  In this paper, we investigate the wave solutions of a stochastic rotating
shallow water model. This approximate model provides an interesting simple
description of the interplay between waves and random forcing ensuing either
from the wind or coming as the feedback of the ocean on the atmosphere and
leading in a very fast way to the selection of some wavelength. This
interwoven, yet simple, mechanism explains the emergence of typical wavelength
associated to near inertial waves. Ensemble-mean waves that are not in phase
with the random forcing are damped at an exponential rate, whose magnitude
depends on the random forcing variance. Geostrophic adjustment is also
interpreted as a statistical homogenization process in which, in order to
conserve potential vorticity, the small-scale component tends to align to the
velocity fields to form a statistically homogeneous random field.
