Abstract : Two distinctive topics are investigated in this dissertation. However, we focus each time on the asymptotic study of the system. The first part deals with the behavior of a flow, constituted of two immiscible, homogeneous fluids under the only influence of gravity. Such a system is widely used in oceanography, as a model for density-stratified fluids. First, we introduce the governing equations of our problem. Then, we construct several asymptotic models, in different regimes. The regimes at stake are the shallow water regime (where the depth of the fluids is assumed to be small when compared with the internal wavelength), and the long wave regime (with the additional smallness assumptions of small deformations at the surface and at the interface). Each of the models is rigorously justified, thanks to a consistency, or a convergence result. Finally, we deal with the so-called dead water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. Again, we construct, justify, and produce numerical simulations of asymptotic models for this problem. The provided analysis allows to predict the behavior of the flow, and in which situations the dead-water effect occurs. The second part is dedicated to the wave propagation in inhomogeneous media, having in mind the applications of photonic crystals. The structure of these materials allow them to shape the flow of light. In particular, we study the effects of the presence of defects in the structure, modeled by singularities, or discontinuities in the microstructure. The effect of such interfaces is predominant in the asymptotic behavior of scattering quantities, such as the transmission coefficient, when the size of the microstructure vanishes.