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Elasticity theory provides an accurate description of the long-wavelength vibrational dynamics of homogeneous crystalline solids, and with supplemental boundary conditions on the displacement field can also be applied to abrupt heterojunctions and interfaces. The conventional interface boundary conditions, often referred to as ''connection rules,'' require that the displacement field and its associated stress field be continuous through the interface. We argue, however, that these boundary conditions are generally incorrect for epitaxial interfaces, and we give the general procedure for deriving the correct conditions, which depend essentially on the detailed microscopic structure of the interface. As a simple application of our theory we analyze in detail a one-dimensional model of an inhomogeneous crystal, a chain of harmonic oscillators with an abrupt change in mass and spring-stiffness parameters. Our results have implications for phonon dynamics in nanostructures such as superlattices and nanoparticles, as well as for the thermal boundary resistance at epitaxial interfaces.

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