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In the present paper we review briefly some of the first works on wave propagation in phononic crystals emphasizing the conditions for the creation of acoustic band-gaps and the role of resonances to the band-gap creation. We show that useful conclusions in the analysis of phononic band gap structures can be drawn by considering the mathematical similarities of the basic classical wave equation (Helmholtz equation) with Schrödinger equation and by employing basic solid state physics concepts and conclusions regarding electronic waves. In the second part of the paper we demonstrate the potential of phononic systems to be used as elastic metamaterials. This is done by demonstrating negative refraction in phononic crystals and subwavelength waveguiding in a linear chain of elastic inclusions, and by proposing a novel structure with close to pentamode behavior. Finally the potential of phononic structures to be used in liquid sensor applications is discussed and demonstrated.


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