(Joint work with Colette Guillopé and Thierry Colin) The lecture will focus upon long-crested waves in relatively shallow water. Examples in view include incoming waves to the surf zone from deep water, bore propagation on rivers and laboratory studies in wave tanks. While the waves are assumed to propagate in mainly in one direction, say the $x$-direction in a standard $xyz$-Cartesian coordinate system, variations in the $y$-direction are not ignored. The assumption is that these die out as $|y|$ becomes large. Theory is developed for model equations derived in this context. Local well-posedness is in fact straightforward. However, to be useful, the models should remain well posed at least on the so-called Boussinesq time scale. It is to this issue that attention is turned.
