  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. We present how to account for
different background mediums for a simple case, to help comprehension: a
halfspace filled with a random particulate, where the background of the
halfspace is different to 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 (QCA). We present some numerical results to
demonstrate that the reflection coefficient can be easily calculated for a
broad range of frequencies and particle properties.
