  In this paper, we first revisit the celebrated Boussinesq approximation in
stratified flows. Using scaling arguments we show that when the background
shear is weak, the Boussinesq approximation yields either (i) $A_t\ll
\mathcal{O}(1)$ or (ii) $Fr_c^2 \ll \mathcal{O}(1)$, where $A_t$ is the ratio
of density variation to the mean density and $Fr_c$ is the ratio of the phase
speed to the long wave speed. The second clause implies, contrary to the
commonly accepted notion, that a flow with large density variations can also be
Boussinesq. Indeed, we show that deepwater surface gravity waves are Boussinesq
while shallow water surface gravity waves are not. However, in the presence of
moderate/strong shear, Boussinesq approximation implies the conventionally
accepted $A_t\ll \mathcal{O}(1)$. To understand the inertial effects of density
variation, our second objective is to explore various non-Boussinesq shear
flows and study different kinds of stably propagating waves that can be present
at an interface between two fluids of different background densities and
vorticities. Furthermore, three kinds of density interfaces - neutral, stable
and unstable - embedded in a background shear layer, are investigated.
Instabilities ensuing from these configurations, which includes
Kelvin-Helmholtz, Holmboe, Rayleigh-Taylor and triangular-jet, are studied in
terms of resonant wave interactions. The effects of density stratification and
the shear on the stability of each of these flow configurations are explored.
Some of the results, e.g. the destabilizing role of density stratification,
stabilizing role of shear, etc. are apparently counter-intuitive, but physical
explanations are possible if the instabilities are interpreted from wave
interactions perspective.
