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Further Remarks on John Scott Russel and on the Early History of His Solitary Wave

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Further Remarks on John Scott Russel and on the Early History of His Solitary Wave Further Remarks on John Scott Russel and on the Early History of His Solitary Wave The first recorded observation of a solitary wave is undoubtedly that made by Russell in the month of August 1834 as described in the quotation from his paper [1.3] given in Chap.1. Russell's fascination for the solitary wave was plain then: that it continued throughout his life is also clear-see for example the quotation from Russell's book of 1865 which appears on the facing page. Russell's career following his discovery of the solitary wave was not uneventful. We mention some points in it. In 1832-1833 he held only the temporary appointment at Edinburgh created by the death of the Professor, Dr. Leslie. He did not apply for permanent appointment to the Chair: Brewster was a candidate and the Chair went finally to J.D. Forbes. In 1838 he did apply for the vacant Chair of Mathematics, but despite a good reference from Hamilton who described Russell as 'a person of active and inventive genius' [1.5], he failed to get it. It is interesting to spe­ culate whether soliton theory would have developed some 100 years earlier if Russell had got either of these Chairs. Forbes's interests were certainly very different from Russell's although he has left us permanent work. His reputation now rests primarily on his early work on glaciers (especially that on the Mer de Glace which he was the first to map [1.10,11]). But we believe he also introduced "Forbes's Bar" used in the measurement of the thermal conductivity of metals. The anomalous thermal behavior of one-dimensional anharmonic lattices predicted from numerical studies by FERMI, PASTA and ULAM in 1955 (the FPU problem [1.12]) stimulated the investigations which led Kruskal to the inverse scattering method for solving the KdV, whilst the soliton solutions of that equation may cause some of the difficul­ ties in deriving from microscopic theory the Fourier law of heat conduction [1.13].1 Russelll had invented a steam trolley in the years before 1834; in that year the Scottish Steam Carriage Company was formed with the proposal to run a regular service between Glasgow and Edinburgh. It had a short life [1.5]. Nevertheless [1.5] it seems to have been in consequence of his association with this Company 1Professor Peierls has pointed out to us that even close to the melting point the magnitudes of excitations in real three-dimensional systems are so small that solitonlike contributions can surely be neglected. 374 that Russell received the invitation2 from the Union Canal Company to investigate the Canal's prospects for steam navigation and so see the first soliton. Subse­ quently Russell made his reputation as a naval architect. His "wave-line" hulls were designed to minimise wave making resistance due to the generation of solitary wave type bow waves. In 1865 he published the first major work on naval architec­ ture [1.14]. And in this he describes his early experiments on the production of low waves and the way they move water at a speed independent of that of the ship [1.5,14]. 8y 1853, however, Russell had been engaged by Brunel to construct the great iron ship the "Great Eastern" [1.5], a direct or indirect cause of many mis­ fortunes. He entered the first of his periods of bankruptcy before that ship could be launched, became embroiled in the controversy surrounding the failure of a steam valve during the trials of that vessel in the English Channel in 1859 and the death of five seamen at which time Brunel also died, whilst in 1867 he was forced to re­ sign (perhaps by Brunel 's family [1.5]) from the Institution of Civil Engineers of which he was a founder member following a charge of unprofessional conduct associ­ ated with a second financial failure whilst acting to purchase guns for the North in the American Civil War. In the last years of his life Russell completed the book The Wave of Translation in the Oceans of Water, Air and Ether published posthumously in 1882 [1.15]. That book contains again reproduced the British Association's 'Report on Waves' (1844) [1.3] where the first observation of the solitary wave is reported. It contains a number of curious speculations on the structure of matter; and it applies the for­ mula (1.7) to compute from the velocity of sound the depth of the atmosphere (5 miles) and from the velocity of light the depth of the universe (5x 10 17 miles)! Russell's point for the former was that, for distortionless transmission, sound must be carried by his solitary waves. This it is the velocity c of (1.7) which is to be interpreted as the sound speed [and not cs = I9h in (1.6) or (1.8)]. The calculated depth of 5 miles proves to be the actual equivalent depth at uniform density. However, for the size of the universe, 5x 10 17 miles is out by at least five orders of magnitude; and in any case it appears to rely on a value of g re­ duced arbitrarily by Russell by a factor 10-5. Nevertheless these early speculations perhaps begin already to hint at the current significance of solitons in modern physics. Russell's 'Report on Waves' [1.3] apparently stimulated work by DE BOUSSINESQ [1.9] and RAYLEIGH [1.16]. Both authors derived the sech form of the solitary wave and the formula (1.7) for its speed. Rayleigh also gave reasons why it breaks for 2See especially Russell's Edinb. Roy. Soc. Trans. XIV (1840) paper listed below. There seems to have been no contract, but the Union Canal Company paid the ex­ penses-according to their report of 27th January, 1835: 'Report on the practical results of experiments on Canal Navigation (Canal Office, Edinburgh). 375 k~h as had been found experimentally by Russell. Rayleigh reviewed the concept of the solitary wave which Russell had introduced in his 'Report on Waves', noted that with lengths 6 or 8 times the canal depth it could apparently be treated by the theory of long waves, noted nevertheless Russell's observations of different be­ haviors for positive and negative waves, quoted Airy's objection, namely that his (Airy's) theory of shallow waves of great lengths admits both 'positive and negative waves' ("We are not disposed to recognise this wave as deserving of the epithets 'great' or 'primary' ... "), quoted Stoke's counter opinion, and then by the impli­ cations of his analysis came down firmly in favour of Russell. By the time KORTEWEG and DE VRIES [1.7] derived their equation, (1.6), the sech 2 sol itary wave was lOwe 11 known" and thei r paper was primarily concerned with re­ futing AIRY's opinion [1.8] that long waves in a canal must necessarily change their form. They showed that Stoke's theory of long waves [1.17] gave the first two terms of the cnoidal wave solution of their KdV equation and that whilst sinusoidal waves become steeper in the front when advancing, other waves behave differently. They showed n = k sech2 px was stationary (in a moving coordinate system) if k = 4op2: for k> 4op2 the waves steepen in front; for k < 4op2 they steepen behind. In our notation a = ~ h3 - yhp-1 g-1. KORTEWEG and DE VRIES [1.7] quoted DE BOUSSINESQ [1.9], RAYLEIGH [1.16] and ST. VENANT [1.18] as establishing the theory of the solitary wave but noted that (in 1895) treatises by Lamb and Basset still assert that Airy was correct in his opinion. They said that even RAYLEIGH and McCOWAN [1.19] do not directly refute Lamb's and Basset's assertions "It is the desire to settle this question definitively which has lead us into the somewhat tedious calculations which are to be found at the end of this paper." The KdV equation it­ self noted as 'very important' nevertheless gets rather less discussion. In surveying the history of the solitary wave one should certainly also mention the paper actually entitled 'On the solitary wave' by J. McCOWAN [1.19] and which is quoted by KORTEWEG and DE VRIES [1.7]. ~1cCOWAN isolated and analysed the error of STOKES [1.17] who in 1847 anyway had concluded that the degradation of a wave is an essential characteristic of its mbtion (and not due to friction therefore). He examined and confirmed the approximations of DE BOUSSINESQ [1.9] and RAYLEIGH [1.16] for the sech 2 form of the sol itary wave, derived (1.7) exactly, and noted the agreement with Russell's deductions and their experimental confirmation by BAZIN [1.20]. He also gave an approximate theory of the breaking of waves passing into shallower water. It is remarkable that despite the work of KOR~EWEG and DE VRIES [1.7] four years later so little other work followed. Indeed the signifi­ cance of the soundly based early work [1.3] of Russell's is only now being re­ cognised as the wide range of application of the concepts of the solitary wave and soliton becomes properly appreciated. The references quoted in this short article appear in the list of references to Chap.1. The reader may care to have other reference to the published scientific 376 work of Russell. The following,. no doubt still incomplete, list was compiled by J.C. Eilbeck from the library of the Royal Society of Edinburgh: Russell's Published Scientific Work Russell, John Scott: Notice of the reduction of an anomalous fact in Hydrodynamics, and of a new law of the Resistance of Fluids to the motion of floating bodies.
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