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Backing effects on the underwater acoustic absorption of a viscoelastic slab with locally resonant scatterers
Applied Acoustics 76 (2014) 48–51
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Backing effects on the underwater acoustic absorption of a viscoelastic slab with locally resonant scatterers Honggang Zhao ⇑, Jihong Wen, Haibin Yang, Linmei Lv, Xisen Wen Vibration and Acoustics Research Group, Science and Technology on Integrated Logistics Support Laboratory, College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, Hunan 410073, PR China
a r t i c l e
i n f o
Article history: Received 19 July 2012 Received in revised form 22 July 2013 Accepted 25 July 2013 Available online 25 August 2013 Keywords: Underwater acoustics absorption Resonance modes Locally resonant scatterers Steel backing
a b s t r a c t Backing effects on the underwater acoustic absorption of a viscoelastic polymer slab embedded with locally resonant scatterers are reported. The polymer slab is embedded with two layers of locally resonant scatterers, i.e. Al spheres coated by soft silicon rubber. Theoretical absorption coefficients of the polymer slab under different backings using a layer multiple scattering method show good agreement with the experimental results, which supports unambiguously the experimental observation. Then relations between the resonance modes and the low-frequency absorption peaks of the composite slab are clarified to address the absorption mechanisms. It shows that the mass of the steel backing affects evidently the low-frequency absorption, the absorption peak shifts to lower frequency range while increasing the backing mass. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Locally resonant sonic materials (LRSMs) are generally known as composite epoxy embedded with various heavy spheres coated by soft rubber with periodic or random position [1]. It has been shown that the soft coating layer and the heavy core are responsible for a band gap at low frequency about two orders of magnitude lower than that predicted by Bragg’s condition. For practical applications, much effort has focused on air sound isolation induced by the gaps of different LRSMs [2]. Our investigations in acoustic dissipation by LRSMs have shown the low frequency acoustic absorption while considering the damping between the components [3,4]. Motivated by understanding of the energy dissipation in LRSMs, the locally resonant scatterers (LRSs) are introduced to improve the low-frequency acoustic absorption of the water impedancematched polymer. Experimental measurement for acoustic absorption of viscoelastic polymer slabs embedded with LRSs has been reported [5]. The fundamental mechanism operating in the localized resonance for acoustic absorption has been investigated by referring the mode conversions during the Mie scattering of a single scatterer. Further, the shape of the scatterer has also been taken into account [6]. In fact, viscoelastic polymers with various shapes of cavities [7– 14] or solid scatterers [15,16] are widely used in modern underwater acoustic applications. In most cases, scattering resonances including monopole and dipole types have been used to enhance ⇑ Corresponding author. E-mail address: [email protected] (H. Zhao). 0003-682X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apacoust.2013.07.022
the low frequency absorption of viscoelastic composites. The monopole resonance may be induced by various cavities [7–14] and the dipole type may occur in case of heavy spheres [5,6,15,16]. For spherical scatterers, the absorption properties can be investigated conveniently by the layer multiple scattering method (LMSM), which is developed for gap analysis in phononic crystals [17,18]. The computation is simplified by using the Bloch theory. In comparison with the finite element method [9–11], although the flexibility in model is limited, there are two important advantages of the LMSM, one is the fast computation, and the other is the convenient understanding of the physical absorption mechanism. For actual applications, the design of viscoelastic polymers embedded with locally resonant scatterers must consider the effect of the backing panel, which is not considered in our previous works [5,6]. The effects of the steel backing on the absorption are shown by the experimental and theoretical results in present paper. To address the absorption mechanisms, the relation of the resonance mode and the absorption peak of the composite slab is clarified based on the theoretical and experimental results. For comparisons, other backings are also considered.
2. Experiment 2.1. Model and sample Fig. 1a shows a sketch model of an absorption structure including a sample slab and a backing under the Cartesian coordinates
49
H. Zhao et al. / Applied Acoustics 76 (2014) 48–51
(a) L1
L2
(b)
Micr.
2
Micr.
sample
1
L3
Pi
s
L backing
Pr Z Steel backing Fig. 2. Schematic diagram of the two-microphone tube apparatus.
Fig. 1. (a) An absorption structure includes a composite slab and a steel backing. A plane wave is normally incident from the left along Z-axis. (b) Cross sections of the array of the scatterers of the sample in X–Y plane.
system. The whole structure can be considered as a sequence of different layers perpendicular to Z-axis. The sample is made of viscoelastic polymer embedded with two layers of LRSs. The interlayer positions of the scaterers are denoted by Li (i = 1, 2, 3), where L1 denotes the center of the first layer of LRSs from the surface of the sample, L2 is the distance between both the layers of LRSs, and L3 denotes the space between the second layer of LRSs and the other surface of the sample. The thickness of the whole sample is 55 mm, and L1 = L3 = 13.75 mm, L2 = 27.5 mm. Considering the limitation of the experiment apparatus, i.e. an impedance tube with an inner diameter d = 120 mm, the sample has an overall shape of short cylinder, which possesses a total diameter D = 118 mm. Each layer contains 30 scatterers, which are arranged in a quasi-triangular lattice with a mean adjacent distance 19.5 mm. The position of the scatterers is shown in Fig. 1b. Each LRS has a uniform spherical shape, in which a core sphere (radius r) coated by soft silicon rubber (with outer radius r1), here r = 5 mm, and r1 = 7.5 mm. In order to reduce the weight of the sample and facilitate the understanding of the resonance mode of the backing, we choose Al material as the core (which separates the resonances induced by the backing and the LRS). The soft coating is made of a type of two-component, room temperature vulcanizing silicone rubbers (RTV-2). It is available in a variety of hardness and suitable for various mold techniques. To begin, the core is coated by the silicon rubber using a self-made mold. Then the sample is fabricated by a hollow cylinder mold with an inner diameter D = 118 mm in the following procedure:
Under normal incidence, unlike the backing, the impedance tube almost does not interact with the testing waves because the first order resonant frequency of the tube walls is higher than that of the upper cut-off frequency in the present test. The experimental approach is based on a frequency response function method (FRFM) using two microphones [19]. One can get the frequency response function H12 = P2/P1, P1 and P2 denoting the acoustic pressure measured by Micr. 1 and Micr. 2 respectively. The complex pressure reflection coefficient (R) is given by:
R¼
H12 ejkL ejkðLþsÞ jkðLþsÞ e H12 ejkL
ð1Þ
;
where k is the wave number, defined as x/c, and c is the acoustic velocity of the water. L denotes the distance between the Micr. 1 and the surface of the sample, and s is the spacing between two microphones. Considering the fully reflective termination in the experiment, the sound energy absorption coefficient (a) can be calculated from the following equation:
a ¼ 1 jRj2 :
ð2Þ
3. Discussion 3.1. Comparison of theoretical and experimental results Generally, the steel backing can be considered as a finite steel slab followed by half infinite air. In order to understand the variation and the formation of the acoustic absorption of the sample under the steel backing, we compare the acoustic absorption of the sample under different backings in the experiment. We choose the backings as half infinite air (air backing in the following), steel
2.2. Absorption measurement method Measurement of the absorption coefficient is conducted in a standard impedance tube apparatus. The steel tube has an inner (outer) diameter 120 mm (180 mm), and 5 m long. The schematic of the apparatus is showed in Fig. 2. The impedance tube is fully filled with water. A plane wave incidence is excited at one end of the tube while the sample is positioned at the other end, where various backings can be attached on the surface of the sample.
Absorption coefficient (α)
1
(1) A polymer slab with thickness of (L1 r1) is made. (2) With a proper interval time, the coated scatterers are arranged on the surface of the polymer slab according to the design, and the rest space of the mold with a height (L1 + L2 r1) is filled by the polymer. (3) The multi-layer sample is fabricated by an iterative procedure (2) according to Li.
E-Air E-5mm E-10mm
0.8 0.6 0.4
T-Air T-5mm T-10mm T-water
0.2 0
500
1000
1500
2000
2500
3000
Frequency /Hz Fig. 3. Comparison of the experimental and theoretical results of the sample’s absorption.
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H. Zhao et al. / Applied Acoustics 76 (2014) 48–51
backings with different thickness (5 mm and 10 mm respectively). Fig. 3 gives the experimental absorption coefficients (labeled by E-) of the sample under different backings. One can readily see that an absorption peak appears under each backing. When the sample has the air backing, 5 mm and 10 mm steel backings in turn, the absorption peak shifts to lower frequencies and the bandwidth of the absorption becomes slightly narrower at the same time. To compare with the experimental results, the acoustic absorption of the sample under the corresponding backing is computed by the LMSM. For simplifying the analysis, the composite slab is assumed to be infinitely along intra-layer (XOY) plane, and the scatterers in each layer are arranged in a triangle lattice. The lattice constant, i.e. the nearest distance between two neighbor scatterers, is 19.5 mm. A plane longitudinal wave is incident from the half infinite water (see Fig. 1a). Material parameters for all components are fixed without including the dependence on frequency for simplicity, and are listed in Table 1. The parameters of the polymer and the silicon rubber are measured using a dynamic mechanical analyzer (DMTA IV), and the parameters at 1 kHz are used. The viscosity of other components is neglected and the corresponding damping factor is set to zero. The density of water (air) is 1000 kg m3 (1.29 kg m3) and the longitudinal wave speed is 1480 m s1(340 m s1). The theoretical results (labeled by T-) are compared with those from the experiment in Fig. 3. One can readily see that the results of the calculation under different backings show good agreement with those of the experiments. Little difference between the theoretical and experimental results exists. For instance, the bandwidth of the absorption peaks by calculation is narrower than those of the experiments. This phenomenon may be induced by little volume of air bubbles (although eliminated by some means, for example, vacuumizing and static pressure) remaining in the actual sample. The bubbles enhance the wave scattering and visco-thermal effects [20,21], then the acoustic absorption in actual experiment. However, the dissipations induced by the bubble scattering and the visco-thermal effects are not considered in the present computation. 3.2. Discussions In order to understand the origin of the above absorption peaks, we also calculate the absorption of the composite slab under water backing, denoted by the long dashed line in Fig. 3. One can see that there is weak absorption peak at 2080 Hz. Here the polymer is water impedance-matched, so the water backing affects little the absorption of the composite slab. The weak absorption peak is induced by the weak resonance of the scatterer determined by the light Al core, which can be verified by the analysis of Mie scattering. For a spherical scatterer, the Mie scattering matrix (T) for each partial wave of l order is given by Tl [5], in which L, M and N denote respectively one longitudinal wave mode and two transverse wave modes. It shows that the L and N modes are coupled to each other, while the M mode is decoupled. Since only longitudinal incidence exists underwater, the M mode is out of our consideration in following discussions. The meanings of the elements in Tl are clear. Table 1 Physical parameters of the materials used in the present paper. Material
q (kg m3)
Polymer Silicon rubber Steel Al
Young’s modulus (Pa)
For example, LL stands for the conversion from a longitudinal incidence into a longitudinal outgoing wave during the scattering procedure. Because the localized resonance at low frequency is a dipole type, we focus on the mode conversions during the dipole (l = 1) scattering. Fig. 4 shows the variation of the absolute values of the T1 elements with frequency for the coated Al sphere in unbounded polymer. From Fig. 4a, one can see that the longitudinal incidence is scattered by the LN channel exceeding the LL channel during the dipole resonance, which suggests that the longitudinal incidence prefers to scatter into transverse wave. From Fig. 4b, it shows that the amplitudes of the T1 elements for the transverse wave incidence present about five orders of those for the longitudinal wave incidence, which hints that the transverse wave dominates the scattering. Generally, the transverse wave damps rapidly in viscoelastic polymer, then the acoustic dissipation is enhanced. Accordingly, the absorption peak of the polymer slab under the water backing appears around the dipole resonant frequency (comparing Figs. 3 and 4). The amplitude of the absorption peak is mainly determined by the amplitude of LN scattering in T1 since only longitudinal wave incidence exists underwater. Under the air backing, however, we can readily see that the absorption coefficient is much enhanced and a new absorption peak comes into being. The acoustic reflection from the polymer/ air interface at the backing forms a partial standing wave in the composite slab because of the energy damping, and there exists corresponding standing wave resonance in the composite slab. The standing wave resonance enhances the mode conversion from longitudinal to transverse waves and the multiple scattering among the scatterers, so the energy dissipation in the composite slab is enhanced and a new absorption peak comes into being. From the dotted line in Fig. 3, one can see that this new absorption peak combines that induced by the localized resonance because both absorption peaks have a close position. For the steel backing, a distinct feature from Fig. 3 is that the absorption peak shifts to lower frequencies while increasing the thickness of the steel slab. The absorption peak induced by the localized resonance appears again because the space between both the absorption peaks increases. However, the absorption peak induced by the localized resonance is smoothed in the experimental results, which may be induced by the variation of the actual parameter with frequency and the bubble dissipation, the thorough analysis is out of our present work. In order to gain more physical insight into the variation of the above absorption peaks under the steel backing, it will be useful to state the input acoustic impedance of the steel backing. Due to the characteristic impedance of air is much less than that of the steel, the input impedance of the steel backing for normal incidence is [22]
Z in ¼ jZ s tanðks dÞ;
where d denotes the thickness of the steel slab, Zs is the characteristic impedance of the steel and ks the wave number, both parameters being defined as Zs = qs cls and ks = x/cls, here qs and cls denotes the density and longitudinal velocity of the steel. Note that the low frequency range are focused and d is less than 10 mm, so the ksd is a small argument, the tangent can be approached as a linear function leading to the expression:
Z in ¼ jZ s ks d ¼ jqs xd:
Poisson’s ratio (m)
Damping factor (tan d)
1100 1300
2.97 107 2.0 106
0.498 0.46
0.3 0.3
7890 2690
2.09 1011 7.44 1010
0.275 0.334
0 0
ð3Þ
ð4Þ
The above equation demonstrates that the input impedance of the steel backing is not dependent on the velocity of the steel for low values of ksd, is proportional to the mass qsd. Based on the absorption peak variation with the different steel backing in Fig. 3, the absorption peak shifts to lower frequencies while increasing the mass (d or qs) of the steel backing. The absorption peak is induced by the standing wave resonance from the overall resonance of the steel backing. The physical mechanism of the resonance
(a)
7
Amplitude of T1- matrix elements
H. Zhao et al. / Applied Acoustics 76 (2014) 48–51
6
by the resonance of the steel backing shifts to lower frequencies while increasing the backing mass. The resonance enhances the mode conversions from longitudinal waves to transverse waves and the multiple scattering among the scatterers, which dominates the low-frequency acoustic absorption.
x10-5 LL LN
5
Acknowledgements
4
Financial support of the National Natural Science Foundation of China under Grant Nos. 11004249 and 51275519 is gratefully acknowledged. The authors are pleased to thank anonymous referees for valuable comments and suggestions to considerably improve the manuscript.
3 2 1 0
References 0
500
1000
1500
2000
2500
3000
2500
3000
Frequency /Hz
Amplitude of T1- matrix elements
(b)
1 NN
0.8
NL
0.6
0.4
0.2
0
51
0
500
1000
1500
2000
Frequency /Hz Fig. 4. The absolute values of Tl-matrix elements (l = 1) for a coated Al sphere in unbounded rubber under longitudinal (a) and transverse (b) incidence. In each plots, the dashed and solid lines present the cases without and with mode conversions between the L and N modes respectively.
phenomenon under the steel backing can be analogous to a massspring system, where the composite polymer and the steel slab contribute to the spring and mass respectively. While increasing the mass of the steel slab by density or thickness, the resonance of the steel slab appears at lower frequency domain, and thus the absorption peak of the composite slab shifts to lower frequencies as well, as showed in Fig. 3. 4. Conclusions In conclusion, we have shown the backing effects on the absorption of the composite slab embedded with locally resonant scatterers. Both the experimental and theoretical results show that the absorption is adjusted significantly by the steel backing. The effects of the steel backing on the acoustic absorption are clarified by the input impedance of the steel slab, and other backings are also investigated for comparison. The standing wave resonance induced
[1] Liu ZY, Zhang X, Mao YW, Zhu Y, Yang Z, Chan CT, et al. Locally resonant sonic materials. Science 2000;289:1734–6. [2] Ho KM, Cheng CK, Yang Z, Zhang XX, Sheng P. Broadband locally resonant sonic shields. Appl Phys Lett 2003;83:5566–8. [3] Zhao HG, Liu YZ, Wen JH, Yu DL, Wen XS. Sound absorption of locally resonant sonic materials. Chin Phys Lett 2006;23:3132–4. [4] Zhao HG, Liu YZ, Wen JH, Yu DL, Wen XS. Tri-component phononic crystals for underwater anechoic coatings. Phys Lett A 2007;367:224–32. [5] Zhao HG, Wen JH, Yu DL, Wen XS. Low-frequency acoustic absorption of localized resonances: experiment and theory. J Appl Phys 2010;107:023519. [6] Wen JH, Zhao HG, Lv LM, Yuan B, Wang G, Wen XS. Effects of locally resonant modes on underwater sound absorption in viscoelastic materials. J Acoust Soc Am 2011;130:1201–8. [7] Mayer E, Brendel K, Tamm K. Pulsation oscillations of cavities in rubber. J Acoust Soc Am 1958;30:1115–24. [8] Gaunaurd GC. One dimensional model for acoustic absorption in a viscoelastic medium containing short cylindrical cavities. J Acoust Soc Am 1977;62:298–307. [9] Hladky-Hennion AC, Decarpigny JN. Analysis of the scattering of a plane wave by a doubly periodic elastic structure using the finite element method: application to Alberich anechoic coatings. J Acoust Soc Am 1991;90:3356–67. [10] Easwaran V, Munjal ML. Analysis of reflection characteristics of a normal incident plane wave on resonant sound absorbers: a finite element approach. J Acoust Soc Am 1993;93:1308–18. [11] Panigrahi SN, Jog CS, Munjal ML. Multi-focus design of underwater noise control linings based on finite element analysis. Appl Acoust 2008;69:1141–53. [12] Gaunaurd GC, Überall H. Resonance theory of the effective properties of perforated solids. J Acoust Soc Am 1982;71:282–95. [13] Ivansson SM. Sound absorption by viscoelastic coatings with periodically distributed cavities. J Acoust Soc Am 2006;119:3558–67. [14] Liang B, Cheng JC. Optimal acoustic attenuation of weakly compressible media permeated with air bubbles. Chin Phys Lett 2007;24:1607–10. [15] Hinders MK, Rhodes BA, Fang TM. Particle-loaded composites for acoustic anechoic coatings. J Sound Vib 1995;185:219–46. [16] Jiang H, Wang YR, Zhang ML, Hu YP, Lan D, Zhang YM, et al. Locally resonant phononic woodpile: a wide band anomalous underwater acoustic absorbing material. Appl Phys Lett 2009;95:104101. [17] Psarobas IE, Stefanou N, Modinos A. Scattering of elastic waves by periodic arrays of spherical bodies. Phys Rev B 2000;62:278–91. [18] Liu ZY, Chan CT, Sheng P, Goertzen AL, Page JH. Elastic wave scattering by periodic structures of spherical objects: theory and experiment. Phys Rev B 2000;62:2446–57. [19] Chung JY, Blaser DA. Transfer function technique method of measuring in-duct acoustic properties. I – Theory. II – Experiment. J Acoust Soc Am 1980;68:907–21. [20] Duclos A, Lafarge D, Pagneux V. Transmission of acoustic waves through 2D phononic crystal: visco-thermal and multiple scattering effects. Eur Phys J Appl Phys 2009;45:11302. [21] Perrot C, Chevillotte F, Hoang MT, Bonnet G, Bécot FX, Gautron L, et al. Microstructure, transport, and acoustic properties of open-cell foam samples: experiments and three-dimensional numerical simulations. J Appl Phys 2012;111:014911. [22] Beranek L. Acoustics. American Institute of Physics; 1996.
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Backing effects on the underwater acoustic absorption of a viscoelastic slab with locally resonant scatterers Honggang Zhao ⇑, Jihong Wen, Haibin Yang, Linmei Lv, Xisen Wen Vibration and Acoustics Research Group, Science and Technology on Integrated Logistics Support Laboratory, College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, Hunan 410073, PR China
a r t i c l e
i n f o
Article history: Received 19 July 2012 Received in revised form 22 July 2013 Accepted 25 July 2013 Available online 25 August 2013 Keywords: Underwater acoustics absorption Resonance modes Locally resonant scatterers Steel backing
a b s t r a c t Backing effects on the underwater acoustic absorption of a viscoelastic polymer slab embedded with locally resonant scatterers are reported. The polymer slab is embedded with two layers of locally resonant scatterers, i.e. Al spheres coated by soft silicon rubber. Theoretical absorption coefficients of the polymer slab under different backings using a layer multiple scattering method show good agreement with the experimental results, which supports unambiguously the experimental observation. Then relations between the resonance modes and the low-frequency absorption peaks of the composite slab are clarified to address the absorption mechanisms. It shows that the mass of the steel backing affects evidently the low-frequency absorption, the absorption peak shifts to lower frequency range while increasing the backing mass. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Locally resonant sonic materials (LRSMs) are generally known as composite epoxy embedded with various heavy spheres coated by soft rubber with periodic or random position [1]. It has been shown that the soft coating layer and the heavy core are responsible for a band gap at low frequency about two orders of magnitude lower than that predicted by Bragg’s condition. For practical applications, much effort has focused on air sound isolation induced by the gaps of different LRSMs [2]. Our investigations in acoustic dissipation by LRSMs have shown the low frequency acoustic absorption while considering the damping between the components [3,4]. Motivated by understanding of the energy dissipation in LRSMs, the locally resonant scatterers (LRSs) are introduced to improve the low-frequency acoustic absorption of the water impedancematched polymer. Experimental measurement for acoustic absorption of viscoelastic polymer slabs embedded with LRSs has been reported [5]. The fundamental mechanism operating in the localized resonance for acoustic absorption has been investigated by referring the mode conversions during the Mie scattering of a single scatterer. Further, the shape of the scatterer has also been taken into account [6]. In fact, viscoelastic polymers with various shapes of cavities [7– 14] or solid scatterers [15,16] are widely used in modern underwater acoustic applications. In most cases, scattering resonances including monopole and dipole types have been used to enhance ⇑ Corresponding author. E-mail address: [email protected] (H. Zhao). 0003-682X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apacoust.2013.07.022
the low frequency absorption of viscoelastic composites. The monopole resonance may be induced by various cavities [7–14] and the dipole type may occur in case of heavy spheres [5,6,15,16]. For spherical scatterers, the absorption properties can be investigated conveniently by the layer multiple scattering method (LMSM), which is developed for gap analysis in phononic crystals [17,18]. The computation is simplified by using the Bloch theory. In comparison with the finite element method [9–11], although the flexibility in model is limited, there are two important advantages of the LMSM, one is the fast computation, and the other is the convenient understanding of the physical absorption mechanism. For actual applications, the design of viscoelastic polymers embedded with locally resonant scatterers must consider the effect of the backing panel, which is not considered in our previous works [5,6]. The effects of the steel backing on the absorption are shown by the experimental and theoretical results in present paper. To address the absorption mechanisms, the relation of the resonance mode and the absorption peak of the composite slab is clarified based on the theoretical and experimental results. For comparisons, other backings are also considered.
2. Experiment 2.1. Model and sample Fig. 1a shows a sketch model of an absorption structure including a sample slab and a backing under the Cartesian coordinates
49
H. Zhao et al. / Applied Acoustics 76 (2014) 48–51
(a) L1
L2
(b)
Micr.
2
Micr.
sample
1
L3
Pi
s
L backing
Pr Z Steel backing Fig. 2. Schematic diagram of the two-microphone tube apparatus.
Fig. 1. (a) An absorption structure includes a composite slab and a steel backing. A plane wave is normally incident from the left along Z-axis. (b) Cross sections of the array of the scatterers of the sample in X–Y plane.
system. The whole structure can be considered as a sequence of different layers perpendicular to Z-axis. The sample is made of viscoelastic polymer embedded with two layers of LRSs. The interlayer positions of the scaterers are denoted by Li (i = 1, 2, 3), where L1 denotes the center of the first layer of LRSs from the surface of the sample, L2 is the distance between both the layers of LRSs, and L3 denotes the space between the second layer of LRSs and the other surface of the sample. The thickness of the whole sample is 55 mm, and L1 = L3 = 13.75 mm, L2 = 27.5 mm. Considering the limitation of the experiment apparatus, i.e. an impedance tube with an inner diameter d = 120 mm, the sample has an overall shape of short cylinder, which possesses a total diameter D = 118 mm. Each layer contains 30 scatterers, which are arranged in a quasi-triangular lattice with a mean adjacent distance 19.5 mm. The position of the scatterers is shown in Fig. 1b. Each LRS has a uniform spherical shape, in which a core sphere (radius r) coated by soft silicon rubber (with outer radius r1), here r = 5 mm, and r1 = 7.5 mm. In order to reduce the weight of the sample and facilitate the understanding of the resonance mode of the backing, we choose Al material as the core (which separates the resonances induced by the backing and the LRS). The soft coating is made of a type of two-component, room temperature vulcanizing silicone rubbers (RTV-2). It is available in a variety of hardness and suitable for various mold techniques. To begin, the core is coated by the silicon rubber using a self-made mold. Then the sample is fabricated by a hollow cylinder mold with an inner diameter D = 118 mm in the following procedure:
Under normal incidence, unlike the backing, the impedance tube almost does not interact with the testing waves because the first order resonant frequency of the tube walls is higher than that of the upper cut-off frequency in the present test. The experimental approach is based on a frequency response function method (FRFM) using two microphones [19]. One can get the frequency response function H12 = P2/P1, P1 and P2 denoting the acoustic pressure measured by Micr. 1 and Micr. 2 respectively. The complex pressure reflection coefficient (R) is given by:
R¼
H12 ejkL ejkðLþsÞ jkðLþsÞ e H12 ejkL
ð1Þ
;
where k is the wave number, defined as x/c, and c is the acoustic velocity of the water. L denotes the distance between the Micr. 1 and the surface of the sample, and s is the spacing between two microphones. Considering the fully reflective termination in the experiment, the sound energy absorption coefficient (a) can be calculated from the following equation:
a ¼ 1 jRj2 :
ð2Þ
3. Discussion 3.1. Comparison of theoretical and experimental results Generally, the steel backing can be considered as a finite steel slab followed by half infinite air. In order to understand the variation and the formation of the acoustic absorption of the sample under the steel backing, we compare the acoustic absorption of the sample under different backings in the experiment. We choose the backings as half infinite air (air backing in the following), steel
2.2. Absorption measurement method Measurement of the absorption coefficient is conducted in a standard impedance tube apparatus. The steel tube has an inner (outer) diameter 120 mm (180 mm), and 5 m long. The schematic of the apparatus is showed in Fig. 2. The impedance tube is fully filled with water. A plane wave incidence is excited at one end of the tube while the sample is positioned at the other end, where various backings can be attached on the surface of the sample.
Absorption coefficient (α)
1
(1) A polymer slab with thickness of (L1 r1) is made. (2) With a proper interval time, the coated scatterers are arranged on the surface of the polymer slab according to the design, and the rest space of the mold with a height (L1 + L2 r1) is filled by the polymer. (3) The multi-layer sample is fabricated by an iterative procedure (2) according to Li.
E-Air E-5mm E-10mm
0.8 0.6 0.4
T-Air T-5mm T-10mm T-water
0.2 0
500
1000
1500
2000
2500
3000
Frequency /Hz Fig. 3. Comparison of the experimental and theoretical results of the sample’s absorption.
50
H. Zhao et al. / Applied Acoustics 76 (2014) 48–51
backings with different thickness (5 mm and 10 mm respectively). Fig. 3 gives the experimental absorption coefficients (labeled by E-) of the sample under different backings. One can readily see that an absorption peak appears under each backing. When the sample has the air backing, 5 mm and 10 mm steel backings in turn, the absorption peak shifts to lower frequencies and the bandwidth of the absorption becomes slightly narrower at the same time. To compare with the experimental results, the acoustic absorption of the sample under the corresponding backing is computed by the LMSM. For simplifying the analysis, the composite slab is assumed to be infinitely along intra-layer (XOY) plane, and the scatterers in each layer are arranged in a triangle lattice. The lattice constant, i.e. the nearest distance between two neighbor scatterers, is 19.5 mm. A plane longitudinal wave is incident from the half infinite water (see Fig. 1a). Material parameters for all components are fixed without including the dependence on frequency for simplicity, and are listed in Table 1. The parameters of the polymer and the silicon rubber are measured using a dynamic mechanical analyzer (DMTA IV), and the parameters at 1 kHz are used. The viscosity of other components is neglected and the corresponding damping factor is set to zero. The density of water (air) is 1000 kg m3 (1.29 kg m3) and the longitudinal wave speed is 1480 m s1(340 m s1). The theoretical results (labeled by T-) are compared with those from the experiment in Fig. 3. One can readily see that the results of the calculation under different backings show good agreement with those of the experiments. Little difference between the theoretical and experimental results exists. For instance, the bandwidth of the absorption peaks by calculation is narrower than those of the experiments. This phenomenon may be induced by little volume of air bubbles (although eliminated by some means, for example, vacuumizing and static pressure) remaining in the actual sample. The bubbles enhance the wave scattering and visco-thermal effects [20,21], then the acoustic absorption in actual experiment. However, the dissipations induced by the bubble scattering and the visco-thermal effects are not considered in the present computation. 3.2. Discussions In order to understand the origin of the above absorption peaks, we also calculate the absorption of the composite slab under water backing, denoted by the long dashed line in Fig. 3. One can see that there is weak absorption peak at 2080 Hz. Here the polymer is water impedance-matched, so the water backing affects little the absorption of the composite slab. The weak absorption peak is induced by the weak resonance of the scatterer determined by the light Al core, which can be verified by the analysis of Mie scattering. For a spherical scatterer, the Mie scattering matrix (T) for each partial wave of l order is given by Tl [5], in which L, M and N denote respectively one longitudinal wave mode and two transverse wave modes. It shows that the L and N modes are coupled to each other, while the M mode is decoupled. Since only longitudinal incidence exists underwater, the M mode is out of our consideration in following discussions. The meanings of the elements in Tl are clear. Table 1 Physical parameters of the materials used in the present paper. Material
q (kg m3)
Polymer Silicon rubber Steel Al
Young’s modulus (Pa)
For example, LL stands for the conversion from a longitudinal incidence into a longitudinal outgoing wave during the scattering procedure. Because the localized resonance at low frequency is a dipole type, we focus on the mode conversions during the dipole (l = 1) scattering. Fig. 4 shows the variation of the absolute values of the T1 elements with frequency for the coated Al sphere in unbounded polymer. From Fig. 4a, one can see that the longitudinal incidence is scattered by the LN channel exceeding the LL channel during the dipole resonance, which suggests that the longitudinal incidence prefers to scatter into transverse wave. From Fig. 4b, it shows that the amplitudes of the T1 elements for the transverse wave incidence present about five orders of those for the longitudinal wave incidence, which hints that the transverse wave dominates the scattering. Generally, the transverse wave damps rapidly in viscoelastic polymer, then the acoustic dissipation is enhanced. Accordingly, the absorption peak of the polymer slab under the water backing appears around the dipole resonant frequency (comparing Figs. 3 and 4). The amplitude of the absorption peak is mainly determined by the amplitude of LN scattering in T1 since only longitudinal wave incidence exists underwater. Under the air backing, however, we can readily see that the absorption coefficient is much enhanced and a new absorption peak comes into being. The acoustic reflection from the polymer/ air interface at the backing forms a partial standing wave in the composite slab because of the energy damping, and there exists corresponding standing wave resonance in the composite slab. The standing wave resonance enhances the mode conversion from longitudinal to transverse waves and the multiple scattering among the scatterers, so the energy dissipation in the composite slab is enhanced and a new absorption peak comes into being. From the dotted line in Fig. 3, one can see that this new absorption peak combines that induced by the localized resonance because both absorption peaks have a close position. For the steel backing, a distinct feature from Fig. 3 is that the absorption peak shifts to lower frequencies while increasing the thickness of the steel slab. The absorption peak induced by the localized resonance appears again because the space between both the absorption peaks increases. However, the absorption peak induced by the localized resonance is smoothed in the experimental results, which may be induced by the variation of the actual parameter with frequency and the bubble dissipation, the thorough analysis is out of our present work. In order to gain more physical insight into the variation of the above absorption peaks under the steel backing, it will be useful to state the input acoustic impedance of the steel backing. Due to the characteristic impedance of air is much less than that of the steel, the input impedance of the steel backing for normal incidence is [22]
Z in ¼ jZ s tanðks dÞ;
where d denotes the thickness of the steel slab, Zs is the characteristic impedance of the steel and ks the wave number, both parameters being defined as Zs = qs cls and ks = x/cls, here qs and cls denotes the density and longitudinal velocity of the steel. Note that the low frequency range are focused and d is less than 10 mm, so the ksd is a small argument, the tangent can be approached as a linear function leading to the expression:
Z in ¼ jZ s ks d ¼ jqs xd:
Poisson’s ratio (m)
Damping factor (tan d)
1100 1300
2.97 107 2.0 106
0.498 0.46
0.3 0.3
7890 2690
2.09 1011 7.44 1010
0.275 0.334
0 0
ð3Þ
ð4Þ
The above equation demonstrates that the input impedance of the steel backing is not dependent on the velocity of the steel for low values of ksd, is proportional to the mass qsd. Based on the absorption peak variation with the different steel backing in Fig. 3, the absorption peak shifts to lower frequencies while increasing the mass (d or qs) of the steel backing. The absorption peak is induced by the standing wave resonance from the overall resonance of the steel backing. The physical mechanism of the resonance
(a)
7
Amplitude of T1- matrix elements
H. Zhao et al. / Applied Acoustics 76 (2014) 48–51
6
by the resonance of the steel backing shifts to lower frequencies while increasing the backing mass. The resonance enhances the mode conversions from longitudinal waves to transverse waves and the multiple scattering among the scatterers, which dominates the low-frequency acoustic absorption.
x10-5 LL LN
5
Acknowledgements
4
Financial support of the National Natural Science Foundation of China under Grant Nos. 11004249 and 51275519 is gratefully acknowledged. The authors are pleased to thank anonymous referees for valuable comments and suggestions to considerably improve the manuscript.
3 2 1 0
References 0
500
1000
1500
2000
2500
3000
2500
3000
Frequency /Hz
Amplitude of T1- matrix elements
(b)
1 NN
0.8
NL
0.6
0.4
0.2
0
51
0
500
1000
1500
2000
Frequency /Hz Fig. 4. The absolute values of Tl-matrix elements (l = 1) for a coated Al sphere in unbounded rubber under longitudinal (a) and transverse (b) incidence. In each plots, the dashed and solid lines present the cases without and with mode conversions between the L and N modes respectively.
phenomenon under the steel backing can be analogous to a massspring system, where the composite polymer and the steel slab contribute to the spring and mass respectively. While increasing the mass of the steel slab by density or thickness, the resonance of the steel slab appears at lower frequency domain, and thus the absorption peak of the composite slab shifts to lower frequencies as well, as showed in Fig. 3. 4. Conclusions In conclusion, we have shown the backing effects on the absorption of the composite slab embedded with locally resonant scatterers. Both the experimental and theoretical results show that the absorption is adjusted significantly by the steel backing. The effects of the steel backing on the acoustic absorption are clarified by the input impedance of the steel slab, and other backings are also investigated for comparison. The standing wave resonance induced
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