Propagation of long-crested water waves
Résumé
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 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.