Solitary Water Waves

Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis by Adrian Constantin was published by SIAM in 2011. This book, which is part of the CBMS-NSF Regional Conference Series in Applied Mathematics, explores some of the main results and recent developments in nonlinear water waves, addresses fundamental aspects of the field, and overviews several current research topics. 

The following text comes from chapter five, entitled “Solitary Water Waves,” and is modified slightly for clarity.

Some two-dimensional traveling water waves have a localized profile (a hump of water that drops smoothly back to a flat water level far ahead and far behind the wave crest). For these solitary waves, the initial profile \(\eta(X,0)=\eta_0(X)\) moves at constant speed \(c>0\) and without distortion in the direction \(X\) of wave propagation; at a later time \(t\), the profile \(\eta(X,t)\) is a translation of the initial profile by an amount \(ct\) in the positive \(X\) direction, \(\eta(X,t)=\eta_0(X-ct)\). If one sets up a solitary wave profile of depression¹—a possibility that is within reach for laboratory experiments—it will immediately collapse into small oscillatory waves.² Solitary waves were first observed by John Scott Russell in 1834 (see Figure 1). He described them in a report to the British Association for the Advancement of Science in 1844 [6]:

“I believe I shall best introduce the phaenomenon by describing the circumstances of my own first acquaintance with it. I was observing the motion of a boat that 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 30 feet long and a foot to a foot and a half in height. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phaenomenon.”

The ability of this water wave to retain its shape for a long period of time is quite remarkable. Russell conducted numerous detailed experiments to investigate the nature of what he called “the great wave of translation,” but which came to be known as the solitary wave.

Solitary waves can be easily obtained experimentally by dropping a weight at one end of a long rectangular tank (see Figure 2). Russell was the first to observe that an initial elevation might—depending on the relation between its height and length—evolve into a pure solitary wave of elevation, a single solitary wave of elevation with some residual oscillatory waves, or two or more solitary waves of elevation with or without residual oscillatory waves. He also attempted to produce solitary waves of depression, but found that an initial depression is transformed into oscillatory waves of gradually increasing length and decreasing amplitude. From his laboratory instruments, Russell concluded that the speed \(c\) of a solitary wave is given by

\[c = \sqrt{g(d+a)},\tag1\]

where \(d\) is the average depth of the water above a flat bed, \(a\) is the wave amplitude, and \(g\) is the acceleration due to gravity. Notice that \((1)\) indicates that higher solitary waves travel faster.

In the early days, the existence of solitary waves excited some controversy because Russell’s conclusions contradicted the predictions of George Biddell Airy [1], who was considered at that moment to be the leading expert in hydrodynamics.³ The conflict between Russell’s observations and the accepted theory was resolved independently by Joseph Valentin Boussinesq in 1871 [2] and Lord Rayleigh in 1876 [5]. Their insight was to incorporate weak nonlinear effects, particularly an appropriate allowance for vertical acceleration that was neglected in Airy’s linear theory. The two showed that suitable approximations to the governing equations for water waves lead, for wave amplitudes \(a>0\) that are sufficiently small, to the solitary wave solution

\[\eta(X,t)=\frac{a}{\cosh^2[\beta(X-ct)]},\tag2\]

with the wave speed \(c\) given by Russell’s formula \((1)\) and \(\beta=\frac{\sqrt{3a}}{2d\sqrt{d+a}}\). They did not, however, derive a model equation (via an approximation procedure that starts from the governing equations for water waves) that admits the solution \((2)\). This final step was completed by Diederik Korteweg and Gustav de Vries in 1895 [4]. For present purposes, we consider not the equation written in physical variables, but its normalized form

\[\eta_t+\eta\eta_X+\eta_{XXX}=0.\tag3\]

We look for a solitary wave solution \(\eta(X,t)=f(X-ct)\), where \(c>0\) and \(f(s)\), \(f'(s)\), and \(f''(s)\) tend to zero as \(|s|\rightarrow\infty\). One can integrate \((3)\) by substituting this form, but it is easier to check that for any constant \(\alpha \in \mathbb{R}\), the wave profile

\[\eta_\alpha(X,t)=\frac{3c}{\cosh^2(\frac{\sqrt{c}}{2}(X-ct)+\alpha)}\tag4\]

represents a solitary wave solution of the Korteweg-de Vries (KdV) equation.⁴ At any fixed time \(t\), the graph of the solitary wave \((4)\) is formed by a hump that drops smoothly and rapidly to zero away from its crest, which is located at \(X=ct-\frac{2\alpha}{\sqrt{c}}\). Russell observed in his experiments that solitary waves with greater amplitude—defined for a solitary wave of elevation as the maximal elevation of the wave profile above the asymptotic flat level⁵—travel faster and are narrower. This is borne out in the solution of \((4)\), as the amplitude \(a\) of the wave is three times the wave speed \(c\), and the width of the wave—defined as the distance between the points of height \(a/2\)—is inversely proportional to the square root of its amplitude. Note the similarity between \((4)\) and \((2)\).

¹ The wave profile would be the reflection in a horizontal line of the profile of a wave of elevation.

² Solitary waves of depression are only possible if one allows for capillary effects; consequently, these are waves of very small amplitude. We consider gravity waves (of elevation) that propagate at the free surface of a layer of water with an average depth that measures from tens of centimeters to hundreds of meters. Displacements of the free surface are of interest from the point of view of waves at sea or in a canal.

³ More details on these fascinating historical aspects are available in [3]. Basically, the linear theory of waves of small amplitude fails to yield any approximation to solitary waves [7].

⁴ It is not difficult to check that these are all of the solitary wave solutions.

⁵ Notice that this level is beneath the mean water level \(Y=\int_\mathbb{R}\eta(x)dx\) — in the absence of elevation, the amount of water raised above it cannot disappear in view of mass conservation.

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References

[1] Airy, G. (1841). Tides and waves. Encyclop. Metrop., 3.
[2] Boussinesq, J.V. (1871). Théorie de l’intumescence appelée ‘onde solitaire’ ou ‘de translation’, se propageant dans un canal rectangulaire. C.R. Acad. Sci. Paris, 72, 755-759. 
[3] Craik, A.D.D. (2004). The origins of water wave theory. Annu. Rev. Fluid Mech., 36, 1-28.
[4] Korteweg, D.J., & de Vries, G. (1895). On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag., 39(240), 422-443.
[5] Rayleigh, B. (1876). On waves. Philos. Mag., 1(4), 257-279.
[6] Russell, J.S. (1844). Report on waves. In Report of the fourteenth meeting of the British Association for the Advancement of Science. York, U.K.
[7] Stoker, J.J. (1957). Water waves: The mathematical theory with applications. New York, NY: Interscience Publishers.