Adrian Constantin talks with
ScienceWatch.com and answers a few questions about
this month's Fast Breaking Paper in the field of
Mathematics.

Field: Mathematics
Article Title: The trajectories of particles in Stokes
waves
Authors: Constantin, A
Journal: INVENT MATH
Volume: 166
Issue: 3
Page: 523-535
Year: DEC 2006
* Trinity Coll Dublin, Sch Math, Dublin 2, Ireland.
* Trinity Coll Dublin, Sch Math, Dublin 2, Ireland.
(addresses may have been truncated; see full
article)

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.

Figure depicting
a typical particle path above the flat
bed.

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!

"The mathematical methods
employed for the simplified linearized equations and
those used in the present paper for the actual
nonlinear governing equations differ
considerably.."

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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