The properties of reflection and transmission of internal gravity waves in planarly stratified, ideal fluids with a parallel shear flow are investigated. The shear flow and the Brunt-Väisälä frequency are modelled by arbitrary, smooth profiles. When the fluid has one critical level, explicit conditions for wave overreflection and for the existence of singular neutral modes can be derived by means of a Wronskian approach.
The same applies when there are two or more critical levels. The theory not only applies to fluids with solid boundaries but also to unbounded flows.
In particular cases, when the profiles of the shear flow and the Brunt-väisälä frequency are such that reduction to an equation of the Fuchsian type is possible, closed-form expressions for the neutral stability curves can be derived by application of a generalized theorem of Miles, derived in this work. For a given configuration of profiles the neutral stability curves determined in this manner prove to be the complete set of curves for these profiles.
C.A. van Duin. Some general properties of ideal fluids with regular critical levels
KNMI number: WR-83-01, Year: 1983, Pages: 36