Acoustic waves in a halfspace material filled with random particulate

Particulate materials include powders, emulsions, composites, and many others. This is why measuring these has become important for both industry and scientific applications. For industrial applications, the greatest need is to measure dense particulates, in-situ, and non-destructively. In theory, this could be achieved with acoustics: the standard method is to send an acoustic wave through the particulate and then attempt to measure the effective wave speed and attenuation. A major obstacle here is that it is not clear how to relate the effective wave speed and attenuation to the reflection and transmission coefficients, which are far easier to measure. This is because it has been very difficult to mathematically account for different background mediums. In this paper, we resolve this obstacle. To help comprehension, we present how to account for different background mediums for a simple case: a halfspace filled with a random particulate, where the background of the halfspace is different from the exterior medium. The key to solving this problem was to derive a systematic extension of a widely used closure approximation: the quasi-crystalline approximation. We present some numerical results to demonstrate that the reflection coefficient can be easily calculated for a broad range of frequencies and particle properties.