FAST BREAKING PAPERS

Why do you think your paper is highly cited?

This paper describes the particle paths within water over a flat bed as a regular wave pattern propagates on the water’s free surface. This is a basic and challenging problem in hydrodynamics and it was widely believed that the particle paths are closed. The surprising fact established in this paper is that the forward motion of a particle is never compensated for by its backward motion. In particular, particle paths are never closed. Thus, the paper offers a better understanding of the flow beneath the waves and opens up new perspectives in the study of water waves.

Does it describe a new discovery, methodology, or synthesis of knowledge?

Let me explain briefly the physical motivation of the paper. Ocean waves are classified as either sea or swell. Irregular patterns made up of various waves with different speeds, wavelengths and amplitudes are called sea. When these waves move past the area of influence of the generating winds, they sort themselves into groups with similar speeds and wavelengths. This process produces swell: a regular pattern of undulation of the ocean surface, often moving thousands of miles away from a storm area to a shore somewhere. These groups of swell with the same wavelength are two-dimensional waves (variations in a direction parallel to the crest line are negligible), periodic, and traveling at constant speed without change of shape at the surface of water with an almost flat bed—a strong variation in the bed topography destroys the wave pattern.

If such waves propagate into a water region previously at rest and with a flat bed, the flow is irrotational (without vorticity). In this case the waves are called Stokes waves. It is widely believed (see, for example, any classical textbook on water waves) that particles in water execute a circular motion as a Stokes wave passes over: individual particles of water do not travel along with the wave, but instead they move in closed, circular or elliptical, orbits (see the free simulation). Support for this conclusion is apparently given by experimental evidence: photographs of small buoyant particles in laboratory wave tanks where almost closed elliptical paths are recognizable.

However, as we show in the present paper, no particle trajectory is closed. Over a wave period, each trajectory of a particle that does not lie on the flat bed consists of a backward/forward and upward/downward movement of the particle, with the path an elliptical-like loop, not closed but with a forward drift. On the flat bed this path degenerates into a back-and-forth horizontal motion. To show that the particle paths are of this form, we investigate certain exact relations satisfied within the fluid (like the conservation of energy and the preservation of the relative mass flux) and we develop a method to analyze the motion below the surface in conjunction with that of the surface. Theoretical analysis is essential since it is very difficult to experimentally measure water waves with accuracy. Also, the highly nonlinear character of the problem and the lack of explicit particular solutions makes it difficult to carry out accurate numerical simulations.

Would you summarize the significance of your paper in layman’s terms?

Its significance is that it offers an understanding of the motion of each water particle as a regular wave pattern propagates at the surface of the sea. Contrary to a possible first impression, what one observes traveling across the sea is not the water but a wave pattern, as enunciated intuitively in the fifteenth century by Leonardo da Vinci in the following form: "...it often happens that 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 place." The waves move much faster than the particles and while the wave propagates in a fixed direction, the particles move for short times opposite to the direction of propagation of the wave, the net movement being however in the direction of wave propagation. The paper marks the beginning of a whole series of investigations devoted to a description of water waves, not only in terms of the pattern propagating on the water's surface but also in terms of the motion induced within the entire fluid.

How did you become involved in this research, and were there any problems along the way?

The problem of particle paths in water waves is more than 200 years old and discussions of it are scattered throughout the literature. The classical approach towards explaining this aspect of water waves consists in analyzing the particle motion after linearizing the nonlinear governing equations for water waves. But even after linearizing the governing equations and obtaining explicit formulas for the free surface and for the fluid velocity field, the system describing particle motion turns out to be again nonlinear. Thus one linearizes again!

Interestingly, the fact that the paths are not closed is lost in the process of performing the second linearization. Indeed, it is only natural to start the investigation of particle paths by simplifying the governing equations for water waves via linearization. In the paper Constantin A and Villari G, "Particle trajectories in linear water waves," J. Math. Fluid. Mech. (doi: 10.1007/s00021-005-0214-2), published online: 19 September 2006, we investigated the linearized governing equations and proved that within this first approximation the particle paths are not closed.

However, while the linearized governing equations are to some extent appropriate to model waves of very small amplitude (meaning practically amplitudes of less than a few centimeters), they do not capture the main features of waves of moderate and large amplitude. After this initial investigation the question was raised: is what we found due to the ample simplification provided by linearization (and thus might be misleading) or is it indicative of what actually goes on? The present paper establishes that the latter is the case.

The mathematical methods employed for the simplified linearized equations and those used in the present paper for the actual nonlinear governing equations differ considerably. While for linear water waves we relied on phase-plane analysis, the nonlinear theory performed in this paper consists in the analysis of a free boundary value problem for harmonic functions. As for problems arising along the way, they were many, ranging from conceptual problems (the findings did not conform to the widely accepted theory) to problems of a technical nature. The way various considerations fit in to provide the general picture still amazes me.

Where do you see your research leading in the future?

The fact that it was possible to provide such information about the motion of each individual water particle is a great motivation to try to understand better the flow pattern in various water waves. An important direction is the case with vorticity: waves interacting with an underlying non-uniform water current. But even within the irrotational setting there are many important aspects not yet fully explored and quite possibly within reach. For example, a widely used approach in ocean engineering to convey information about Stokes waves is by recording pressure data. But very little is known about the pressure within the fluid and conclusions from linear theory lead to frequent large inaccuracies. It is thus important to understand the pressure fluctuations within the framework of nonlinear theory. Presently, in collaboration with Prof. Walter Strauss from Brown University, we have elucidated some properties of the pressure in the water below a Stokes wave.

Do you foresee any social or political implications for your research?

I could comment on the relevance of mathematics to physics. A hallmark of the scientific method is the interdependence of analytical theory on one hand and laboratory experimentation and observation on the other. An appreciation of mathematical rigor and elegance, combined with the power of meaningful abstraction, often leads to breakthroughs in physical insight, while mathematics draws considerable inspiration and stimulation from physics.

Adrian Constantin
Erasmus Smith’s Chair of Mathematics (1762)
School of Mathematics
Trinity College
Dublin, Ireland

RELATED> Also see: Adrian Constantin in ESI Special Topics, March 2005

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