  When atmospheric storms pass over the ocean, they resonantly force
near-inertial waves (NIWs); internal waves with a frequency close to the local
Coriolis frequency $f$. It has long been recognised that the evolution of NIWs
is modulated by the ocean's mesoscale eddy field. This can result in NIWs being
concentrated into anticyclones and provide an efficient pathway for their
propagation to depth. Whether mesoscale eddies are effective at modulating the
behaviour of NIWs depends on the wave dispersiveness $\varepsilon^2 =
f\lambda^2/\Psi$, where $\lambda$ is the deformation radius and $\Psi$ is a
scaling for the eddy streamfunction. If $\varepsilon\gg1$, NIWs are strongly
dispersive, and the waves are only weakly affected by the eddies. We calculate
the perturbations away from a uniform wave field and the frequency shift away
from $f$. If $\varepsilon\ll1$, NIWs are weakly dispersive, and the wave
evolution is strongly modulated by the eddy field. In this weakly dispersive
limit, ray-tracing emerges as a valid description of the NIW evolution even if
the large-scale atmospheric forcing apparently violates the requisite
assumption of a scale separation between the waves and the eddies. The
large-scale forcing excites many wave modes, each of which varies on a short
spatial scale and is amenable to asymptotic analysis analogous to the
semi-classical analysis of quantum systems. The strong modulation of weakly
dispersive NIWs by eddies has the potential to modulate the energy input into
NIWs from the wind, but under oceanic conditions, this effect should be small.
