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Adrian Constantin & Walter A. Strauss
answer a few questions about this month's
new hot paper in the field of Mathematics.
From
•>>March 2005
Field:
Mathematics
Article Title: Exact steady periodic water waves with vorticity
Authors: Constantin,
A;Strauss, W
Journal: COMMUN PURE APPL MATH
Volume: 57
Page: 481-527
Year: APR 2004
* Lund Univ, Dept Math, POB 118, S-22100 Lund, Sweden.
* Lund Univ, Dept Math, S-22100 Lund, Sweden.
* Brown Univ, Dept Math, Lefschetz Ctr Dynam Syst, Providence, RI 02912 USA.
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 Why do you think your paper is highly cited?
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“...the connection between the shape of the waves and the properties of the underlying current is much more intricate: in our paper we show that regular waves can occur for an arbitrary vorticity distribution.”
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This paper is concerned with periodic waves which propagate
steadily in a fixed direction on water running over an impermeable
flat bed. They are plane waves on the water's surface, the motion
being identical in any direction parallel to the crest line. Almost
all previous mathematical studies for water waves have assumed that
there is no vorticity at all; that is, that there are no eddies in the
flow. However, there are many circumstances where vorticity plays an
essential role, for example in the interaction of water waves and
nonuniform currents. The paper opens the door to a general study of
water waves that have vorticity and large amplitude. Even though our
paper does not specifically consider tsunamis, tsunamis do possess a
lot of vorticity and the recent tragic tsunami in Asia may also have
heightened interest in our paper.
Does it describe a new discovery or a new methodology that's useful
to others?
We discovered many new shapes of water waves, in fact, an infinite
number of them. We developed a mathematical method to analyze the
motion of the water particles below the surface in conjunction with
the motion of the surface. It is very difficult to experimentally
measure water waves with any accuracy. Therefore theoretical analysis
plays a central role. Furthermore, the highly nonlinear character of
the problem makes it difficult to carry out accurate numerical
simulations. For example, some recent numerical studies have suggested
that regular water waves (having one crest per period and a profile
that is decreasing from crest to trough, symmetric with respect to the
crest line) could not propagate on underlying currents of variable
vorticity. This turns out to be false. In fact, the connection between
the shape of the waves and the properties of the underlying current is
much more intricate: in our paper we show that regular waves can occur
for an arbitrary vorticity distribution.
Could you summarize the significance of your paper in layman's
terms?
Its significance is that it opens the door to the study of water
waves with eddies (little whirlpools). Such waves are commonly seen in
nature, for example as waves generated by wind which propagate over
running water. Tidal waves and tsunamis also possess eddies.
How did you become involved in this research?
Initially, both Adrian Constantin and Walter A. Strauss
were studying water waves of very
small amplitude. The propagation of such waves can be modeled by
certain nonlinear partial differential equations, such as the
Korteweg-deVries equation, which are approximations to the basic
governing equations for water waves. However, in order to study
large-amplitude waves it became necessary to work with the exact
governing equations. There is a very particular example by Gerstner of
a large-amplitude water wave with vorticity. We wanted to study such
waves from a modern viewpoint and go far beyond Gerstner's particular
example. This led us to the present paper. We are now studying to what
extent these waves are stable.
Adrian Constantin
Erasmus Smith's Chair of Mathematics
Department of Mathematics
Trinity College
Dublin, Ireland
Walter A. Strauss,
L. Herbert Ballou University Professor
Lefschetz Center for Dynamical Systems
Brown University
Providence, RI, USA
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ESI Special Topics,
March 2005
Citing URL - http://www.esi-topics.com/nhp/2005/march-05-AdrianConstantin.html
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