In situ sound absorption estimation with the discrete complex image source method

TL;DR

This work advances in situ sound absorption estimation by introducing the Discrete Complex Image Source Method (DCISM), which represents the reflected field as a distribution of complex image sources along a line and couples it with a regularized inverse problem to estimate absorption coefficients from microphone-array data. Compared to the traditional Image Source Model (ISM), DCISM yields more accurate reconstructions of surface pressure and particle velocity, and absorption coefficients that align with spherical-wave incidence results, particularly when Gauss-Legendre discretization is used. The approach is validated through simulations of infinite and finite non-locally reacting absorbers and through in situ measurements on PET, Melamine, and Helmholtz absorbers, demonstrating feasibility with compact hardware and modest sensor counts. The findings highlight the importance of including a distributed monopole representation for accurate in situ characterization and provide practical guidelines for array design and future work on edge-diffraction effects and finite-size behavior.

Abstract

Estimating the sound absorption in situ relies on accurately describing the measured sound field. Evidence suggests that modeling the reflection of impinging spherical waves is important, especially for compact measurement systems. This article proposes a method for estimating the sound absorption coefficient of a material sample by mapping the sound pressure, measured by a microphone array, to a distribution of monopoles along a line in the complex plane. The proposed method is compared to modeling the sound field as a superposition of two sources (a monopole and an image source). The obtained inverse problems are solved with Tikhonov regularization, with automatic choice of the regularization parameter by the L-curve criterion. The sound absorption measurement is tested with simulations of the sound field above infinite and finite porous absorbers. The approaches are compared to the plane-wave absorption coefficient and the one obtained by spherical wave incidence. Experimental analysis of two porous samples and one resonant absorber is also carried out in situ. Four arrays were tested with an increasing aperture and number of sensors. It was demonstrated that measurements are feasible even with an array with only a few microphones. The discretization of the integral equation led to a more accurate reconstruction of the sound pressure and particle velocity at the sample's surface. The resulting absorption coefficient agrees with the one obtained for spherical wave incidence, indicating that including more monopoles along the complex line is an essential feature of the sound field.

Figures (12)

• Figure 1: Schematic of the sound scene. The point source (red) at $\textbf{r}_{\text{s}}$ excites the sound field. The absorber of dimensions $L_x \times L_y \times d$ (light blue) is baffled at $z=-d$. The Arrays 0--3 are also shown. Note that Array 0 contains Arrays 1--3, Array 1 contains Arrays 2--3, and Array 2 contains Array 3.
• Figure 2: (a) The schematics of the FEM/BEM multiphysics simulation used to model finite absorbers and the reconstruction grid of receivers; and (b) schematics of the sound field 3D scanner used in the measurements to position the microphone automatically.
• Figure 3: (a) The illustration of the windowing process used to separate the incident and scattered sound field components from the room's reflections; and (b) schematics of the Helmholtz resonant absorber with a slit width, $w$, and length, $l$, indicated.
• Figure 4: The NMSE of the reconstructions for the DCISM (GLE, GLA, and MP discretizations) and the ISM for the simulation of the measurement above an infinite PET porous absorber: (a) NMSE for the sound pressure reconstruction; and (b) NMSE for the $\hat{z}$ component of the particle velocity reconstruction.
• Figure 5: Simulation of the measurement of an infinite PET porous absorber: (a) sound absorption coefficient obtained from Array 0 with the DCISM (GLE, GLA, and MP discretizations) and with the ISM (reconstruction and reflection coefficient) --- for reference, the plane-wave absorption coefficient and the one from spherical wave incidence; and (b) magnitude, $|\tilde{s}|$, of the sources for the DCISM (GLE, GLA, and MP discretizations) for 500 Hz (top) and 1 kHz (bottom).