Abstract
We show that in general there is not only one effective wave, but there is a series of effective waves in an ensemble averaged particulate material. By particulate material we mean a medium filled with randomly placed particles and consider waves governed by the scalar wave equation. Although most of these waves decay rapidly, they make a significant contribution to reflection and transmission beyond the low-frequency regime. In two spatial dimensions, we develop an efficient method to calculate all these waves, which gives highly accurate predictions when compared to a finite-difference scheme. This method is, to the authors knowledge, the first of its kind to give accurate predictions across a broad frequency range and particle volume fraction.
To the right of the dashed line are particles, whose position has been numerically ensemble averaged. To the left is just a homogenious medium. What is shown here (though faintly) is that more than one transmitted wave exists in an ensemble averaged medium. The paper shows this with much more clarity!