By R. S. Johnson
For over 100 years, the idea of water waves has been a resource of interesting and infrequently tough mathematical difficulties. nearly each classical mathematical approach looks someplace inside of its confines. starting with the creation of the fitting equations of fluid mechanics, the outlet chapters of this article think of the classical difficulties in linear and nonlinear water-wave conception. This units the degree for a research of extra glossy facets, difficulties that provide upward push to soliton-type equations. The booklet closes with an creation to the consequences of viscosity. all of the mathematical advancements are offered within the simplest demeanour, with labored examples and easy instances conscientiously defined. workouts, additional examining, and old notes on the various very important characters within the box around off the booklet and make this an awesome textual content for a starting graduate path on water waves.
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Extra resources for A Modern Introduction to the Mathematical Theory of Water Waves
It is not our purpose to provide an engineering or physical appraisal of the usefulness of these theories as they apply to the many and varied types of water waves that are encountered in nature. The importance of these considerations should not be underestimated though; they are paramount in the design of ships, offshore platforms, breakwaters, and dams, in the prediction and avoidance of catastrophes following earthquakes or storms, and a host of other areas of significance to mankind. Nevertheless, we shall extend our methods to some more obviously relevant and practical applications, such as flows with shear (rotational flows) and propagation over variable depth.
Nevertheless, we shall extend our methods to some more obviously relevant and practical applications, such as flows with shear (rotational flows) and propagation over variable depth. It is clear that our field of discussion will be somewhat restricted, but even so we shall still face immensely difficult mathematical problems that we wish to overcome. The most natural way forward is to develop a suitable - but systematic - approximation procedure. To this end we need to characterise problems in terms of the sizes of various fundamental parameters.
1 There are many texts - and many good texts - on fluid mechanics; readers may have their favourites, but we list a few that can be recommended. A wide-ranging and well-written text is Batchelor (1967); more recent texts are Paterson (1983) and Acheson (1990), this latter including an introduction to waves in fluids. A more descriptive approach is provided by Lighthill (1986), and there are the classical texts: Lamb (1932), Schlichting (1960), Rosenhead (1964) and Landau & Lifschitz (1959). 3 We shall provide many references to research papers and texts later, but two texts that can be mentioned at this stage are Stoker (1957) and Crapper (1984).
A Modern Introduction to the Mathematical Theory of Water Waves by R. S. Johnson