## 4 Acoustic Properties of MCPs and MPPs Resonant Absorbers
The lowest frequency at which the MCP1 material achieves its highest absorption (0.92) is 942 Hz, making it a \(\lambda/7\) sub-wavelength absorber at this frequency. At a frequency of 2400 Hz, at which the absorption of MCP1 stays above 0.8, it is still a \(\lambda/2.5\) sub-wavelength absorber. By comparison, the UMPP is a \(\lambda/8\) resonance absorber when it fully dissipates the incident energy at 800 Hz, albeit over a narrower bandwidth than MCP1. Because the design of thin resonance absorbers that are efficient over a broad bandwidth is a current challenge, it is of interest to examine how the absorption properties of MCP1 are modified when decreasing the cavity depth from 55 mm down to 10 mm. The experimental and simulated absorption results are shown in Figure 9 with an acceptable correlation between the measured and modeled performance. The performance is summarized in Table 2.  
As the cavity depth \(D\) decreases, the equivalent stiffness of the air cavity increases so that the cut-off frequency, below which the absorption of MCP1 drops below 0.5, increases from 365 Hz to 910 Hz. This results in an upshift by 3500 Hz of the first resonance frequency, therefore turning a \(\lambda/4\) resonance absorber at 1500 Hz (\(\alpha_{\rm max}=0.92,\ D=55\) mm) into a \(\lambda/7\) sub-wavelength absorber at 5000 Hz (\(\alpha_{\rm max}=0.82,\ D=10\) mm). Moreover, the half-bandwidth of absorption is significantly increased from 2700 Hz for \(D=55\) mm up to 5800 Hz for \(D=10\) mm, with a steady value of the total absorption (0.69-0.76). Figure 10 examines how the absorption properties of a UMPP are modified when decreasing the cavity depth from 55 mm to 20 mm, when compared to those of MCP1. This produces an upshift by 660 Hz of the first resonance frequency of the UMPP, therefore turning a \(\lambda/8\) resonance absorber at 800 Hz into a \(\lambda/11\) sub-wavelength absorber at 1500 Hz. When \(D=20\) mm, the first resonance peak still exhibits a near-unit absorption value, and its half-bandwidth increases from 1830 Hz to 2200 Hz. However, the full absorption property is lost if one further decreases the cavity depth beneath 20 mm. Like MCP1, the total absorption of the UMPP is almost unchanged (from 0.51 to 0.54) when decreasing the cavity depth. Interestingly, the efficiency ranges (\(\alpha>0.8\)) of the UMPP and MCP1, backed by a 20 mm cavity, complete each other without overlapping (Figure 10, plain gray and  
orange curves), covering ranges of 1-2 kHz for the UMPP and 2-6.5 kHz for MCP1. A parallel arrangement of these materials would lead to a compact ultra-broadband absorber. Figure 10: Simulated (dotted) and measured (plain) absorption properties of UMPP when backed by a rigid cavity with depths of 55 mm (thick black) and 20 mm (thick gray), under normal incidence; also shown are the measured absorption properties of MCP1 when backed by a rigid cavity with depths of 55 mm (thick red) and 20 mm (thick orange). Figure 9: Simulated (dotted) and measured (plain) absorption properties of MCP1 when backed by a rigid cavity with depths of 55 mm (thick red), 35 mm (dark gray), 25 mm (gray), 20 mm (light gray) and 10 mm (thin red), assuming a normal incidence. By removing the rigid bottom of the cavity, a stiffness-controlled closed cavity can be turned into a mass-controlled open cavity, which may potentially remove the absorption dip observed in Figures 8-10 at very low frequencies. This is verified in Figure 11 according to both simulations and experiments, with a good correlation between both types of results. The maxima of absorption are then caused by half-wavelength resonances with a maximum particle velocity over the materials' surface, whereas absorption dips are due to quarter-wavelength resonances with a low particle velocity over the materials. Below 1800 Hz, MCP1 and UMPP clearly outperform MCP2 and MPP, with high absorption values (greater than 0.8) observed at the lower frequency limit (80 Hz) for the Kundt tube measurements.