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An inertia-gravity wave (IGW) propagating in a vertically sheared, rotatingstratified fluid interacts with the pair of inertial levels that surround thecritical level. An exact expression for the form of the IGW is derived here inthe case of a linear shear and used to examine this interaction in detail. Thisexpression recovers the classical values of the transmission and reflectioncoefficients $|T|=\exp(-\pi \mu )$ and $|R|=0$, where $\mu^2=J(1+\nu^2)-1/4$,$J$ is the Richardson number and $\nu$ the ratio between the horizontaltransverse and along-shear wavenumbers.  For large $J$, a WKB analysis provides an interpretation of this result interm of tunnelling: an IGW incident to the lower inertial level becomesevanescent between the inertial levels, returning to an oscillatory behaviourabove the upper inertial level. The amplitude of the transmitted wave isdirectly related to the decay of the evanescent solution between the inertiallevels. In the immediate vicinity of the critical level, the evanescent IGW iswell represented by the quasi-geostrophic approximation, so that the processcan be interpreted as resulting from the coupling between balanced andunbalanced motion.  The exact and WKB solutions describe the so-called valve effect, a dependenceof the behaviour in the region between the inertial levels on the direction ofwave propagation. For $J < 1$ this is shown to lead to an amplification of thewave between the inertial levels. Since the flow is inertially unstable for$J<1$, this establishes a correspondence between the inertial-level interactionand the condition for inertial instability.

Inertia-gravity waves in inertially stable and unstable shear flows