  Wave localization is ubiquitous in disordered media -- from amorphous
materials, where soft-mode localization is closely related to materials
failure, to semi-conductors, where Anderson localization leads to
metal-insulator transition. Our main understanding, though, is based on
discrete models. Here, we provide a continuum perspective on the wave
localization in two-phase disordered elastic media by studying the scalar wave
equation with heterogeneous modulus and/or density. At low frequencies, soft
modes arise as a result of disordered elastic modulus, which can also be
predicted by the localization landscape. At high frequencies, Anderson-like
localization occurs due to disorder either in density or modulus. For the
latter case, we demonstrate how the vibrational dynamics changes from plane
waves to diffusons with increasing frequency. Finally, we discuss the
implications of our findings on the design of architected soft materials.
