Details of Grant 

EPSRC Reference:	GR/R32109/01
Title:	Open Mathematical Questions About Water Waves
Principal Investigator:	Toland, Professor JF
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Researcher Co-Investigators:
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Department:	Mathematical Sciences
Organisation:	University of Bath
Scheme:	Standard Research (Pre-FEC)
Starts:	25 September 2001	Ends:	24 December 2001	Value (£):	6,570
EPSRC Research Topic Classifications:	Mathematical Analysis Non-linear Systems Mathematics
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Summary on Grant Application Form
1. Over many years an enormous effort has been expended in an attempt to understand the properties of the Stokes steady wave of limiting form. Recently it has offered quite a challenge to refined numerical methods, and extensive computational effort continues to be devoted to illucidating the very delicate features of this wave in a neighbourhood of its crests. However further experimentaion always seems to be desirable to resolve the delicate issues which this highly singular phenomenon represents. In addition to speculating about the crest singularity, Stokes suggested that the extreme wave (exists and) is convex between consecutive crests. Numerical experiments seem to bear this out, but so far there is no analytic proof of convexity, which is a global property, although existence was settled 20 years ago. We propose to study the convexity question.2. A standing wave here is one in which the surface of an infinitely deep flow oscillates up and down periodically in time while at the same time remaining periodic in space. Thus it is like an uncountablely infinite number of coupled oscillators (the particles in the surface) in periodic motion. The existence question here is completely open. The proposal is to attempt an extension of our theory for standing waves on flows of finite depth to the infinite depth case where, in mathematical terms, the analysis is even more degenerate.
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