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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.

The Sleeping Beauty problem is a probability riddle with no definite solution for more than two decades and its solution is of great interest in many fields of knowledge. There are two main competing solutions to the problem: the halfer approach, and the thirder approach. The main reason for disagreement in the literature is connected to the use of different probability spaces to represent the same probabilistic riddle. In this work, we analyse the problem from a mathematical perspective, identifying probability distributions induced directly from the thought experiment's rules. The precise choices of probability spaces provide both halfer and thirder solutions to the problem. To try and decide on which approach to follow, a criterion involving the information available to Sleeping Beauty is proposed., Comment: 14 pages, 1 figure

Microwave remote sensing is significantly altered when passing through clouds or dense ice. This phenomenon isn't unique to microwaves; for instance, ultrasound is also disrupted when traversing through heterogeneous tissues. Understanding the average transmission in particle-filled environments is central to improve data extraction or even to create materials that can selectively block or absorb certain wave frequencies. Most methods that calculate the average transmitted field assume that it satisfies a wave equation with a complex effective wavenumber. However, recent theoretical work has predicted more than one effective wave propagating even in a material which is statistical isotropic and for scalar waves. In this work we provide the first clear evidence of these predicted multiple effective waves by using high fidelity Monte-Carlo simulations that do not make any statistical assumptions. To achieve this, we also had to fill in a missing link in the theory for particulate materials: we prove that the incident wave does not propagate within the material, which is usually taken as an assumption called the Ewald-Oseen extinction theorem. By proving this we conclude that the extinction length - the distance it takes for the incident wave to be extinct - is equal to the correlation length between the particles.