  The spatial structure of the inhomogeneity in a disordered medium determines
how waves scatter and propagate in it. We present a theoretical model of how
the Fourier components of the disorder control wave scattering in a
two-dimensional disordered medium, by analyzing the disordered Green's function
for scalar waves. By selecting a set of Fourier components with the appropriate
wave vectors, we can enhance or suppress wave scattering to filter out unwanted
waves and allow the robust coherent transmission of waves at specific angles
and wavelengths through the disordered medium. Based on this principle, we
propose an approach for creating selective transparency, band gaps and
anisotropy in disordered media. This approach is validated by direct numerical
simulations of coherent wave transmission over a wide range of incident angles
and frequencies and can be experimentally realized in disordered photonic
crystals. Our approach, which requires neither nontrivial topological wave
properties nor a non-Hermitian medium, creates opportunities for exploring a
broad range of wave phenomena in disordered systems.
