  In this work, we study the nonlinear traveling waves in density stratified
fluids with depth varying shear currents. Beginning the formulation of the
water-wave problem due to [1], we extend the work of [4] and [18] to examine
the interface between two fluids of differing densities and varying linear
shear. We derive as systems of equations depending only on variables at the
interface, and numerically solve for periodic traveling wave solutions using
numerical continuation. Here we consider only branches which bifurcate from
solutions where there is no slip in the tangential velocity at the interface
for the trivial flow. The spectral stability of these solutions is then
determined using a numerical Fourier-Floquet technique. We find that the
strength of the linear shear in each fluid impacts the stability of the
corresponding traveling wave solutions. Specifically, opposing shears may
amplify or suppress instabilities.
