transfer. Positive waves of this kind can besolitary. Negative ones are always accompanied by an undulating series of secondarywaves (see Fig. 2.13).(ii) Oscillatory waves. These do not involve mass transfer. They appear as groups ofsuccessively positive and negative waves. They are the most commonly seen waves, createdby wind for instance.
They can be progressive or standing.(iii) Capillary waves. These only involve a minute-depth agitation of the water. Theydepend on the surface tension of the water.(iv) Corpuscular waves. These are rapid successions of solitary waves. Sound waves arethe prime example.Although Russell focused on the first order, he also performed careful experiments onthe second and third kind. For instance, he showed that the Kelland-Airy formulac? = (gjk) tanh kh correctly represented the velocity c of progressive oscillatory waves,even when their amplitude was not small. 51 He illustrated the evolution of such wavesFig. 2.1 1 .Self-drawn solitary-wave profiles of various heights (Russell [1 845] plate).51 Ibid.
p. 67.56WORLDS OF FLOWwhen approaching a shore (see Fig. 2.14). He drew the shape of steady waves produced byan obstacle in the bed of the stream (see Fig. 2.15). He obtained a beautiful pattern ofcapillary waves by plunging a rod vertically in a stream of water (see Fig. 2.1 7).52Strangest to Russell's readers must have been the fourth order of waves, supposed torepresent sound waves. For any physicist at that time, sound corresponded to the propagation of small-amplitude vibrations through an elastic medium. No special kind of wavewas needed.
As appears from a posthumously published manuscript, Russell rejected thisexplanation for he believed it could not explain the ability of sound to propagate far fromits source. From the fact that the sound of a tuning fork or the vibrations of a string couldbe heard at a non-negligible distance only if the fork or string was attached to a hollow casewith an aperture, he inferred that sound was not the harmonic vibration of the fork andsurrounding air but the repeated emission of solitary waves through the aperture of thecase. As solitary waves are surface waves, Russell needed to imagine an open surface for themedium of propagation.
For sound in water, the free water surface did the job. For soundin air, he imagined an ocean of air oflarge but fmite depth around the Earth. Most daringly,he proposed that light was a wave of fourth order in an even larger ocean of ether.53These suggestions ouly confirm Russell's ignorance of elementary principles of mechanics. The Royal Society never published the series of manuscripts it received from him onthis theme. Yet the elite of British natural philosophers often praised Russell's early workson waves and ship forms, for they admired the quality of his experiments and the frequentvalidity of his intuitions.2.3. Tides and waves2.3.1Russell's illuminationBetween Russell's careful experiments on water waves and his hair-raising speculation oncorpuscular waves, there was a middle ground which seems to have perplexed his learnedsupporters, namely, the notion that tides were essentially solitary waves of very largeextent.
As Russell recounts, he submitted this idea to William Whewell in 1835 togetherwith a plan for observations. Whewell had then been working for several years on tidalobservations and prediction, and was with John Lubbock, the leading British expert onthis topic. He approved Russell's project, which thus became part of the duties of the'Committee on Waves'.54At the Liverpool meeting of 1837, Russell reported the tidal observations thecommittee had made on the rivers Dee (Cheshire) and Clyde (Scotland). He als,o promotedhis own theory of tides. The general idea was to divide the problem into two parts:the general elevation of water in the Pacific and Atlantic Ocean as ruled by celestialmechanics, and the propagation of this elevation in smaller basins, channels, and rivers52Russell (ibid. p.
78) was aware of similar observations by Poncelet ((1831] p. 78)."Russell (1 885].54Russell [1837c] p. 420. In 1838 (BAR p. 20), Whewell praised Robison and Russell for 'highly valuablematerials, likely to assist us in the further prosecution of the subject [the theory of tides].' On Lubbock, Whewell,and tides, cf. Deacon [1971] Chap.
12.'[mmtu--:�l - -m�:---� ""m��U "Uuu_,, ..· · -..The separation of two solitary waves (Russell [ 1845] plate).WAVES� l Jn7er ll..HUXFig. 2.12.-"'" _.:;_,_;:: :..-·-·-·t----�-��Fig. 2.13.'Vitl.rdllatli�tJ TVaYr.r.·,-���v�_ "{E5Negative wave of translation and the accompanying oscillatory wave (Russell [1845] plate)._�w.Fig. 2.14.Fig. 2.15.Waves approaching a shore and evolving into solitary waves (Russell [1845] plate).Standing wave created by an obstacle in running water (Russell [1845] plate).J�--�-�b::: := �<� " ' �-�-Fig. 2.16.��s?;=.The evolution of a compound solitary wave according to Russell ([1845] plate).-<WATER WAVESWAVES .
. _(l,�ifn· Dl..Car•if!,tl} ' mz,,.-e.r.59� -r--<�1Fig. 2.17.Waves generated by a vertical rod (0 = l/16 inch) moving along the water surface with a uniformvelocity. The smaller waves in front of the rod are capillarity waves (Russell [1 845] plate).WORLDS OF FLOW60as ruled by terrestrial hydrodynamics. Russell described the latter mechanism asfollows: 55The Tide Wave appears to be .
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