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The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline, and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the Higher Order Spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both in theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of studying these kinds of problems, to incorporate piecewise constant background shear.

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)is the ratio of density variation to the mean density and F r 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 deep water surface gravity waves are Boussinesq while shallow water surface gravity waves arent. However, in the presence of moderate/strong shear, Boussinesq approximation implies the conventionally accepted A t 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.

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