Solid-solid thermal boundary resistance $Rb$ plays an important role in determining heat flow, both in cryogenic and room-temperature applications, such as very large scale integrated circuitry, superlattices, and superconductors. The acoustic mismatch model (AMM) and the related diffuse mismatch model (DMM) describe the thermal transport at a solid-solid interface below a few Kelvin quite accurately. At moderate cryogenic temperatures and above, $Rb$ is dominated by scattering caused by various sources, such as damage in the dielectric substrates and formation of an imperfect boundary layer near the interface, making $Rb$ larger than that predicted by AMM and DMM. From a careful review of the literature on $Rb,$ it seems that scattering near the interface plays a far more dominant role than any other mechanism. Though scattering near the interface has been considered in the past, these models are either far too complicated or are too simple (i.e., inaccurate) for engineering use. A new model, called the scattering-mediated acoustic mismatch model (SMAMM), is developed here that exploits the analogy between phonon and radiative transport by developing a damped wave equation to describe the phonon transport. Incorporating scattering into this equation and finding appropriate solutions for a solid-solid interface enable an accurate description of $Rb$ at high temperatures, while still reducing to the AMM at low temperatures, where the AMM is relatively successful in predicting $Rb.$

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