Propagation of long-crested water waves

The present essay is concerned with a model for the propagation ofthree-dimensional, surface water waves. Of especial interest will be long-crestedwaves such as those sometimes observed in canals and in near-shore zones oflarge bodies of water. Such waves propagate primarily in one direction, taken tobe the x−direction in a Cartesian framework, and variations in the horizontaldirection orthogonal to the primary direction, the y−direction, say, are oftenignored. However, there are situations where weak variations in the secondaryhorizontal direction need to be taken into account.Our results are developed in the context of Boussinesq models, so they areapplicable to waves that have small amplitude and long wavelength when comparedwith the undisturbed depth. Included in the theory are well-posednessresults on the long, Boussinesq time scale. As mentioned, particular interestis paid to the lateral dynamics, which turn out to satisfy a reduced Boussinesqsystem. Waves corresponding to disturbances which are localized in thex−direction as well as bore-like disturbances that have infinite energy are takenup in the discussion