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Propagation of long-crested water waves

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  • The present essay is concerned with a model for the propagation of three-dimensional, surface water waves. Of especial interest will be long-crested waves such as those sometimes observed in canals and in near-shore zones of large bodies of water. Such waves propagate primarily in one direction, taken to be the $x-$direction in a Cartesian framework, and variations in the horizontal direction orthogonal to the primary direction, the $y-$direction, say, are often ignored. However, there are situations where weak variations in the secondary horizontal direction need to be taken into account.
        Our results are developed in the context of Boussinesq models, so they are applicable to waves that have small amplitude and long wavelength when compared with the undisturbed depth. Included in the theory are well-posedness results on the long, Boussinesq time scale. As mentioned, particular interest is paid to 1000 the lateral dynamics, which turn out to satisfy a reduced Boussinesq system. Waves corresponding to disturbances which are localized in the $x-$direction as well as bore-like disturbances that have infinite energy are taken up in the discussion.
    Mathematics Subject Classification: Primary: 35A01, 35A02, 35A09, 35Q35, 35Q51; Secondary: 35Q86, 45G15, 76B03, 76B15, 76B25.

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