The theory of water waves has a long and distinguished history, with important contributions from Euler, Stokes, Kelvin, Zakharov and others. The subject has had practical application throughout history; its recent significance was demonstrated dramatically by the tsunami of 2004, the storm surge that battered New Orleans in 2005, and "rogue waves" that seem to appear and disappear mysteriously. In each of these cases, the large amplitudes of these waves were predicted poorly or not at all and caused enormous damage. For these reasons and others, much of the emphasis in current research on water waves is on the dynamics of nonlinear waves, with finite amplitude. More abstractly, this subject is a prototype of a dynamical system that exhibits interesting nonlinear phenomena: resonant interactions, solitons, wave breaking and more.
In a series of ten lectures, Professor Harvey Segur (University of Colorado at Boulder) will develop the mathematical theory of water waves, starting from basic principles and ending at forefronts of research in several directions. The lectures are intended to be accessible to someone with a basic knowledge of partial differential equations and of classical physics. The lecture series has two objectives: (i) to identify mathematical methods that have proven useful in solving important problems in this subject; and (ii) to identify important open problems.
Some of the topics to be discussed are:
-What are the governing equations of water waves? Stokes' formulation (from 1847) is the most commonly accepted statement of the problem, and it describes very many (but not all) of the phenomena observed. -Which physical phenomena can be predicted easily from these equations? Some examples are phase velocity, group velocity, dispersive waves, wave focusing, deep water vs. shallow water, and more. -How can participants at the conference observe these physical phenomena?
This lecture series will differ from others in the CBMS series in that a laboratory will be set up, where Professor Diane Henderson (Penn State University) will create a series of physical experiments on water waves. Participants at the conference will be encouraged to 'play' in the laboratory, and to see how mathematical concepts from the lectures appear naturally in physical experiments.
-Both deterministic and statistical models are used to predict oceanic events like storms, dangerously high seas and tsunamis. What are the advantages and disadvantages of each kind of model? -Careful analysis, applied to Stokes' formulation, has led to impressive new results, both in terms of well-posedness and in terms of existence of wave patterns of permanent form. What are these results? -What have we learned from approximate theories of water waves? Is there more to learn from them? For example, the interlocking miracles of soliton theory were first discovered in the KdV equation, which was derived by Korteweg and de Vries as an approximate model of wave propagation in shallow water. -What important physical phenomena are not described by Stokes' formulation? -What are some of the open problems in this subject?
