## 4 Acoustic Testing and Results  
### Experimental Setup, Data Acquisition and Processing  
In order to assess the acoustic performance of the liner samples, measurements were conducted in the test section of the DUct aCoustic Test rig- Rectangular cross section (DUCT-R) facility of the German Aerospace Centre (DLR) in Berlin. The test rig has a length of about 8 m and consists of two symmetrical sections; each containing several microphones and a loudspeaker for acoustic excitation (see sketch in Figure 7). The cross section of the rig was 60 mm \(\times\) 80 mm. A radial compressor was connected to the upstream part of the duct and provides a grazing flow across the test object with a maximum Mach number of 0.3 at the duct centreline. The rig has been used extensively during the last years for the investigation of acoustic scattering coefficients and impedances of several liner types. The basic setup for the acoustic measurements is identical to the setup for a conventional single-degree-of-freedom (SDOF) liner. Consecutively, six multi-tones are excited and fed into the rig by either the upstream or the downstream speakers. Thereby, a frequency range from 204 to 2091 Hz with 51 Hz resolution is covered. The overall sound pressure level (SPL) of these multi-tones was adjusted to about 130 dB in maximum, but this could not be fully ensured for all frequencies due to the coupling of the speaker driver and the speaker characteristics. The acoustic investigation was limited to the plane wave regime (cut-on frequency for higher-order modes is about 2150 Hz for the no-flow case). For each liner test configuration, two different sound fields are excited consecutively in two separate measurements. Speaker A is used in the first measurement and in the second measurement the same signal is fed into speaker B. Then, the data of Sections 1 and 2 (index 1 and 2) are analysed separately. Thereby, the sound field is decomposed into upstream and downstream traveling waves and their respective complex sound pressure amplitudes for each section and measurement. By a combination of all measurements, the influence of end reflections can be cancelled out, and the reflection coefficient \(r\) and transmission coefficient \(t\) for the acoustic pressure can be calculated. The energetic quantities are obtained applying the acoustic energy flux in a moving medium as given by [PERSON] (2006). After integrating over the duct cross section \(A\), this yields for the power of the downstream traveling wave \(P^{+}\), respectively the upstream traveling wave \(P^{-}\):  
\[P^{\pm}=\frac{A}{2\rho c}(1\pm M)^{2}{|p^{\pm}|}^{2} \tag{3}\]  
where \(M\) is the mean Mach number of the flow, \(\rho\) the density, \(c\) the speed of sound, and \(p^{\pm}\) the complex pressure amplitude of the downstream/upstream traveling wave. Then, the energy coefficients for reflection \(R^{\pm}\) and transmission \(T^{\pm}\) can be given relative to the pressure coefficients. The dissipation of acoustic energy is expressed by the dissipation coefficient \(\Delta^{\pm}\). The dissipation coefficient can be calculated directly from the reflection and transmission coefficients via an energy balance:  
\[R^{\pm}+T^{\pm}+\Delta^{\pm}=1 \tag{4}\]  
Figure 7: Sketch of acoustic test rig DUCT-R. The energy of the incident wave is partly reflected, partly transmitted, and partly dissipated inside the damping module. \(R\) and \(T\) are the power quantities of the reflection and transmission coefficients, while \(r\) and \(t\) denoted the pressure quantities. Assuming the same flow conditions and cross-sectional area in Sections 1 and 2, finally, the dissipation coefficients of the test object for the downstream \(\Delta^{+}\) and upstream direction \(\Delta^{-}\) can be obtained:  
\[\Delta^{\pm}=1-\frac{\left(1\mp M\right)^{2}}{\left(1\pm M\right)^{2}}{\left| r^{\pm}\right|^{2}}+{\left|t^{\pm}\right|^{2}} \tag{5}\]  
This is an integral value of the acoustic energy that is absorbed while a sound wave is passing the damping module.