# Acoustic Characterization of Some Steel Industry Waste Materials
## 2 Materials and Methods
### Experimental Characterization--Four-Microphone Impedance Tube  
In this paragraph, the experimental set-up used for the characterization of the samples is described. The measurements of the acoustic properties have been performed by means of the four-microphone impedance tube method, following the process given by the ASTM E2611 standard [48]. On one end of the apparatus features, a loudspeaker generates a plane wave field inside the tube. The other end can be configured with two different types of termination (anechoic and/or reflecting), to perform the investigation with two different boundary conditions. Two microphones are mounted in front of the sample, at the \"emitting side\" of the tube, and the other two microphones are placed close to the sample at the \"receiving side\" of the tube. A transfer matrix approach can be used, allowing to relate the particle velocities (\(u_{i}\)) and the sound pressures (\(p_{i}\)) at both surfaces of the tested sample. Denoting the front surface of the sample with the coordinate \(x=0\) and the back surface with \(x=d\), the resulting transfer matrix can be written as  
\[\left[ \begin{array}{c}p_{0}\\ u_{0}\end{array} \right]=\left[ \begin{array}{cc}T_{11}&T_{12}\\ T_{21}&T_{22}\end{array} \right]\left[ \begin{array}{c}p_{d}\\ u_{d}\end{array} \right] \tag{31}\]  
Thanks to the comparison between the signals measured by the four microphones, it is possible to apply the decomposition technique: referring to Figure 2, the upstream and downstream sound field can be distinguished in two forward travelling waves (A and C) and two backward travelling waves (B and D). The wave components A-D represent the complex amplitudes of the incident and reflected waves on both sides of the sample and can be derived from the complex transfer functions \(H_{i,ref}\) measured between the -\(i\)th microphone (\(i=1,\,\ldots,\,4\)) and the reference (_ref_) microphone. In this study, the first microphone was selected as the reference microphone, but generally any of the four microphones can be chosen for this role. At this point, an interchanging procedure must be applied between the transducers to correct the measured transfer functions for amplitude and phase mismatches. Once the corrected transfer functions are obtained by dividing the measured transfer functions by relative correction transfer functions, the four components A, B, C and D can be obtained. These coefficients are used for the derivation of the transfer matrix terms. Pressures and particle velocities at both sides of the sample can be determined in terms of incident and reflected plane wave components. In the case of geometrically symmetric specimens, since the physical properties are the same on either side, reciprocity and symmetry can be applied and a single set of measurements is sufficient to characterize the material. The acoustical properties of the sample can thus be calculated as a function of the transfer matrix elements, the acoustic impedance of air, the sample thickness and the wavenumber in air. In particular, the following properties are obtained:  
Figure 2: Schematic drawing of a four-microphone impedance tube. - Normal incidence sound absorption coefficient, \(\alpha\);
- Normal incidence sound transmission loss, \(TL\);
- Propagation wavenumber inside the material, \(k_{c}\);
- Characteristic impedance inside the material, \(Z_{c}\). The Applied Acoustic Laboratory impedance tube at the University of Brescia is composed of two 1200 mm long-segments, with an internal diameter of 46 mm, determining a cross-section that ensures that the plane-wave assumption is verified up to approximately 3700 Hz. The loudspeaker is installed in an isolated and sealed volume at the source endpoint of the tube. Through the connection to the generator of a multichannel analyzer, a wide-band white noise test signal (50 Hz-5 kHz) is created inside the tube.