By R. S. Johnson

For over 100 years, the idea of water waves has been a resource of interesting and infrequently tough mathematical difficulties. almost each classical mathematical method looks someplace inside of its confines. starting with the creation of the suitable equations of fluid mechanics, the outlet chapters of this article examine the classical difficulties in linear and nonlinear water-wave concept. This units the level for a learn of extra sleek elements, difficulties that supply upward push to soliton-type equations. The ebook closes with an advent to the results of viscosity. the entire mathematical advancements are offered within the simplest demeanour, with labored examples and easy situations rigorously defined. workouts, extra analyzing, and historic notes on a number of the vital characters within the box around off the booklet and make this a terrific textual content for a starting graduate direction on water waves.

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Additional info for A Modern Introduction to the Mathematical Theory of Water Waves

Example text

One simple way to do this is simply to set g == 0; an alternative is to suppose that the initial data is on bounded (or compact) support. 40). In either event, it is then possible to follow just the one component. 76) which has the general solution u(x,t)=f(x-ct). 77) This is then completely determined, given the function u(x, 0) =/(x). 76). 78) ut + ux-uxx = 0. 80) where A: is a real parameter. 79) if co^k-ik2. 81), we see that kx-o)t = k{x - (1 - k2)t}, so that the speed of propagation, is a function of k.

Thus, if the solution is initially on compact support, the two components will move apart, or disperse. The separate components do not change shape, although the observed sum does give the appearance of a changing profile. The speed, co/k, is called the phase speed of the wave; this describes the motion of each individual component. However, as we shall discuss later, another speed, defined by dco/dk, describes the motion of a group of waves. This is called the group speed and, as we shall explain later, it is the speed at which energy is propagated.

7 where u = u(x, 0 Incompressible flows. 6 (a) all satisfy the condition for an incompressible flow, namely V • u = 0.

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