Forbidden Transmission of Broadband Duct Noise Realized by Compactly Placed Detuned Resonators

Abstract

Effective broadband duct sound propagation control is highly required in many practical engineering applications. In this study, a compact structure constituted by multiple detuned resonators is proposed for broadband duct noise transmission control. The coupling characteristics of two detuned resonators flush-mounted on the sidewall of a duct are firstly investigated. Results show that a coherent perfect absorption (CPA) is induced when these two resonators are precisely designed. Meanwhile, a nearly flat transmission forbidden band is formed, which is very beneficial for duct noise control. Furthermore, it is found that the appearance of the forbidden band is insensitive to the distance between resonators. On this basis, a customized broadband CPA-based structure constructed by detuned resonators is developed, in which the geometric parameters of each adjacent resonator satisfying the CPA condition and the resonators are closely placed. By overlap** the forbidden band of adjacent resonators, a broad duct sound transmission forbidden band is attained. The acoustic performance of the proposed compact design is demonstrated experimentally.

Appendices

Appendix A: End-Correction, Complex Wave Number and Impedance

This appendix gives the expressions of the end-correction and the equivalent parameters. The end-correction length of HR $\Delta l$ is due to the radiation effects induced by the discontinuities from neck to the cavity of HR and the duct, i.e., $\Delta {l_1}$ and $\Delta {l_2}$. They are calculated by [35, 36]

$$\begin{aligned} \Delta {l_1}= & {} 0.82\left[ {1 - 1.35\frac{{{R_{\mathrm {n}}}}}{{{R_{\mathrm {c}}}}} + 0.31{{\left( {\frac{{{R_{\mathrm {n}}}}}{{{R_{\mathrm {c}}}}}} \right) }^3}} \right] {R_{\mathrm {n}}} , \end{aligned}$$

(9)

$$\begin{aligned} \Delta l_{2}= & {} 0.82 \left[ 1 - 0.235 \frac{ R _ { \mathrm { n } } }{ R _ { \mathrm { w } } } - 1.32 \left( \frac{ R _ { \mathrm { n } } }{ R _ { \mathrm { w } } } \right) ^ { 2 } \right. + 1.54 \left( \frac{ R _ { \mathrm { n } } }{ R _ { \mathrm { w } } } \right) ^ { 3 } \nonumber \\&-\, 0.86 \left( \frac{ R _ { \mathrm { n } } }{ R _ { \mathrm { w } } } \right) ^ { 4 } ] R _ { \mathrm { n } } , \end{aligned}$$

(10)

The expressions of the effective wave number and the characteristic impedance can be written as [35, 36]

$$\begin{aligned} {k_{\mathrm {e}}}= & {} \frac{\omega }{{{c_0}}}{}\left( {1 + \frac{{(1 - i)}}{{\sqrt{2} (r/\sqrt{2\mu /{\rho _0}\omega } )}}(1 + (\gamma - 1)/ \sqrt{\mathrm {Pr}} } \right) , \end{aligned}$$

(11)

$$\begin{aligned} {Z_{\mathrm {e}}}= & {} \frac{{{\rho _0}{c_0}}}{{\pi {{r}^2}}}\left( {1 + \frac{{(1 - i)}}{{\sqrt{2} (r/\sqrt{2\mu /{\rho _0}\omega } )}}(1 - (\gamma - 1)/\sqrt{\mathrm {Pr}} )} \right) , \end{aligned}$$

(12)

where $\rho _0$ is the density of air; $\mu$ and $\gamma$ are the viscosity of air and the ratio of specific heats; r is the radius of the waveguide, or the neck and cavity of HR; $\mathrm {Pr}$ is the Prandtl number.

Fig. 12

Schematic of reflection and transmission measurement based on two-load method

Appendix B: Reflection and Transmission Measurement

This appendix presents the measurement setup for measuring the reflection and transmission coefficients of test sample based on the two-load method [42]. As shown in Fig. 12, test sample with a length of d is placed between four microphones, where A/C and B/D are the coefficients of the planar waves propagating in positive and negative x directions at left/right side of test sample, respectively. The pressures at different microphone positions from $x_1$ to $x_4$ can be expressed as

$$\begin{aligned}&{P_{x_1}} = A{e^{ - ik{x_1}}} + B{e^{ik{x_1}}}, \end{aligned}$$

(13)

$$\begin{aligned}&{P_{x_2}} = A{e^{ - ik{x_2}}} + B{e^{ik{x_2}}}, \end{aligned}$$

(14)

$$\begin{aligned}&{P_{x_3}} = C{e^{ - ik{x_3}}} + D{e^{ik{x_3}}}, \end{aligned}$$

(15)

$$\begin{aligned}&{P_{x_4}} = C{e^{ - ik{x_4}}} + D{e^{ik{x_4}}}. \end{aligned}$$

(16)

By rearranging Eqs. (13)–(16), the coefficients A, B, C, and D can be determined once the pressures $P_{x_1}$, $P_{x_2}$, $P_{x_3}$, and $P_{x_4}$ are measured. Then, the measured pressures and particle velocities in the x direction at the surfaces of the test sample can be expressed as

$$\begin{aligned}&\left[ \begin{array}{l} P \\ V \end{array}\right] _{x=0}=\left[ \begin{array}{c} A+B \\ A-B \\ \hline \rho _{0} c_{0} \end{array}\right] , \end{aligned}$$

(17)

$$\begin{aligned}&\left[ \begin{array}{l} P \\ V \end{array}\right] _{x=d}=\left[ \begin{array}{l} C e^{-i k d}+D e^{i k d} \\ \frac{C e^{-i k d}-D e^{i k d}}{\rho _{0} c_{0}} \end{array}\right] . \end{aligned}$$

(18)

Two different terminations, i.e., anechoic back and rigid back, are used in experiment. The measured pressures and particle velocities for two different terminations are related by the total transfer matrix

$$\begin{aligned} {\left[ {\begin{array}{*{20}{c}} {{P_1}}&{}{{P_2}} \\ {{V_1}}&{}{{V_2}} \end{array}} \right] _{x = 0}} = \mathbf {T_t} {\left[ {\begin{array}{*{20}{c}} {{P_1}}&{}{{P_2}} \\ {{V_1}}&{}{{V_2}} \end{array}} \right] _{x = d}}, \end{aligned}$$

(19)

where the subscripts 1 and 2 represent the tested results with anechoic back and rigid back, respectively. The total transfer matrix can be calculated based on Eq. (19)

$$\begin{aligned} \mathbf {T_t}= & {} \frac{1}{{{{\left. {{P_1}} \right| }_{x = d}}{{\left. {{V_2}} \right| }_{x = d}} - {{\left. {{P_2}} \right| }_{x = d}}{{\left. {{V_1}} \right| }_{x = d}}}} \nonumber \\&\times \left[ \begin{array}{l} \left. \left. P_{1}\right| _{x=0} V_{2}\right| _{x=d}-\left. \left. P_{2}\right| _{x=0} V_{1}\right| _{x=d}\\ \qquad -\left. \left. P_{1}\right| _{x=0} P_{2}\right| _{x=d}+\left. \left. P_{2}\right| _{x=0} P_{1}\right| _{x=d} \\ \left. \left. V_{1}\right| _{x=0} V_{2}\right| _{x=d}-\left. \left. V_{2}\right| _{x=0} V_{1}\right| _{x=d}\\ \qquad -\left. \left. P_{2}\right| _{x=d} V_{1}\right| _{x=0}+\left. \left. P_{1}\right| _{x=d} V_{2}\right| _{x=0} \end{array}\right] . \end{aligned}$$

(20)

Then, the reflection and transmission coefficients can be calculated from the elements of $\mathbf {T_t}$ based on Eqs. (5) (6).