Acoustic gradient barriers (exactly solvable models)

This paper reviews the physical fundamentals and mathematical formalism for problems concerning acoustic waves passing through gradient wave barriers formed by a continuous one-dimensional spatial distribution of the density and/or elastic parameters of a medium in a finite-thickness layer. The physical mechanisms of such processes involve nonlocal (geometric) normal and anomalous dispersion determined by the profiles and geometric parameters of the gradient barrier. The relevant mathematics relies on exactly solvable gradient barrier models with up to three free parameters and on the auxiliary barrier method with which the exactly solvable models found can be used to build new, also exactly solvable, models for such barriers. The longitudinal and shear wave transmission spectra through the gradient barriers considered are presented, and the dependence of these spectra on the gradient and curvature of the density distribution and on the elastic parameters of the barrier is expressed using general formulas corresponding to the geometrical and abnormal geometric dispersion. Examples of reflectionless tunneling of sound through gradient barriers formed either by the elastic parameter distribution in an inhomogeneous layer or by curvilinear boundaries of a homogeneous layer are considered. It is also shown that by using subwavelength gradient barriers and periodic structures composed of them, phonon crystal elements can be fabricated.