We are interested in a model of rotating fluids, describing the motion of the

ocean in the equatorial zone. This model is known as the Saint-Venant, or

shallow-water type system, to which a rotation term is added whose amplitude

is linear with respect to the latitude ; in particular it vanishes at the equator.

After a physical introduction to the model, we describe the various waves

involved and study in detail the resonances associated to those waves. We

then exhibit the formal limit system (as the rotation becomes large), obtained

as usual by filtering out the waves, and prove its wellposedness. Finally we

prove three types of convergence results: a weak convergence result towards a

linear, geostrophic equation, a strong convergence result of the filtered solutions

towards the unique strong solution to the limit system, and finally a "hybrid"

strong convergence result of the filtered solutions towards a weak solution to

the limit system. In particular we obtain that there are no confined equatorial

waves in the mean motion as the rotation becomes large.