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By R. S. Johnson

For over 100 years, the idea of water waves has been a resource of interesting and sometimes tricky mathematical difficulties. nearly each classical mathematical process seems someplace inside its confines. starting with the creation of the correct equations of fluid mechanics, the hole chapters of this article give some thought to the classical difficulties in linear and nonlinear water-wave conception. This units the level for a learn of extra glossy points, difficulties that provide upward thrust to soliton-type equations. The e-book closes with an advent to the consequences of viscosity. all of the mathematical advancements are provided within the simplest demeanour, with labored examples and straightforward circumstances rigorously defined. workouts, additional interpreting, and ancient notes on a few of the vital characters within the box around off the publication and make this a great textual content for a starting graduate direction on water waves.

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Extra resources for A Modern Introduction to the Mathematical Theory of Water Waves

Example text

65) in nondimensional variables. 2 Scaling of the variables We have described the nondimensionalisation of the governing equations, but another equally important transformation is also required. 64), yields the observation that both w and p (on z — 1 + erj) are essentially proportional to s; that is, proportional to the wave amplitude. This makes good sense, particularly as e -> 0, for then w -> 0 and p -> 0: there is no disturbance of the free surface - it becomes a horizontal surface on which w = 0 = p.

The arguments that suggest that such simplifying assumptions lead to problems worthy of consideration will be rehearsed later. However, we take this opportunity to emphasise that the main thrust of our work will be towards an understanding of the equations (and boundary conditions), and what they imply for wave propagation. 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 arguments that suggest that such simplifying assumptions lead to problems worthy of consideration will be rehearsed later. However, we take this opportunity to emphasise that the main thrust of our work will be towards an understanding of the equations (and boundary conditions), and what they imply for wave propagation. 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.

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