  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.
