A method is proposed for measuring the angle-dependent absorption coefficient of a boundary material *in situ*. The method relies on decomposing a non-uniform three-dimensional pressure distribution, measured in the vicinity of a boundary, into plane-wave components (i.e., via estimation of its wavenumber transform). The incident and reflected plane-wave components at the boundary are separated in the wavenumber domain, from which it is possible to deduce an absorption coefficient for each angle of incidence simultaneously. The technique is used to verify theoretical predictions of the angle-dependent absorption coefficient of an absorbing ceiling, based on *in situ* measurements in a conventional room.

## 1. Introduction

The absorption coefficient of an acoustical material is not itself a fundamental property of the material but is related to the specific conditions of its use. As such, measured absorption values change when the material is placed in different rooms^{1} and with the angle of incidence of sound. The *Sabine absorption coefficient*, as determined in the reverberation chamber, is an averaged property for the particular condition of isotropic sound incidence. This condition is not satisfied in the reverberation chamber,^{2} and is even less well approximated in conventional rooms. Hence, absorption coefficients established from reverberation-chamber measurements do not agree with coefficients determined in the field and may not be meaningful for specific rooms or configurations.^{3,4} In addition, other factors such as diffraction at the sample edges and differences in mounting conditions can cause the absorption coefficient *in situ* to differ from that measured in the laboratory.^{4}

There is great interest in measuring the angle dependence of the absorption coefficient. Instead of attempting to average the absorption coefficient for random angles of incidence, it may be profitable to determine the absorption coefficient for the material at each angle of incidence, and reconstruct this coefficient into a single, angle-independent value, by considering the directional distribution of sound incidence on the material (numerically or experimentally). Another example is the use of directional coefficients in computer simulation algorithms (see, e.g., Ref. 5).

A number of measurement procedures have been developed to characterize absorption *in situ* instead of under idealized laboratory conditions (see, e.g., Ref. 6). These methods can estimate the angle dependence of the absorption coefficient, which cannot be measured with the reverberation chamber method. In many cases the absorption coefficient is derived from separating the incident and reflected fields, which can be done temporally^{7} or spatially.^{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. Notably, Tamura^{8} used a double-layer regular array lying close to the surface of a sample material to determine angle-dependent reflection coefficients. 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. Because it relies on explicit spatial Fourier inversions [via Fast Fourier Transform (FFT)], the technique suffers from replicated aperture errors due to the finiteness of the measurement area.^{11,12} In Ref. 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. The method was tested in free-field, for successive incidence directions.

This paper describes a new method for measuring the angle-dependent absorption coefficient of a boundary material *in situ*. As in Ref. 8, the proposed method relies on decomposing a measured sound field into plane-wave components traveling in multiple directions. 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. Additionally, non-uniform spatial sampling is used to mitigate aliasing errors at high frequencies. The amplitudes of the plane-wave components are determined algebraically by regularized matrix inversion rather than via FFT. 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 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. The method is validated *in situ* in a reverberant environment, based on measurements in a conventional classroom with absorbing ceiling.

## 2. Theoretical background

### 2.1 Wavenumber transform

Let us consider a harmonic pressure field in the vicinity of a boundary. In the far-field of the source, the pressure at the point characterized by the vector **r** = (*x*, *y*, *z*) can be decomposed into plane-wave components of variable complex amplitudes $P(k)$,^{12}

where the integrals represent a 3 D Fourier transform in *k _{x}*,

*k*, and

_{y}*k*, respectively, and the amplitude term $P(k)$ is the so-called

_{z}*wavenumber*(or

*angular*)

*spectrum*.

^{12}Each plane wave in Eq. (1) travels in a direction specified by the wavenumber vector

**k**= (

*k*,

_{x}*k*,

_{y}*k*), and satisfies the condition

_{z}*k*

^{2}= $kx2+ky2+kz2$ with $k2\u2265kx2+ky2$, i.e., exponentially attenuated waves (evanescent waves) are not considered. In a conventional room, evanescent waves do appear at the measuring aperture if diffracting elements (such as furniture, light fixtures, etc.) are located in close proximity to the boundary. In addition, exponentially attenuated waves can be caused by partial covering of the room boundary with the absorber. In the following, it is assumed that no diffraction phenomena reach the measuring aperture and, consequently, that the wave expansion in Eq. (1) is fundamentally due to

*propagating*terms only. Besides, the boundary material is assumed to reflect purely specularly (i.e., scattering or redirection phenomena are not present).

In practice, the wavenumber spectrum is estimated over a finite number *L* of plane waves distributed over a spherical domain, based on a discrete approximation of Eq. (1). The pressure field is sampled at a discrete number *M* of positions, and can be expressed with the linear model

where **p**$\u2208\u2009\u2102M$ is the complex-valued data vector from the measurements at the *M* microphone positions, **H**$\u2208\u2009\u2102M\xd7L$ is a transfer matrix containing the plane-wave functions $e\u2212jk\u22c5r$, and **x**$\u2208\u2009\u2102L$ is the unknown vector of the complex plane-wave amplitudes. Practically, there are often more unknown coefficients than measurement points (*M *<* L*), and the problem is underdetermined. A way of solving such problem is to constrain the possible solutions with prior information. In the following, the underdetermined problem in Eq. (2) is solved by seeking the solution with the minimum *ℓ*_{2}-norm which best fits the data. The *ℓ*_{2}-norm leads to the conventional *least-squares* minimization problem, which has the well-known closed form analytical solution^{2}

where the superscript *H* denotes the conjugate transpose, **I** is the identity matrix, and $\lambda $ is a regularization parameter. Equation (3) is the least-squares solution of the problem with Tikhonov regularization.^{2} It is important to remark that although the transfer matrix **H** contains an inverse discrete Fourier transform for the space-frequency representation, the solution to the underdetermined problem in Eq. (2) is sought via a matrix inversion, instead of explicitly computing a Fourier transform, as in Ref. 8. As such, the proposed method makes it possible to reduce errors introduced by the limited size of the measurement area, and the appearance of replicated apertures (i.e., wraparound errors).^{11,12} Besides, it should be noted that no specific array configuration is required for the estimation of the wavenumber spectrum, and the pressure field can be sampled arbitrarily over the measurement volume.

### 2.2 Sound field separation and absorption coefficient estimation

The wavenumber spectrum $P(k,\u2009\theta ,\u2009\varphi )$, here expressed in spherical coordinates, may be subdivided into two hemispheres representing the incident and reflected wave fields, corresponding to waves traveling in the positive or negative *z*-directions, respectively (see Ref. 2). Based on this sound field separation, the angle-dependent absorption coefficient is determined according to

where $|P(k,\u2009\pi \u2212\theta ,\u2009\varphi )|2$ corresponds to the reflected power, and $|P(k,\u2009\theta ,\u2009\varphi )|2$ to the incident power. The estimation of absorption as in Eq. (4) follows from the ratio of absorbed to incident power in the measurement aperture and, as such, does not assume a specific waveform incidence. It is easy to see that the absorption coefficient may be singular in directions with no incident energy [i.e., $|P(k,\u2009\theta ,\u2009\varphi )|2=0$], which may bias the final estimate. The individual wavenumber spectra are therefore integrated over the azimuth angle $\varphi $, as such averaging is expected to guarantee a finite amount of incident energy at all elevation angles, thus preventing biases at angles with no incidence.

## 3. Experimental results

Experiments are conducted in a classroom (9.3 m × 6.2 m × 2.9 m) with absorbent ceiling at the Technical University of Denmark, using a programmable robotic arm to scan the sound field [see Fig. 1(a)]. Two configurations are considered: (i) the empty classroom with absorbing ceiling; and (ii) the classroom with absorbing ceiling and furniture (9 tables and 18 chairs). The ceiling consists of a 20 mm porous layer (FOCUS DS 20 mm, Saint-Gobain Ecophon, Hyllinge, Sweden), backed by a 600 mm air cavity. A scanning robot UR5 (Universal Robots, Odense, Denmark) is programmed to move a free-field microphone (Brüel & Kjær, Nærum, Denmark) within a rectangular volume of dimensions 0.6 m × 0.8 m × 0.25 m located in the vicinity of the absorbing ceiling (the closest microphone position is located 10 cm away from the ceiling), creating a random array of 290 sequential measurement positions. Considering the minimum and maximum distances between transducers, the effective frequency range of the array is approximately 200 Hz to 2.5 kHz. The room is excited by a loudspeaker [located in the lower corner of the room in Fig. 1(a)] driven with a logarithmic sine sweep ranging from 20 Hz to 20 kHz with duration of 10 s (although a shorter signal would have been sufficient). The frequency response is measured at the 290 positions with a spectral resolution of 0.1 Hz. The duration of a measurement sequence (for one configuration of the room) is approximately 2 h 45 mins, including 10 s for the arm to stabilize after each displacement. The sequence runs automatically, based on the programmed positioning path. For the estimation of the wavenumber spectrum in front of the absorbing ceiling, a plane-wave basis of 2000 plane waves is considered, whose directions of propagation are distributed uniformly across all propagation angles.^{2} Tikhonov regularization is used for the inversion of Eq. (2) and the regularization parameter chosen with the L-curve criterion.^{2} Unlike Refs. 8–10, the results presented in the following determine the angle-dependent absorption coefficient simultaneously for all directions.

Theoretical values of the angle-dependent absorption coefficient are computed for comparison, based on a Transfer Matrix Model (TMM) in the case of a single layer of porous material backed by an air cavity.^{14} The characteristic impedance *Z _{c}* and wavenumber

*k*for the material are assumed to depend mainly on the angular frequency $\omega $ and on the flow resistivity $\sigma $ of the material, and are therefore obtained from the one-parameter Delany-Bazley-Miki model (DBM),

^{15}which corresponds to the well-known Delany-Bazley model further modified by Miki to avoid non-physical predictions of the surface impedance at low frequencies (i.e., for

*f*/$\sigma $ < 0.01). A flow resistivity value $\sigma $ = 106 000 N m

^{−4}s, measured in the direction normal to the material surface, is provided by the manufacturer.

Figure 1(b) shows the magnitude of the wavenumber spectrum estimated in the empty classroom (without furniture) for the third-octave band centered at 630 Hz—the pure-tone wavenumber spectra are averaged over the third-octave band. Because the wavenumber spectrum corresponds to the directions of propagation (and not the directions of arrival), the upper and lower hemispheres correspond to directions of incident and reflected components, respectively. Figure 1(c) shows the corresponding distribution of incident vs reflected energy as a function of the angle of incidence $\theta $ averaged over the azimuth angle $\varphi $.

Some dominant incident directions are identified, i.e., some of the incident waves carry more energy, seemingly corresponding to the direct radiation from the source and a few reflections from the walls [Fig. 1(b)]. Most incident directions (with the exception of the direct sound) are detected toward grazing angles of incidence (close to the equator) and represent standing waves oscillating back and forth between the walls, parallel to the absorbing ceiling. These results confirm that concentration of absorption on one single surface of the otherwise rigid-walled classroom favors the separation of grazing waves from non-grazing waves that are greatly affected by the absorption. It can further be noticed that fewer energy is propagating in the *z*-axis (at normal incidence, close to the poles), as sound is being absorbed effectively by the ceiling. This is clearly seen in Fig. 1(c).

Figure 2 compares the angle-dependent absorption coefficients measured *in situ* in the classroom with and without furniture with the TMM prediction. The results are displayed for the third-octave bands ranging from 315 Hz to 2 kHz. The absorption coefficient is averaged over the azimuth angle (see Sec. 2) and calculated as mean values per third-octave bands. The frequency averaging is performed on the wavenumber spectrum to avoid biased estimates, as a pure-tone sound field is not isotropic, and not all frequencies contain energy in every direction. The resulting third-octave band wavenumber data are interpolated onto a rectangular grid evenly spaced in elevation and in azimuth, and the averaging over angle is performed on the interpolated spectrum. The markers represent the experimental data obtained every 3° ($\pi $/60 rad).

The experimental values compare reasonably well with the TMM prediction (solid line). The agreement is particularly good in the frequency range between 315 Hz and 1.25 kHz and for angles close to grazing incidence, which is typically challenging. Below 315 Hz, the spatial resolution of the array, determined by its size, is limited (the 3 dB bandwidth at 250 Hz is about 15°, estimated numerically). Above 2 kHz, the final estimate starts to be biased, due to the spacing between microphones.

At large angles of incidence [between 60° ($\pi $/3 rad) and 90° ($\pi $/2 rad)], the absorption coefficient is slightly underestimated at low frequencies (*f *<* *500 Hz), due to difficulties distinguishing between incident and reflected components. This is a result of the limited wavenumber resolution of the measurement system: some of the incident energy on the absorber at grazing incidence (close to the equator) leaks into the lower half, appearing to be reflected sound instead (this is seen in the wavenumber spectrum estimations, not shown for conciseness).

Toward small angles of incidence [between 0° and 30° ($\pi $/6 rad)], deviations at high frequencies (*f *>* *1.25 kHz) can be attributed to insufficient sound incidence at those angles, which cannot be improved significantly by averaging over the azimuth angle, nor over frequencies. This may be due to (i) the measurement system itself: the spatial resolution of the array is higher at high frequencies, leading to narrow lobes in the wavenumber spectrum representations; (ii) the angular distribution of normal modes in the room:^{16} in the empty classroom, the number of modes impinging on the ceiling close to normal incidence is rather low, and these same modes are highly affected by the absorption [as also seen in the wavenumber representation in Figs. 1(b) and 1(c)]. This results in limited incoming energy close to normal incidence, leading to a bias. It is anticipated that averaging over multiple source positions would enforce a more uniform sound incidence and, in turn, improve the coefficient estimation. As a matter of fact, when furniture is added to the room, the measured absorption at high frequencies (*f *>* *1 kHz) is closer to prediction at small angles of incidence, suggesting that the furniture redirects the incoming sound energy in a way that is more uniform over all directions.

Finally, it is apparent in Fig. 2 that the TMM prediction exhibits multiple peaks and dips, which are not observed in the measurements. This is seemingly due to additional physical phenomena, which are not considered in the prediction model. Although the DBM model is widely used and can generally provide reasonable orders of magnitude for *Z _{c}* and

*k*, the model is restricted to describing wave dissipation in the interstitial fluid only. A more physically complete approach would require an examination of the structural effects related to the deformation of the solid phase of the material. Besides, fibrous materials are generally anisotropic, because most fibres lie in planes parallel to the surface of the material. Therefore, the flow resistivity depends on the direction of the flow, and the DBM model must in principle be used to evaluate characteristic impedances and wavenumbers in both the

*normal*and

*planar*directions.

^{14}As such, it is not expected that the model used in this study provides a perfect prediction of the properties of the material under test. Yet, the use of more elaborate models generally depends on the available parameters for the tested material. Although the TMM prediction cannot be considered as a ground truth, it is encouraging to see that the predictions agree well with the measured data at most angles of incidence (the relative deviation is between 10% and 20%, and mostly depends on the rapid fluctuations that the theoretical model predicts, and which are not observed in reality). In particular, it can be seen that the measured data follows rather accurately the general trend of the predicted angle-dependent behavior.

## 4. Conclusions

A wavenumber-transform method of measuring *in situ* the angle-dependent absorption coefficient of a boundary material is proposed in this study. The proposed method makes use of an elementary plane-wave expansion to model an arbitrary pressure field, measured in the vicinity of the boundary over a 3 D volume. The incident and reflected plane-wave components are separated in the wavenumber domain, which leads to the determination of the absorption coefficient as a function of the angle of incidence. This coefficient is determined simultaneously for all angles. Experimental results obtained in a classroom with and without furniture are compared with theoretical values calculated from a TMM. A good agreement is found at most angles of incidence, demonstrating that the method can successfully estimate the angle-dependent absorption coefficient in a reverberant, complex environment. Finally, it should be remarked that the wavenumber representation not only allows separating incident from reflected wave fields at the boundary but also gives extensive information in terms of distribution of sound incidence. As such, it is anticipated that knowledge of the wavenumber spectrum may lead to a proper weighting of a “random” incidence absorption coefficient. Necessary future work includes: a controlled comparison with existing approaches, a better characterization of the material properties as a ground truth, and a systematic quantification of errors and uncertainties.

## Acknowledgments

This work is jointly supported by the Innovation Fund Denmark, under individual postdoctoral Grant No. 8054-00042B, and Saint-Gobain Ecophon. The author would like to thank Erling Nilsson and Efren Fernandez-Grande for valuable discussions and Samuel A. Verburg for assistance with the experimental arrangement.