Directional reflection modeling via wavenumber-domain reflection coefficient for 3D acoustic field simulation

TL;DR

The paper addresses direction-dependent acoustic reflection by eschewing interior material modeling in favor of a boundary operator framework. It defines a wavenumber-domain reflection coefficient $\mathbf{C}_{\mathrm{r}}$ that maps incident to reflected plane-wave components in $\bm{k}_{xy}$ space, and shows how to estimate this operator from multi-source measurements, convert it into a normalized acoustic admittance $\mathbf{B}$, and apply it as a nonlocal boundary condition in Boundary Element Method simulations. The key contributions include a practical estimation approach (with optional sparsity regularization), a closed-form admittance representation, and thorough 3D validation demonstrating accurate reproduction of specular and multi-directional scattering with substantially reduced meshing effort. The results indicate potential for efficient, data-driven boundary modeling in architectural acoustics, material characterization, and noise control.

Abstract

This study proposes a framework for incorporating wavenumber-domain acoustic reflection coefficients into sound field analysis to characterize direction-dependent material reflection and scattering phenomena. The reflection coefficient is defined as the amplitude ratio between incident and reflected waves for each propagation direction and is estimated from spatial Fourier transforms of the incident and reflected sound fields. The resulting wavenumber-domain reflection coefficients are converted into an acoustic admittance representation that is directly compatible with numerical methods such as the Boundary Element Method (BEM), enabling simulation of reflections beyond simple specular components. Unlike conventional extended reaction models, the proposed approach avoids explicit modeling of the material interior. This significantly reduces computational cost while allowing direct use of measured data, empirical models, or user-defined directional reflection characteristics. The validity of the proposed formulation was previously demonstrated by the authors through two-dimensional sound field simulations, in which accurate reproduction of direction-dependent reflection behavior was confirmed. In the present work, the framework is extended to three-dimensional analysis, demonstrating its applicability to more realistic and complex acoustic environments. The proposed approach provides a practical and flexible tool for simulating direction-dependent acoustic reflections and scattering, with potential applications in architectural acoustics, material characterization, and noise control.

Figures (8)

• Figure 1: Schematic illustration of reflection in a three-dimensional acoustic field. A plane wave is incident from the upper half-space onto the boundary surface at $z=0$, where it is reflected back into the same half-space.
• Figure 2: Numerical analysis setup for the two boundary conditions (units in meters). Omnidirectional point sources (white triangles) are placed on a plane 0.10 m above the boundary surface. A total of 100 sources are distributed with a fixed radial distance of $\bm{r=0.75}$ m using Fibonacci sampling to ensure an approximately uniform angular distribution without directional bias. Receiver points (white circles) are located on a plane 0.01 m above the boundary and arranged in a regular grid of 400 points with a spacing of 0.04 m. (a) Flat plate boundary condition; (b) Periodic slit boundary condition. Both boundaries are modeled as acoustically rigid (Neumann) surfaces.
• Figure 3: Estimated wavenumber-domain reflection matrices $\mathbf{C}_\mathrm{r}$. For the flat plate, energy is concentrated along the main diagonal, indicating dominant specular reflection. In contrast, the periodic slit exhibits pronounced off-diagonal components, corresponding to multi-directional scattering.
• Figure 4: Reconstructed reflection directivity maps $|P_\mathrm{r}|$ computed using the estimated $\mathbf{C}_\mathrm{r}$. Blue crosses indicate the incident directions.
• Figure 5: Conventional BEM (Full geometry mesh, flat plate)