![]() ![]() Where the integrals represent a 3 D Fourier transform in k x, k y, and k z, respectively, and the amplitude term P ( k ) is the so-called wavenumber (or angular) spectrum. The method is validated in situ in a reverberant environment, based on measurements in a conventional classroom with absorbing ceiling. The angle-dependent absorption coefficient is obtained by directly separating the incident from the reflected components in the 3 D wavenumber domain and can be determined at all angles simultaneously. As such, the explicit computation of a discrete Fourier transform is avoided, circumventing errors (such as wraparound) due to the finiteness of the measurement area. The amplitudes of the plane-wave components are determined algebraically by regularized matrix inversion rather than via FFT. Additionally, non-uniform spatial sampling is used to mitigate aliasing errors at high frequencies. Unlike prior techniques that operate in the nearfield of a sample surface, 8–10 the pressure field is measured over a three-dimensional (3 D) volume to better resolve the 3 D space. 8, the proposed method relies on decomposing a measured sound field into plane-wave components traveling in multiple directions. This paper describes a new method for measuring the angle-dependent absorption coefficient of a boundary material in situ. The method was tested in free-field, for successive incidence directions. 13, this effect could be minimized by using a very large measurement aperture (1.8 m diameter) in combination with a dipolar source to focus the incident field inside the aperture area. Because it relies on explicit spatial Fourier inversions, the technique suffers from replicated aperture errors due to the finiteness of the measurement area. The method consists in decomposing the pressure field generated over the array into plane-wave components using spatial Fourier transforms, to separate the incident and resulting reflected waves for each incidence direction. Notably, Tamura 8 used a double-layer regular array lying close to the surface of a sample material to determine angle-dependent reflection coefficients. 4,8–10 Temporal separation is possible with a single measurement point, whereas spatial separation relies on measurements at several locations, e.g., with an array of microphones. In many cases the absorption coefficient is derived from separating the incident and reflected fields, which can be done temporally 7 or spatially. These methods can estimate the angle dependence of the absorption coefficient, which cannot be measured with the reverberation chamber method. A number of measurement procedures have been developed to characterize absorption in situ instead of under idealized laboratory conditions (see, e.g., Ref.
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