  An inertia-gravity wave (IGW) propagating in a vertically sheared, rotating
stratified fluid interacts with the pair of inertial levels that surround the
critical level. An exact expression for the form of the IGW is derived here in
the case of a linear shear and used to examine this interaction in detail. This
expression recovers the classical values of the transmission and reflection
coefficients $|T|=\exp(-\pi \mu )$ and $|R|=0$, where $\mu^2=J(1+\nu^2)-1/4$,
$J$ is the Richardson number and $\nu$ the ratio between the horizontal
transverse and along-shear wavenumbers.
  For large $J$, a WKB analysis provides an interpretation of this result in
term of tunnelling: an IGW incident to the lower inertial level becomes
evanescent between the inertial levels, returning to an oscillatory behaviour
above the upper inertial level. The amplitude of the transmitted wave is
directly related to the decay of the evanescent solution between the inertial
levels. In the immediate vicinity of the critical level, the evanescent IGW is
well represented by the quasi-geostrophic approximation, so that the process
can be interpreted as resulting from the coupling between balanced and
unbalanced motion.
  The exact and WKB solutions describe the so-called valve effect, a dependence
of the behaviour in the region between the inertial levels on the direction of
wave propagation. For $J < 1$ this is shown to lead to an amplification of the
wave between the inertial levels. Since the flow is inertially unstable for
$J<1$, this establishes a correspondence between the inertial-level interaction
and the condition for inertial instability.
