The recently proposed concept of metamaterials has opened exciting venues to control wave-matter interaction in unprecedented ways. Here we demonstrate the relevance of metamaterials for inducing acoustic birefringence, a phenomenon which has already found its versatile applications in optics in designing light modulators or filters, and nonlinear optic components. This is achieved in a suitably designed acoustic metamaterial which is non-Eulerian, in the sense that at low frequencies, it

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... be homogenized to a uniform acoustic medium whose behavior is characterized by Euler equation. Thanks to the feasibility of engineering its subwavelength structure, such non-Eulerian metamaterial allows one to desirably manipulate the birefringence process. Our findings may give rise to generation of innovative devices such as tunable acoustic splitters and filters.