Westward Ho! Musing on Mathematics and Mechanics

Westward Ho! Musing on Mathematics and Mechanics


A previous Westward Ho! [1] considered some of the fascinating mathematics associated with the mechanics of water waves. Here, inspired by a visit to the Gloucester Waterways Museum, Alan Champneys continues that story by describing how canals can provide effective laboratories for understanding the behaviour of solitary waves. We shall learn of a remarkable scientist and engineer, John Scott Russell, and observations he made in 1834 on a canal near the present-day campus of Heriot-Watt University. The pioneers of fluid mechanics originally disbelieved Scott Russell’s work, as his experiments did not fit the theory. This and many instances of bad luck throughout his career, have led to Scott Russell’s many achievements being largely unknown outside the specific community of mathematicians and physicists working in soliton theory. We shall thus ponder on the importance of giving weight to empirical observation and the need to challenge current orthodoxy when creating new mathematical models to explain physical phenomena.

Canal Dreams

I can recall several memorable canal holidays as a teenager, where we merrily pootled along in a narrow boat, through the idyllic British countryside with faint smells of warm beer and diesel fumes \dots . Today I enjoy cycling along the towpath of the Kennet and Avon Canal, dodging the twin hazards of puddles and those out for an afternoon stroll, four abreast, oblivious to the etiquette of shared-use paths.

The Kennet and Avon is part of a network of canals that criss-cross England and Wales and central Scotland. Most of these were built in the golden age of British canal building in the 1770s to 1830s. Over the last 70 years, it has become a passion for local volunteer groups to seek to restore such canals to their former glory. In 1970, the Waterway Recovery Group was founded as the national coordinating body for all these groups, and in 2012 the Canal and Rivers Trust was established to manage the network for the nation. Not used for haulage anymore, narrowboats are available for hire by holidaymakers and day-trippers alike. It has also become fashionable for bohemian types to own a narrowboat as their main home. The National Waterways Museum pictured in Figure 1 is situated in the docks of the city of Gloucester. But why Gloucester, which seems largely disconnected from the UK canal network?

Westward-Ho-Musing-on-Mathematics-and-Mechanics-figure-1
Figure 1: The ‘National’ Waterways Museum (left) and the Gloucester and Sharpness Canal (right). Photographs by the author.

In fact, since 2010, the museum has officially been known as the National Waterways Museum Gloucester, to avoid confusion with the other two ‘National Waterways Museums’, one being at the junction between the Grand Union and Manchester Ship Canals at Ellesmere Port and the other being beside the Grand Union Canal between Milton Keynes and Northampton at Stoke Bruerne (the latter is now in fact known as the Canal Museum). Nevertheless, the restored, Grade II listed, red-brick former timber, grain and alcohol warehouse that contains Gloucester’s museum still bears the moniker of the {National Waterways Museum} in capitals in a bold mock-3D font (see Figure 1, left).

The museum is situated in the recently restored (and gentrified) Gloucester Docks, the most inland port in the UK. The port sits at the far end of the Gloucester and Sharpness Canal which was was once the broadest and deepest canal in the world. At 25km, it connects Gloucester to Sharpness on the Severn estuary, allowing boats to bypass a treacherous tidal stretch of the River Severn. The canal was opened in 1827, and like the much later Manchester Ship Canal, was designed to take large vessels; it is over 25m wide and 8m deep and even today can be used by ships up to 64m long. Its large swing-bridges and ornate bridge-keepers’ cottages add to the uniqueness, as does its delineation of the boundary of the Slimbridge Wildfowl and Wetlands Trust Centre. Another unique feature is the Purton Hulks, the largest ship graveyard in inland Britain, consisting of abandoned boats that were deliberately beached in the 1950s to shore up the barrier between the Severn and the canal.

I recall a visit to the museum, quite a few years ago, in which I spent rather too long playing with a table-top exhibit that allowed visitors to manipulate flows of water along a small network of canals, locks and sluices. I recall my pleasure (and the impatience of the rest of my family) as I was able to launch a wave by suddenly opening and then closing a set of lock gates. This was a solitary wave (or soliton), an isolated single heap of water that travelled at a regular speed around the circuit I had created. What I was trying to do, in my own small way, was to recreate an experiment I had seen pictured, reproduced here in Figure 2.

That in itself was an emulation of an event that took place some 150 years previously by the Scottish polymath John Scott Russell. This recreation of Scott Russell’s solitary wave on the aqueduct that bears his name near the site of present-day Heriot-Watt University, was performed by participants of the Conference on Nonlinear Coherent Structures in Physics and Biology, in July 1995. One can see the solitary wave propagating a few metres in front of the boat. When the boat is subsequently brought to a halt, the wave will continue on its merry way along the canal.

Westward-Ho-Musing-on-Mathematics-and-Mechanics-figure-1a
John Scott Russell (1808–82)

John Russell was born near Glasgow in 1808, the son of a schoolmaster and clergyman – the Scott of his surname was added later to mark the maiden name of his mother who died shortly after his birth. During his lifetime, Scott Russell was principally known as a naval architect and shipbuilder, pioneering the use of steam power and iron hulls. He was Brunel’s shipbuilder of choice (more of that later). He built the boat that raced (and lost) to the yacht America, for a trophy that thereafter became known as the now famous America’s Cup. He designed the Admiralty’s first steam warship, HMS Warrior now lovingly restored in Portsmouth Harbour, and built a ship that held the prize for the fastest crossing from London to Australia. In 1865, he wrote the definitive three-volume The Modern System of Naval Architecture [2].

Westward-Ho-Musing-on-Mathematics-and-Mechanics-figure-2
Figure 2: Recreation in July 1995 of Scott Russell’s solitary wave on the aqueduct that bears his name near the site of present-day Heriot-Watt University, by participants of the Conference on Nonlinear Coherent Structures in Physics and Biology. One can see the solitary wave propagating a few metres in front of the boat. When the boat is subsequently brought to a halt, the wave will continue on its merry way along the canal. Photograph by Chris Eilbeck, copyright, Heriot-Watt University.

In common with many great Victorian scientists and engineers, Scott Russell’s contributions extend way beyond one domain. He wrote on subjects as diverse as sound-wave propagation, railway rolling stock resistance, social politics and military planning. He was almost certainly responsible for suggesting to Prince Albert what became the Great Exhibition of 1851, serving in a capacity the Prince referred to as the ‘indefatigable secretary’. Another notable achievement was his design of what was then the world’s largest roof, for Vienna’s International Exhibition. And yet, of this great man there is only one, relatively obscure, biography [3].

It seems I am not alone in enjoying creating solitons in model water features. The title ‘Canal Dreams’ of this piece is borrowed from the name of a novel by one of my favourite authors (although not one of my favourites of his books), the Scottish author Iain Banks, who died in 2013. In his only work of non-fiction, Raw Spirit [4], Banks documents a series of road trips in the summer of 2003 in which he visits every whisky distillery in Scotland. He describes a stopover in which he and a friend get to play in the water park on the esplanade in Montrose [4, pp. 146–147]:

I am convinced I see a soliton. \ldots A soliton is a single wave that propagates along a channel over long distances, losing very little energy. It’s all to do with the width and depth of the channel and the wavelength and height of the wave itself; if these figures are all in a certain proportion to each other a kind of harmony is established that sets up a soliton, and it’ll just sweep calmly along a channel for a long, long time. They were first noticed in the Dutch canals where they could keep going for kilometres. This miniature example \ldots eventually hits the shallows at the far end of one of the little concrete channels after ten meters or so, but while it lasts it’s beautiful.

I shout ‘Woo-hoo!’ \ldots Everybody else is looking at me like I’m a bit of a mad fellow, but hey, I’m used to that.

It is an interesting reflection on Scott Russell’s relative obscurity in the public consciousness that Banks, who by his writings identified himself as a proud Scotsman, appears to think the solitary wave was a Dutch discovery. But what did John Scott Russell do, and why is he not better remembered?

John Scott Russell had aspirations to be a scientist, but never quite made it. His forte was not to theorise in mathematical terms, but to construct a plausible argument based on physical analogies and painstaking experimental investigation. He started at the University of Glasgow at the startlingly early age of 13. Aged 23, he held the position of acting Professor of Natural Philosophy at the University of Edinburgh. However, it seems John’s talents lay in making things. As an 18-year-old, he started a unique venture of running steam-powered coaches between Edinburgh and Glasgow. The business was unfortunately ruined after the combined effects of what was possibly the world’s first fatal car crash and the underhand tactics of the owners of the stage coaches and turnpikes. Before starting his shipbuilding career in earnest, he was also offered a contract by the Scottish Union Canal Company to investigate whether a steam-powered canal barge could transport passengers between Edinburgh and Glasgow.

This work led to an incident recalled by Scott Russell in an oft-quoted passage written some 10 years later [5, pp. 319–320]:

I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, \ldots

The term ‘wave of translation’ did not really catch on, and soon became known as a solitary wave. It was a startling discovery. Until this point in the history of fluid mechanics, waves were thought to be fundamentally oscillatory with a speed depending only on the depth of the fluid; see [6]. Specifically, in Lagrange’s theory, the wave speed c = \sqrt{gh} where g is gravitational acceleration and h is the depth of the canal. To see this, let us start with Laplace’s equation for the velocity potential form of motion of a wave of height w, in a channel of height h (see [1, equations (1)–(4)]):

(1)   \begin{equation*} \frac{\partial^2 \phi }{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0, \quad \mbox{for all} \quad 0<y<h+w. \end{equation*}

(2)   \begin{equation*} (v=) \quad \frac{\partial \phi}{\partial y} = 0 \quad \mbox{for} \quad y = 0. \end{equation*}

(3)   \begin{equation*} (v=) \quad \frac{ \partial \phi}{\partial y} = \frac{\partial w}{\partial t} \quad \mbox{at} \quad y=h+w. \end{equation*}

(4)   \begin{equation*} \frac{\partial \phi}{\partial t} + g w + \frac{1}{2} \left [ \left ( \frac{\partial \phi}{\partial x} \right )^2 + \left ( \frac{\partial \phi}{\partial y} \right )^2 \right ] = \mbox{const} \quad \mbox{at} \quad y = h + w. \end{equation*}

Suppose that the depth h of the channel is small compared to the wavelength \lambda and the wave height w \ll h. Now suppose that the
variation of \phi in the y-direction is weak and so we can write to leading order

(5)   \begin{equation*} \phi = \phi_0(x,t) + y^2 \phi_2(x,t). \end{equation*}

Note that there cannot be a linear term in y, otherwise we could not satisfy the bottom boundary condition (2). Feeding this form of \phi into the Laplace equation (1), we find

(6)   \begin{equation*} \phi_2 = - \frac{1}{2}\frac{\partial^2 \phi_0}{\partial x^2}. \end{equation*}

The main difficulty is how to simplify the nonlinear, dynamic boundary condition (4). If we now suppose that the velocities are all of the same order of magnitude as w, and keep only the leading-order terms, this condition becomes

(7)   \begin{equation*} \frac{\partial \phi_0}{\partial t} + g w = \mbox{const} \quad \mbox{at} \quad y = h + w, \end{equation*}

which differentiates to

(8)   \begin{equation*} \frac{\partial^2 \phi_0}{\partial t^2} + g \frac{\partial \phi}{\partial y} = 0 \quad \mbox{at} \quad y = h + w. \end{equation*}

To obtain (8), we have used the kinematic boundary condition (3) to substitute for \mathrm{d} \phi/\mathrm{d} y for \mathrm{d} w/\mathrm{d} t on on the free surface.

Combining (8) with (5) and keeping only the leading-order terms gives

(9)   \begin{equation*} \frac{\partial^2 \phi_0 }{\partial t^2} - gh \frac{\partial^2 \phi_0}{\partial x^2} = 0. \end{equation*}

Differentiation of this equation with respect to t and substitution of \partial \phi_0/\partial t = g w\ + const. from (7) we find that the height w(x,t) satisfies the linear wave equation:

    \[\frac{\partial^2 w}{\partial t^2} - gh \frac{\partial^2 w}{\partial x^2} =0.\]

This equation has the well-known d’Alembert solution

(10)   \begin{equation*} w(x,t) = F_1(x-ct) + F_2 (x+ct), \quad \mbox{where} \quad c = \sqrt{gh}, \end{equation*}

for any arbitrary functions F_1 and F_2. Earlier analysis by Lagrange, which was later improved upon by Airy, showed that the only physically realisable forms for the wave profiles F_1 and F_2 are periodic functions, specifically \sin(k X) or \cos(k X), for some as yet undetermined wavelength \ell=2\pi/k.

Scott Russell’s real scientific talents lay in painstaking experimental observation, and for the next 10 years he carried out over 20,000 distinct observations of solitary waves including full-scale tests on a 30-mile stretch of canal, in which there was greatly varying depth. He also made observations on the rivers Clyde and Dee and in a 20-foot tank he constructed in his garden. The results were written up in some detail [5]; see also Figure 3. What Scott Russell discovered from these experiments was that the creation of a solitary wave was highly repeatable. The speed of the wave was dependent only on the depth of the canal and the height of the wave, according to the formula c=\sqrt{g(h+w)}.

In his garden he constructed a raised reservoir, from which water was suddenly released into a channel. According to the linear theory of waves, one would expect to see that the water level in the remainder of the channel would oscillate. Yet, if sufficient water was placed in the raised reservoir, a solitary wave would be produced, with its characteristic hyperbolic secant squared (sech^2) shape. With too much water though, two solitary waves were released, with different heights and different speeds.
He also recorded many other interesting features of solitary waves, including what appears to be the first mention of the group velocity of a train of waves.

Westward-Ho-Musing-on-Mathematics-and-Mechanics-figure-3
Figure 3: Some of Scott Russell’s illustrations in [5] of his experimental findings.
Following his observations, Scott Russell was keen to publicise his work. Although received as a fellow of the Royal Society of Edinburgh, his repeated attempts to get his work published by London’s Royal Society were rejected. His painstaking experimental results were, in fact, widely ignored or dismissed for many years, as told by Oliver Darrigol in his history of 19th century fluid mechanics [6]. First, in 1845, the British Astronomer Royal, George Airy published a lengthy article on waves and tides for the Encyclopedia Metropolitan. Among other things, Airy introduced the notion of dispersion. That is waves of different wavelength travel at different speeds. Short waves travel slower, see Figure 4, but provided the wavelength is about three times as long as the depth of the channel, then one recovers Lagrange’s formula, equation (10), to good accuracy.

Airy was particularly scathing of Scott Russell’s observations and dismissed true solitary waves as being impossible both mathematically and on physical grounds – rather claiming that what Scott Russell had observed were just examples of periodic waves with particularly long wavelengths. For his part, Scott Russell when republishing his 1844
report was able to insert a comment on Airy’s work (quoted in [6, p. 53]):

This paper I have long expected with anxiety, in the hope that it would furnish a final solution of this difficult problem [the discrepancy between wave theory and wave phenomena], a hope justified by the reputation and position of the author, as well as the clear views and elegant processes that characterise some of his former papers. \ldots It is deeply to be deplored that the methods of investigation employed with so much knowledge, and applied with so much tact and dexterity, should not have led to a better result.

Ouch!

Next to enter the debate was the Irish rising star of water-wave theory, George Gabriel Stokes. He showed that a solitary wave initial profile would, in effect, be composed of many different sine waves, each with a different wavelength which, due to dispersion, would cause the solitary wave to break up. However, Stokes did leave the door open, because he could see the compelling nature of Scott Russell’s experimental evidence. Much later, in 1879, Stokes wrote to Lord Kelvin to say that he was now of an opinion that, by taking the expansion in (9) to higher order in h, solitary waves really could propagate as steady waves, without degradation. Kelvin wrote back to say that he disagreed.

Westward-Ho-Musing-on-Mathematics-and-Mechanics-figure-4
Figure 4: The sine-function shape of a small amplitude wave according to the Laplace–Lagrange–Airy theory (left), defining the wavelength lambda, frequency omega and wave speed c. The graph of Airy’s dispersion relation that links wave speed to wavelength for a channel of depth 1m (right).

But seemingly unknown to both of them, the new kid on the block Lord Rayleigh had already shown in 1876 that solitary waves are indeed a valid solution, if one carefully chooses the right balance at the next order. Moreover, seemingly unknown to all three, the same conclusion, albeit by a different method, had already been reached a few years earlier by a young Frenchman, Joseph Boussinesq. By a quirk of fate, Boussinesq’s analysis was also largely forgotten in favour of a much later derivation of the same set of mathematical equations by Korteweg and de Vries in 1895, as I described in [1]. For Scott Russell though, this vindication came much too late.

Other bad luck befell Scott Russell throughout his career. His biographer, Emmerson [3], claims he did not receive a knighthood for his role in the 1851 Exhibition, because, as the lowly son of a Scottish non-conformist, he was of the wrong social class. Then there is the whole episode with Brunel. History, quite unfairly, tends to regard Scott Russell as Brunel’s shipbuilder. But as Emmerson points out, really they were collaborators – many of the novel features ascribed to Brunel had already featured on Scott Russell’s previous ships – but Brunel did not recognise the notion of a collaborator, ever. The failure of Brunel’s last great ship, the Great Eastern, coming soon after a fire and subsequent bankruptcy of Scott Russell’s shipyard, coincided with Brunel’s untimely death in 1859. This led to a besmirching of Scott Russell’s reputation and indirectly, via some rather murky doings, him being expelled from the Institution of Civil Engineers. Scott Russell was forced to spend the rest of his days on overseas commissions.

It is fair to say though that Scott Russell is not entirely blameless for the poor reception of his science during his lifetime. Unlike other experimentalists and empiricists, such as Michael Faraday who was well known for clear and simple language, Scott Russell was rather prone to grandiose over-claiming. For example, the above oft-quoted passage (see e.g. [7]), is generally terminated mid sentence after Wave of Translation. The completion in the original is [5, p. 320]:

a name which it now very generally bears; which I have since found to be an important element in almost every case of fluid resistance, and ascertained to be the type of that great moving elevation of the sea, which, with the regularity of a planet, ascends our rivers and rolls along our shores.

In fact, not only did Scott Russell claim that the presence of solitary waves explains the tides and fluid drag, but he also suggested they form the basis of the propagation of light.

Nevertheless, since the 1970s and the popularity of integrable systems and soliton theory, John Scott Russell’s reputation has begun to be repaired. This effort has been spearheaded by Chris Eilbeck, now emeritus professor of mathematics at Heriot-Watt University; see [8]. Eilbeck was behind the recreation of the soliton depicted in Figure 2 on the Scott Russell Aqueduct. This aqueduct was so named when it was pointed out to Eilbeck, by fellow soliton pioneer Al Scott, that Scott Russell’s soliton observations had been carried out at his Hermiston Experimental Station. Moreover an aqueduct was about to be constructed at Hermiston to carry the Union Canal over the newly built Edinburgh city by-pass.

Is there a moral? Perhaps it is about the need for a greater mutual appreciation between empirical and mathematically driven science. How often in today’s metrics-driven publication culture, do we mathematicians balk at so-called high profile publications from experimentalists that announce startling new data in glossy graphics? A mathematical model, if there is one – or, more likely, some beautiful but misguided simulation – is like an afterthought, with the role of the mathematician or computer scientist on the paper being merely to provide a service to the ‘real’ science. Sometimes the ‘correct’ theoretical explanation comes much later, with far less fanfare, after painstaking work in mathematical modelling and analysis.

But if Scott Russell’s story teaches us anything, it is that we mathematicians too can be guilty of dismissing the entirely empirical approach to science, especially if it contradicts the prevailing mathematical orthodoxy. We often forget, when attempting to make sense of the real world, the adage attributed to George Box that ‘all models are wrong’. It would seem to me that we need to encourage a more mature discourse, where mathematicians learn to be humble before real data, and empiricists learn the value of true dialogue with mathematicians.

Alan Champneys CMath FIMA
University of Bristol

References

  1. Champneys, A.R. (2017) Westward Ho! Boardmasters, Math. Today, vol. 53, no. 4, pp. 180–184. Available at https://tinyurl.com/MT-Boardmasters.
  2. Scott Russell, J. (1865) The Modern System of Naval Architecture, Day and Son, London.
  3. Emmerson, G.S. (1977) John Scott Russell. A Great Victorian Engineer and Naval Architect, John Murray, London.
  4. Banks, I. (2004) Raw Spirit: In Search of the Perfect Dram, Random House, London.
  5. Scott Russell, J. (1845) Report on Waves, 14th Meeting of the British Association for the Advancement of Science, York, September 1844, John Murray, London, pp. 311–390, Plates XLVII–LVII.
  6. Darrigol, O. (2005), Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, Oxford.
  7. Remoissenet, M. (1999) Waves Called Solitons: Concepts and Experiments, 3rd edition, Springer, Berlin.
  8. Eilbeck, J.C. (2007) John Scott Russell, www.macs.hw.ac.uk/~chris/scott_russell.html (accessed 15 February 2018).

Reproduced from Mathematics Today, April 2018

Download the article, Westward Ho! Musing on Mathematics and Mechanics

Image credit: National Waterways Museum © Alan Champneys
Image credit: Gloucester and Sharpness Canal © Alan Champneys
Image credit: Recreation in July 1995 by Chris Eilbeck © Heriot-Watt University
Image credit: John Scott Russell © Science Photo Library
Published

Leave a Reply

Your email address will not be published. Required fields are marked *