In this article, a general method is proposed to model layered systems with two-by-two transfer matrices, and further, to solve for the acoustic absorption, reflection, and transmission coefficients. Since the proposed method uses the matrix representation of various layers and interfaces from the Transfer Matrix Method (TMM), the equation system can be established efficiently. However, the traditional TMM can lose stability when there is a large disparity between the magnitudes of the waves traveling in opposite directions within the layers (i.e., at higher frequencies, for a thick layer, or for extreme parameter values). In such cases, the contribution of the most attenuated wave can be masked by numerical errors and can induce instability when solving the system. Therefore, in the proposed method, to stabilize the calculated acoustic properties of the system, the principle is to ensure the accuracy of the wave attenuation terms by decomposing each layer's transfer matrix and reformulating the equation system. This method can couple different layer types in a general way and is easy to assemble and implement with numerical code. The predicated acoustic properties of layered systems calculated using the proposed method have been validated by comparison with those predicted by other existing methods.

The physical properties of particle stacks have been of interest for a long time, and the acoustic properties of such materials have been actively investigated in recent times. Traditional acoustic theories, such as the Biot theory, can serve as a guide for determining the general properties of the stacks, but they do not allow the identification of the differences between the particle stacks and traditional acoustic materials, which are usually modeled as homogeneous continua. Recent research suggests that the Biot theory, combined with depth-dependent stiffness and equivalent density, can be used to model such materials. In this work, a finite difference (FD) scheme based on the Biot theory has been developed based on the idea that the apparent stiffness of the particle stack varies with depth within of the stack. This FD scheme is two-dimensional in cylindrical coordinates with an axisymmetric condition imposed at the axis of the cylinder. This approach is thus suitable for modeling common scenarios in which particle stacks are tested in a cylindrical standing wave tube. The results of several example cases are shown in this paper.