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Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

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Nonlinear Water Waves

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2158))

Abstract

Most of the waves that are observed on the surface of the world’s oceans, seas and lakes are wind generated. Once initiated, these water waves propagate substantial distances before their energy is dissipated—propagation distances in excess of hundreds or thousands times a wavelength are needed for the occurrence of a significant energy loss. We address some fundamental aspects of water-wave propagation once waves have been generated, within the framework of inviscid two-dimensional flow theory and in the absence of underlying currents. The emphasis is placed upon periodic travelling gravity water waves of large amplitude. These wave patterns can only be fully understood in terms of nonlinear effects, linear theory being not adequate for their description. An in-depth mathematical study is made possible by uncovering the rich structure of the hydrodynamical free-boundary problem under investigation, taking advantage of insights from physical observation, experimental evidence and numerical simulations. The interdisciplinary nature of this classical research subject is also reflected in the fact that its theoretical investigation relies on an interplay between methods and techniques from dynamical systems, complex analysis, functional analysis, topology, harmonic analysis, and the calculus of variations.

…fluid dynamicists were divided into hydraulic engineers who observed things that could not be explained and mathematicians who explained things that could not be observed.

Sir James Lighthill (1924–1998)

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Notes

  1. 1.

    For example, large parts of the Pacific Ocean floor are very flat. Relevant for the discussion are abyssal plains—vast sediment-covered regions of the sea floor that are the flattest areas on Earth, with an almost total absence of geographic features, presenting variations in depth in the range of 10–100 cm/km of horizontal distance. Abyssal plains result from the blanketing of a pre-existing irregular ocean floor topography by accumulated land-derived sediment. They are found in all major sea and ocean basins, usually adjacent to a continent and at depths between 2 and 6 km, covering overall almost a third of the Earth’s surface (about as much as all the exposed land combined). The largest are hundreds of km wide and thousands of km long. For example, in the North Atlantic the Sohm Plain, located to the south of Newfoundland, has an area of approximately 900,000 km2. The flat areas of the bed of the Pacific Ocean cover a greater area than those of the Atlantic Ocean, but the proportion due to abyssal plains is considerably smaller since deep trenches (with lengths of thousands of km, and generally hundreds of km wide) near the continents trap most of the sediment before it reaches the open ocean. In the Mediterranean Sea the most extensive abyssal plains are the Balearic Abyssal Plain (at 2800 m depth, with a total area of about 240,000 km2, its major sources of sediment being the Ebro and Rhone rivers) and the Tyrrhenian Abyssal Plain (at 3500 m depth and with a total area of about 30,000 km2, it is composed by two plains separated by an undersea ridge, with sediment feeded partly by the Tiber river but mostly composed of volcanic materials—several terrestrial and submarine volcanoes being located in this area).

  2. 2.

    For a prescribed wavelength, this is the highest two-dimensional periodic travelling wave [26]. However, the highest wave is not the fastest nor the most energetic [53].

  3. 3.

    The choice of the coefficient ρ g d is due by calibration relative to the standard unit of pressure measurement, so that p takes on numerical values.

  4. 4.

    This is very useful since \(u_{x} + v_{y} = 0\) ensures the existence of a stream function ψ(x, y, t) such that u = ψ y and \(v = -\psi _{x}\) throughout the simply connected non-dimensionalised fluid domain, thus reducing the number of unknown functions. Indeed, the vector field (u, v) is uniquely determined by ψ.

  5. 5.

    This denomination of the regime is well-established, despite being slightly confusing when first encountered. A shallow-water regime is not indicative of a small water depth but of a water depth that is small relative to the typical wavelength. For example, the devastating December 2004 tsunami was generated by earthquakes in the Indian Ocean that produced waves with wavelengths in excess of 100 km in water of average depth d ≈ 4 km (see [13]). Despite this large average depth, these tsunami waves qualify as shallow-water waves since δ ≪ 0. 04 in this setting. Out in the open sea the typical amplitude a of these waves was less than 1 m. With the corresponding \(\varepsilon \approx 25 \times 10^{-5}\), far from the shore it is therefore reasonable to model these waves within the linear shallow-water regime.

  6. 6.

    Since the time-dependence amounts to a simple translation of the horizontal spatial variable x by ct, it suffices to investigate the problem at time t = 0.

  7. 7.

    That is, a linear combination of solutions is again a solution, so that the overall wave motion is simply the sum of all its parts. The previous discussion shows that, in general, the speeds of the individual wave components are different, so that the resulting interaction pattern is not a travelling wave.

  8. 8.

    That is, setting δ = 0, and therefore also \(\varepsilon = 0\), due to (34).

  9. 9.

    In particular, the wave speed c should not be confused with the horizontal component U of the fluid velocity: energy is moving at the speed of the wave, but water is not (see the discussion in Sect. 4.4).

  10. 10.

    For example, in linear water wave theory energy is roughly proportional to amplitude squared (see [45]); each linear metre of a wave 2 m above the sea level represents an energy flow of about 25 kW (34 horsepower), cf. [35].

  11. 11.

    This terminology is used since variations of such parameters bring about qualitative changes in the flow regime even for fixed geometry and boundary conditions. If (U 0 − c, 0) is the uniform subcritical flow with relative mass flux \(\mathcal{M}\) and head \(\mathcal{H}\), then \(\mathcal{M} =\rho (U_{0} - c)\xi _{-}\), so that \(\frac{1} {\mu } = \frac{(U_{0}-c)^{2}} {g\xi _{-}}\) is the square of the Froude number \(Fr = \frac{\vert U_{0}-c\vert } {\sqrt{g\xi _{-}}}\). In hydraulics the range Fr > 1 defines the supercritical uniform open channel flows (U 0 − c, 0), to be avoided in design, if possible, since they are highly unstable.

  12. 12.

    For a discussion of the effects of a current (that is, κ ≠ 0) we refer to [23].

  13. 13.

    Allowing the parameter \(\lambda\) to be multi-dimensional corresponds to “multiparameter bifurcation theory”.

  14. 14.

    For genuine waves no explicit formulas are available and the separation of variables that is manifest in (98) does not occur.

  15. 15.

    The corresponding solution to (56) is \(\psi (x,y) = y - 1\), in view of (99).

  16. 16.

    Since \(\mathcal{C}_{w}\) is the sum of the periodic Hilbert transform \(\mathcal{C}\) and the operator of convolution with a specific smooth 2π-periodic function, cf. [24], this replicates a feature which is well-established for \(\mathcal{C}\) (see [18]). By the same token, \(\mathcal{C}_{w}\) shows some “traces of continuity”, in that it has the intermediate mean-value property on the set in which it is finite, as it is the case for \(\mathcal{C}\), cf. [66].

  17. 17.

    Due to the third relation in (80), the analytic extension to a neighbourhood of the line \(b = -w\) is a consequence of Schwarz’s reflection principle.

  18. 18.

    Generally, in the calculus of variations, a change in the geometric or topological type of the level sets of the functional is revealed by the existence of a critical point.

  19. 19.

    For the relevant basic results in Fourier analysis we refer to the discussion in [18].

  20. 20.

    It is easy to see that the unique solution of (125) is \(\psi (x,y) = y - 1\).

  21. 21.

    Note that the expression under the square root in (129) is \(\frac{1} {4} -\frac{Z^{3}} {27} <0\), so that imaginary numbers are involved although all three roots are real.

  22. 22.

    We recall from Sect. 3.2.1 that the distinction between deep and shallow water waves has nothing to do with absolute water depth, being determined by the ratio δ of the water’s depth to the wavelength of the wave.

  23. 23.

    However, \(\mathcal{A}\) is not necessarily a maximal connected subset of the solution set \(F(\lambda,f) = 0\), since the branching out of solutions from \(\mathcal{A}\) is not excluded.

  24. 24.

    We choose τ > 0 for definiteness; all the arguments can be easily adapted to deal with the case τ < 0.

  25. 25.

    This function is actually also dependent on the parameter τ. Since only its value at τ = τ 0 is relevant to our considerations, we do not keep track of this parameter, in order to ease the notation.

  26. 26.

    For example, it is convenient to derive a priori estimates in the Sobolev spaces W 1, r[−π, π] of 2π-periodic functions with weak derivatives in L r[−π, π] for 1 < r < 3.

  27. 27.

    English translation: “⋯ the wave flees the place of its creation, while the water does not; like the waves made in a field of grain by the wind, where we see the waves running across the field while the grain remains in its place”.

  28. 28.

    Each streamline [Ψ = constant] intersects such a curve in precisely one point, and each curve lies between adjacent vertical lines below the crest and the trough, respectively, without intersecting these lines.

  29. 29.

    The primary information from any wave observation technique consists of an estimate of wave height, wave speed and wave period (see the discussions in [3, 15, 31, 39, 49].

  30. 30.

    In the absence of waves we have equality in (165) and in (166).

  31. 31.

    As suggested by numerical simulations and casual observations with the naked eye (see the discussions in [25, 28]).

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  1. Department of Mathematics, King’s College London, Strand, London, WC2L 2RS, UK

    Adrian Constantin

  2. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Adrian Constantin

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Constantin, A. (2016). Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows. In: Constantin, A. (eds) Nonlinear Water Waves. Lecture Notes in Mathematics(), vol 2158. Springer, Cham. https://doi.org/10.1007/978-3-319-31462-4_1

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