Prasad, Phoolan and Ravindran, Renuka (1977) Theory of Non-linear Waves in Multi-dimensions: with Special Reference to Surface Water Waves. In: Institute of Mathematics and its Applications, 20 (1). pp. 9-20.
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The surface water waves are "modal" waves in which the "physical space" (t, x, y, z) is the product of a propagation space (t, x, y) and a cross space, the z-axis in the vertical direction. We have derived a new set of equations for the long waves in shallow water in the propagation space. When the ratio of the amplitude of the disturbance to the depth of the water is small, these equations reduce to the equations derived by Whitham (1967) by the variational principle. Then we have derived a single equation in (t, x, y)-space which is a generalization of the fourth order Boussinesq equation for one-dimensional waves. In the neighbourhood of a wave froat, this equation reduces to the multidimensional generalization of the KdV equation derived by Shen & Keller (1973). We have also included a systematic discussion of the orders of the various non-dimensional parameters. This is followed by a presentation of a general theory of approximating a system of quasi-linear equations following one of the modes. When we apply this general method to the surface water wave equations in the propagation space, we get the Shen-Keller equation.
| Item Type: | Journal Article |
|---|---|
| Publication: | Institute of Mathematics and its Applications |
| Publisher: | Academic Press Ltd |
| Additional Information: | Copyright of this article belongs to Academic Press Ltd. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 21 Jan 2010 07:11 |
| Last Modified: | 21 Jan 2010 07:11 |
| URI: | http://eprints.iisc.ac.in/id/eprint/24386 |
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