Abstract:
A device with simultaneous negative effective mass density and bulk modulus has at least one tubular section and front and back membranes sealing the tubular section. The front and back membranes sealing the tubular sections seal the tubular section sufficiently to establish a sealed or restricted enclosed fluid space defined by the tubular section and the membranes, and restrict escape or intake of fluid resulting from acoustic vibrations. A pair of platelets are mounted to the membranes, with the individual platelets substantially centered on respective ones of the front and back membranes.

Description:
RELATED APPLICATION(S) 
       [0001]    The present patent application claims priority to U.S. Provisional Patent Application No. 61/796,024 filed Nov. 1, 2012, which is assigned to the assignee hereof and filed by the inventors hereof and which is incorporated by reference herein. 
     
    
     BACKGROUND 
       [0002]    1. Field 
         [0003]    This disclosure relates to acoustic metamaterial that exhibits negative-valued effective mass density and effective bulk modulus in an overlapping frequency regime. 
         [0004]    2. Background 
         [0005]    Acoustic metamaterials are man-made structures that aim to achieve acoustic/elastic properties which are not available in tradition materials. In particular, negativity in effective dynamic mass density was demonstrated in various different designs. Materials with negative acoustic properties present a negative mass density and bulk modulus, and therefore a negative index of refractivity. Negative effective bulk modulus was also realized in fluid channels with cavity resonators. Other effects such as focusing, image magnifying, acoustic cloaking, total absorption were also realized experimentally. Currently, simultaneous negativity in both effective mass density and bulk modulus was only achieved by a composite structure of membranes and pipe with side-holes. 
         [0006]    Current panels do not offer simultaneous negative-valued effective mass density and bulk modulus in acoustics. An existing recipe for acoustic double negativity relies on coupling of two resonating structures. Additionally, a chain of unit cells is required for the demonstration of sufficient effect. Finally, the side-shunting holes are a significant source of dissipation. 
       SUMMARY 
       [0007]    A device with simultaneous negative effective mass density and bulk modulus, has at least one tubular section and front and back membranes sealing the tubular section. The membranes seal the tubular section sufficiently to establish a sealed or restricted enclosed fluid space defined by the tubular section and the membranes, so that the sealing or restriction restricting escape or intake of fluid resulting from acoustic vibrations. A pair of platelets are mounted on the membranes, with each platelet mounted to and substantially centered on respective ones of the front and back membranes. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]      FIGS. 1A  and B are schematic depictions of a structural unit in perspective ( FIG. 1A ) and cross-sectional ( FIG. 1B ) views. 
           [0009]      FIG. 2  is a graphic diagram showing calculated and measured transmission and reflection amplitudes. The left side (a) shows transmission amplitude and the right side (b) shows corresponding reflection amplitude. 
           [0010]      FIGS. 3A-C  are graphic images showing displacement of the metamaterials of  FIGS. 1A  and B in one-dimensional imagery. 
           [0011]      FIG. 4  is a graphic diagram showing properties of materials described by Green&#39;s function. 
           [0012]      FIGS. 5A and 5B  are graphical depictions representing experimental transmission amplitude (left axis) and phase (right axis) as a function of frequency.  FIG. 5A  shows the functions for plastic wrap membrane.  FIG. 5B  shows the functions for Al foil membrane. 
           [0013]      FIGS. 6A  and B are graphical depictions of numerical simulations of the structures with Acrylonitrile Butadiene Styrene (ABS) membranes having different dimensions. 
           [0014]      FIG. 7  is a graphical depiction showing numerical simulations of an all-aluminum structure. 
           [0015]      FIGS. 8A  and B are graphical depictions of numerical simulations of structures with working frequencies in the ultrasound regime with membranes having different dimensions. 
           [0016]      FIG. 9  is a schematic drawing showing an alternate structures in which two smaller hollow cylinders are attached onto the middle of the large membrane, one on each side. 
           [0017]      FIG. 10  is a schematic drawing showing an alternate structure having an outer cylinder supporting an inner cylinder suspended by a membrane supported by an outer cylinder. 
           [0018]      FIG. 11A-D  are diagrams showing eigenmodes of the example alternate structure of  FIG. 9 . 
           [0019]      FIGS. 12 and 13  are graphical depictions of calculated transmission coefficient and effective parameters for the structure of  FIG. 9 .  FIG. 12  shows the transmission spectrum.  FIG. 13  shows the calculated effective parameters. 
           [0020]      FIGS. 14 and 15  are graphical depictions of calculated transmission coefficient and effective parameters for the structure of  FIG. 10 .  FIG. 14  shows the calculated transmission coefficient.  FIG. 15  shows the calculated effective parameters. 
           [0021]      FIG. 16  is a schematic diagram showing a two-level hierarchically scaled repeating architecture based on the configuration of  FIG. 9 . 
           [0022]      FIG. 17  is a schematic diagram showing a two-level hierarchically scaled repeating architecture based on the configuration of  FIG. 10 . 
           [0023]      FIG. 18  is a diagram modeling the hierarchically scaled repeating architecture of  FIG. 17 , and is used in eigenmode representation. 
           [0024]      FIGS. 19A-G  are diagrams showing the eigenmodes obtained by numerical simulations of the structure in  FIG. 17 . 
       
    
    
     DETAILED DESCRIPTION 
     Overview 
       [0025]    The present disclosure implements a technique that reduces a complex system to a fictitious homogenous material that is characterized by a small set of effective constitutive parameters. This perspective greatly simplifies the description of wave propagation in metamaterials, and also exposes fresh physics and new possibilities for wave manipulations. This approach is used to tackle the problem of double negativity media for of low frequency sound, a traditionally very difficult problem. 
         [0026]    The present disclosure describes a type of acoustic metamaterial that can exhibit simultaneously negative effective mass density and bulk modulus in a finite but tunable frequency regime. The described configuration comprises two identical membranes sealing the two opening ends of a hollow cylindrical tube. Two identical platelets of certain rigid material are attached to the center of each said membrane. The two membranes are connected by a second hollow cylinder tube of certain rigid material. It is seen that the low-frequency behavior of the metamaterial is governed by three eigenmodes. A laser vibrometer is used to acquire the displacement fields as well as the relative phases of the two membranes, through which the three modes by their associated symmetry can be unambiguously discriminated. In addition, the effective parameters are extracted directly from the experimentally measured displacement fields. Double negativity in both the effective mass density and effective bulk modulus is found in a frequency regime of 500-800 Hz. In terms of functionality, negative effective mass density can be realized by membrane structures. Negative bulk modulus can be realized using Helmholtz resonators. Making the two effective parameters overlap is not ordinarily achieved in prior art acoustic metamaterials. 
         [0027]    An acoustic metamaterial is described that exhibits simultaneously negative effective mass density and bulk modulus in a finite but tunable frequency regime. The design features two elastic membranes augmented by rigid disks or platelets that are placed close together and joined by a rigid ring. The side surface of the structure is enclosed in an air-tight manner. The resultant structure is a resonator that displays double negativity. 
         [0028]    The disclosed technology provides an acoustic device that exhibits extraordinary double negativity for low frequency airborne sound. 
         [0029]    Structure of Metamaterials 
         [0030]      FIGS. 1A  and B are schematic depictions of structural unit  101  in perspective ( FIG. 1A ) and cross-sectional ( FIG. 1B ) views. The diagrams show a structure of metamaterial that comprises two identical circular membranes. Depicted is outer cylinder  103  supporting two membranes  111 ,  112 . Membranes  111 ,  112  support inner cylinder  115 . Inner cylinder  115  is fixed to membranes  111 ,  112  which results in membranes  111 ,  112  forming outer ring portions of the membrane  121 ,  122  and inner circular portions  125 ,  126 , separated by inner cylinder. A pair of disks or platelets  131 ,  132  are fixed to respective inner circular portions  125 ,  126 . 
         [0031]    By way of non-limiting example, the typical sample here is with total membrane radius R=14 mm (outer ring portion; same as total radius of the membrane), thickness 0.2 mm, and augmented by a circular rigid platelet  131 ,  132  (radius of 4.5 mm and mass of 159 mg) attached to the center. The two membranes  111 ,  112  are each fixed to a rigid cylindrical side wall with a radial tensile stress 1.3×10 6  Pa. They are connected by a poly(methyl methacrylate) (PMMA) cylinder which forms the inner cylinder  115 , which has a thickness of 1.5 mm, inner radius 10 mm and is 6.0 mm in height. The ring  115  has a mass of 395 mg, and the materials parameters of the membranes may include, by way non-limiting example, may be any solid materials, as long as their thickness and elasticity is such that with proper dimensions of cylinders and platelets the structures can give rise to the desired eigenmodes. The amplitude and phase of the transmission and reflection were measured in a modified impedance tube apparatus, comprising two Brüel and Kjær type-4206 impedance tubes with the sample sandwiched in between. The front tube has two sensors, plus a loud speaker at one end to generate a plane wave in the tube. The back tube has one sensor to measure the transmitted wave. 
         [0032]    While a cylindrical tube and identical platelets are described, it is possible, within the scope of this disclosure, to use a variation on a cylindrical tube, such as a frustoconical tube or a complex shaped tube. It is also possible to use platelets which are either non-identical but sharing at least one eigenmode or eigenfrequency when mounted on the membrane or non-identical and not sharing an eigenmode or eigenfrequency. It is also possible to select the shape of the tube and/or the sizes of the platelets such that the eigenmodes or eigenfrequencies of the platelets are close but still differing enough to interact with each other as a result of resonant differences. 
         [0033]      FIG. 2  is a graphic diagram showing calculated and measured transmission and reflection amplitudes of a sample structural unit  101  constructed according to  FIGS. 1A  and B. The left side (a) shows transmission amplitude and the right side (b) shows corresponding reflection amplitude. The calculated values are depicted by the solid lines and the measured values are depicted by the circles. Three transmission peaks, located at 290.1 Hz, 522.6 Hz, 834.1 Hz, are seen in both the measured and reflected amplitudes and corresponding peaks occur inversely in the transmission and reflection amplitudes. 
         [0034]    The relevant acoustic angular frequency ω is limited by the condition 2πv 0 /ω=λ&gt;2R, where v 0 =343 msec is the speed of sound in air. Thus we have ω&lt;7.79×10 4  Hz under this constraint. An immediate consequence is that as far as the radiation modes are concerned, i.e., transmission and reflection, the system may be accurately considered as one-dimensional. This can be seen as follows. The normal displacement u of the membrane may be decomposed as u=         u         +δu, where          u          represents the piston-like motion of the membrane (with           representing surface averaging) and δu the fine details of the membrane motion. In the air layer next to the membrane surface, the acoustic wave must satisfy the dispersion relation k ∥   2 +k ⊥   2 =(2π/λ) 2 , where k ∥/(⊥)  represents the wave vector component parallel (perpendicular) to the membrane surface. Since the two dimensional fine pattern of k ∥  can be described by a linear superposition of k ∥ &#39;s, all of which must be greater than 2π/2R&gt;2π/λ, it follows that the relevant k ⊥   2 &lt;0. That is, the displacement component δu leads only to evanescent, non-radiating modes. The displacement component          u         , on the other hand, has k ∥  components peaked at k ∥ =0; hence it is coupled to the radiation modes. 
         [0035]      FIGS. 3A-C  are graphic images showing displacement of the metamaterials forming the structural unit  101  of  FIGS. 1A  and B, depicted in one-dimensional imagery. The images show measured (circles or blocks) and calculated (curves) displacement profiles of the metamaterial at three eigenfrequencies. 
         [0036]    Simplification to a one-dimensional system greatly facilitates the visualization of the relevant symmetries of the two types of resonances, involving either the in-phase or the out-of-phase motion of the two membranes. An important element of the experimental measurements is the use of laser vibrometer (Graphtec AT500-05) to map the normal displacement across the membrane on the transmission side, plus the relative phases of the two membranes. For simplicity, this relative phase can be detected by the relative motion between the two platelets. In  FIG. 2  it is shown that three displacement fields of the coupled-membrane system at the transmission peaks, i.e., resonance frequencies (ω 1     +   =290.1 Hz, ω 2     −   =522.6 Hz, and ω 3     +   =834.1 Hz). The continuous curves delineate the simulated results by using the COMSOL Multiphysics finite element package, whereas the circles represent the measured results using laser vibrometer. Excellent agreement is seen. For the first mode, both membranes oscillate in unison, carrying the ring together in a translational motion. For the second eigenmode, the ring is motionless and only the membranes vibrate. Since the PMMA ring is rather rigid, it is impossible for the soft membrane to compress the ring at such low frequencies. Consequently, the ring acts like an anchor, and the central portions of the two membranes vibrate in an out-of-phase manner. For the third eigenmode, the ring and the platelets vibrate in opposite phase. It is seen that the simulated phase relation between the two platelets agrees with the experimental results almost perfectly. 
         [0037]    While the first and third eigenmodes are clearly dipolar in character and hence mass-density-type (MDT), the second mode has the monopolar symmetry and hence bulk-modulus-type (BMT). For the dipolar resonance, the total mass of the ring and the platelets serves as the most important parameter for tuning its frequency. For the monopolar resonance, the membranes&#39; separation and transverse dimension are the crucial parameters. The fourth eigenmode is noted to be at a much higher frequency of 2976.3 Hz. Its effect in the frequency range of interest was minimal, and thereby ignored in the following analysis. 
         [0038]    The average displacement of the two coupled membranes may be denoted by          {right arrow over (w)}         =[         u(x 0 )         ,         u(−x 0 )         ], where −x 0  and x 0  denote the positions of the two membranes.          {right arrow over (w)}          can be decomposed into two distinct modes discriminated by symmetry, i.e.,          {right arrow over (w)}         =ξ         {right arrow over (w)}           + ±η         {right arrow over (w)}           − . Here ξ and η are arbitrary coefficients. Symmetric mode          {right arrow over (w)}           +  denotes the motion in which the two membranes move in unison, i.e.,          u(x 0 )         =         u(−x 0 )         . Anti-symmetric mode          {right arrow over (w)}           −  is characterized by          u(x 0 )         =−         u(−x 0 )         , i.e., the two membranes moving out of phase with each other. 
         [0039]    The two relevant effective material parameters are the dynamic mass density  ρ  (associated with the symmetric mode) and the effective bulk modulus  κ  (associated with the anti-symmetric mode). To extract these two effective parameters, a homogenization scheme is established, based on the fact that the behavior of the system is dictated by the resonant eigenmodes. The scheme needs only the 3 relevant eigenfunctions to delineate the correlated motions on the two ends (i.e., the two membranes). This aspect is distinct from the homogenization schemes in which matching the response of the entire frequency range of interest is required. 
         [0040]    Consider the eigenfunction expansion of Green&#39;s function: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    where ρ α ≡∫ Ω u α *({right arrow over (x)})ρ({right arrow over (x)})u α ({right arrow over (x)})d{right arrow over (x)} denotes the averaged mass density for the α th  eigenfunction u α ({right arrow over (x)}), and ω α  and β α  are the resonant frequency and dissipation coefficient that can be experimentally determined. By using the experimentally measured eigenfunctions, as shown in  FIGS. 3A-3C , the relevant ρ α  can be evaluated. On the other hand, the dissipation coefficients β α  are determined by comparing the magnitudes of the measured and simulated eigenfunctions. For the frequency range of interest, it turns out that only three eigenfunctions are needed, i.e., α ranges from 1 to 3 only. We are interested in the cross-sectional averaged motion of the two membranes. By carrying out the cross-sectional average on G, we obtain 
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         [0000]    where the two coordinates are now specified at the positions of the two coupled membranes.          G          can always be decomposed into a symmetric component  G   +  and an anti-symmetric component  G   − , where 
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         [0041]    Now consider a homogeneous one-dimensional system of length 2 x 0 . Green&#39;s function of such a one-dimensional system is uniquely determined by the two material parameters  ρ  and  κ . In particular, we can have the similar quantities  G   ±   (1D)  that are given by the formulas: 
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         [0042]    By requiring  G   ± =  G   ±   (1D) , we obtain two equations which determines  ρ  and  κ  as a function of frequency. 
         [0043]      FIG. 4  is a graphic diagram showing properties of the materials described by Green&#39;s function. The properties depicted are the real part of effective mass density, shown in panel (a), real part of effective bulk modulus, shown in panel (b), real part of effective wave vector, shown in panel (c), and the magnitude of the effective impedance of the metamaterial, shown in panel (d). Since there can be multiple solutions to (equation 4), the solution branch with the longest wavelength is selected. The results are shown in  FIG. 4 , at curves (a) and (b). For the purpose of clarity, only the real parts of the effective parameters are plotted. The darkly shaded frequency range denotes the double negativity regime. 
         [0044]    From  ρ  and  κ , we can use the transport matrix method to calculate the transmission and reflection coefficients T and R from the one-dimensional model. The results are displayed in  FIG. 2  as solid curves. They agree remarkably well with experimental results, even beyond the usual long wavelength regime (e.g., around ω 1     +    and ω 2     −   ). 
         [0045]    The transmission properties of the metamaterial are determined by two factors: impedance matching with air and the values of effective wave-vectors. We note that  ρ  crosses zero precisely at the two eigenfrequencies ω 1     +    and ω 3     +   , as depicted in  FIG. 4 , at curve (a), arising from the dipolar resonances. A direct consequence is that the effective impedance 
         [0000]    
       
         
           
             
               
                  
                 
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                   _ 
                 
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                       ρ 
                       _ 
                     
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         [0000]    as depicted in  FIG. 4 , at curve (d), matches well with the background air. Two transmission peaks, accompanied by reflection minima shown in  FIG. 2  (right side), are seen at ω 1     +    and ω 3     +   . The anti-resonance frequency, represented by ω 1     +   , is between the two MDT eigen-frequencies. The anti-resonance is due to the out-of-phase hybridization of the two neighboring MDT eigenmodes that leads to          {right arrow over (w)}           + =0, at which point  ρ  must diverge. 
         [0046]    To simplify the picture, the BMT frequency ω 2     −    has been tuned to coincide with the anti-resonance ω 1     +    (within several Hertz). Due to the monopolar resonance, the volumetric pulsation (anti-symmetric motion) is large, thereby leading to a small but finite  κ . (In the absence of loss, namely β 2     +   =0,  κ  vanishes at ω 2     −   .) The calculation shows that because of the large  ρ , |  Z | still takes a very large value, and consequently the impedance mismatches with air. This raises the question of why we see a transmission peak at this frequency, instead of a dip. The reason lies in the effective wave-vector  k , depicted in  FIG. 4  at curve (c), which takes the value of  k =π/2x 0  at ω 2     −   , with 2x 0  being the thickness of the metamaterial (as well as the homogenized slab). This indicates that the effective wavelength  λ =2π/  k =4x 0 , is twice the thickness of the slab. As a result, Fabry-Perot-like multiple reflections of the wave inside the slab constructively interferes at the transmission end of the homogenized slab, eventually enhancing the overall transmission. 
         [0047]    The key frequencies discussed above: ω 1     +   , ω 1     +   , (ω 2     −   ), and ω 3     +   , divide the spectrum into two passbands. The first one is a conventional double-positive band, found in ωε(ω 1     +   , ω 1     +   (ω 2     −   )). This is depicted in  FIGS. 2 and 4  as the white regions  203  (third band from the top). The second one, residing in ωε(ω 1     +   (ω 2     −   ), ω 3     +   ) is due to the overlapping of the negative  ρ  and negative  κ  bands. This is depicted in  FIGS. 2 and 4  as the darkly shaded regions  202  (second band from the top). In the doubly negative frequency regime, the instantaneous acceleration of the homogenized medium is always opposing the external excitation. In the meantime, it is expanding upon compression, and contracting upon release. Medium with such properties can support the propagation of acoustic wave, since effective wave-vector 
         [0000]    
       
         
           
             
               k 
               _ 
             
             = 
             
               ω 
                
               
                 
                   
                     ρ 
                     _ 
                   
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                     κ 
                     _ 
                   
                 
               
             
           
         
       
     
         [0000]    is real. This is depicted in  FIG. 4C . Its response is out-of-phase to the double-positive medium, which is demonstrated in the negative group velocity as can be seen from the slope of the dispersion in  FIG. 4C . 
         [0048]    Single-negative bandgaps are found in two regimes: ω&lt;ω 1     +   , and ω&gt;ω 3     +   . This is depicted in  FIGS. 2 and 4  as the lightly shaded regions  201 ,  204  (top and bottom bands). The first gap is due to negative-valued  ρ , whereas the second gap is due to the negative  κ . Single-negativity in the effective parameters gives rise to pronounced imaginary part of the effective wave-vectors within the bandgaps, so that the acoustic wave must be evanescent. Here, the transmission coefficients within the band gaps are not necessarily small. This is due to the relatively long decay length, given by d=Im(  k ) −1 . The minimum of d is around 13 mm, which is still larger than the thickness of the material. Hence the sound wave is penetrative. 
         [0049]    Alternative Membrane Materials 
         [0050]    The membranes used in the structures in this invention can in fact be of any solid materials, as long as their thickness and elasticity is such that with proper dimensions of cylinders and platelets the structures can give rise to the desired eigenmodes. This is because Hook&#39;s law of elasticity is generally held for any solid membranes as long as they are held tightly but not necessarily pre-stressed. It should preferably be crease-free but the functionality does not go away if the amount of creases or wrinkle is small. They are just imperfections caused by imperfect fabrication processes. The membrane can have thickness variation across the cell, as the general principle still applies. 
         [0051]      FIGS. 5A and 5B  are graphical depictions representing experimental transmission amplitude (left axis) and phase (right axis) as a function of frequency for ( FIG. 5A ) plastic wrap membrane, and ( FIG. 5B ) Al foil membrane. Both types of membranes are the familiar types of materials frequently used for food packaging in home kitchens, e.g., 0.1 mm thick by way of non-limiting example. 
         [0052]    Both spectra exhibit typical transmission minimum anti-resonances between two transmission maximum resonances. The anti-resonance principle for the occurrence of transmission minimum works in structures containing membranes made of solids other than rubber. The aluminum foil was held tightly but not pre-stressed. The basic unit of the structures of the disclosed technology is the fixed membrane plus weight structure, and so if the basic properties of such structure are the same regardless of the type of materials used as membrane, it is possible to construct the disclosed structures using materials other than rubber for the elastic membranes and without pre-stress. 
         [0053]      FIGS. 6A  and B are graphical depictions of numerical simulations of the structures with Acrylonitrile Butadiene Styrene (ABS) membrane.  FIG. 6A  depicts a simulation with an ABS membrane radius=50 mm, thickness=0.1 mm, Pb weight radius=8 mm, thickness=1.1 mm.  FIG. 6B  depicts a simulation with an ABS membrane radius=100 mm, thickness=0.5 mm, ABS weight radius=40 mm, thickness=2.25 mm. 
         [0054]      FIGS. 6A  and B and  FIG. 7  show numerical simulation transmission spectra for the structures with an acrylonitrile butadiene styrene membrane and an aluminum membrane, respectively. These membranes behave according to the experimental results depicted in  FIGS. 5A  and B.  FIGS. 8A and 8B  shows numerical simulation transmission spectra for the structures with working frequency in the ultrasound regime. It is evident that by adjusting the design parameters one can cover a much wide frequency range. As the eigenfrequencies of the basic structure can be changed by adjusting the dimensions and materials used over a wide frequency range, it is possible to construct the disclosed structures for use in other frequency ranges, such as ultrasound. 
         [0055]      FIG. 7  is a graphical depiction showing numerical simulations of an all-aluminum structure. Membrane radius=50 mm, thickness=0.1 mm, weight radius=20 mm, thickness=0.1 mm. 
         [0056]      FIGS. 8A  and B are graphical depictions of numerical simulations of structures with working frequencies in the ultrasound regime.  FIG. 8A  depicts a simulation with Al membrane radius=0.5 mm, thickness=0.1 mm, Pb weight radius=0.15 mm, thickness=0.1 mm.  FIG. 8B  depicts a simulation with Si membrane radius=0.5 mm, thickness=0.1 mm, Si weight radius=0.2 mm, thickness=0.3 mm. 
         [0057]    Structure with Cylinder Suspended by Primary Membrane 
         [0058]      FIGS. 9 and 10  are schematic drawings showing two alternate structures. The alternative structures are both characterized by a large and rigid hollow cylinder with a large elastic membrane attached, and with a smaller cylinder arrangement supported by the large elastic membrane. In the configuration of  FIG. 9 , two smaller hollow cylinders  911 ,  912  are attached onto the middle of the large membrane  915 , one on each side. A smaller elastic membrane is attached to the open end of each smaller cylinder. Finally, a rigid platelet is attached onto the center of each smaller membrane. The two smaller cylinders can be separate, and joined through the membrane, or can be a single cylinder, with the membrane separating the single cylinder into two halves. 
         [0059]      FIG. 10  is a schematic drawing showing an alternate structure having an outer cylinder  1011  supporting an inner cylinder  1012  suspended by membrane  1015  supported by outer cylinder  1011 . The alternative structure is characterized by a hollow cylinder with both ends sealed by elastic membranes  1021 ,  1022 . A platelet  1025 ,  1026  is attached onto the center of each membrane  1021 ,  1022 . The whole sub-structure is then attached to larger hollow cylinder  1011  by membrane  1015 , which is separate from membranes  1021 ,  1022 . This results in an interruption in the continuity of the large elastic membrane within the smaller cylinder arrangement. The configuration of  FIG. 10  is similar to that of  FIG. 9 , except that the smaller cylinder arrangement is configured as a single cylinder. 
         [0060]    In one non-limiting example, for each of the configurations of  FIGS. 9 and 10 , the inner diameter of the large cylinder and the large membrane is 20 mm, while that of the smaller cylinder and membrane is 14 mm. The thickness of the membranes is 0.20 mm, and they are made of rubber. The wall thickness of the small cylinders is 0.5 mm, and its height is 1.5 mm. The diameter of the platelet is 4 mm, its thickness 0.2 mm. The mass density of the cylinders is 1.0 g/cm 3 , while that of the platelet is 13.6 g/cm 3 . 
         [0061]    In one non-limiting example, the large cylinder in either of  FIG. 9  or  10  has an inner diameter of 20 mm, with the diameter of the large membrane being the same. The inner diameter of the smaller cylinders and the diameters of their membranes is 12 mm. In this example, the thickness of the membranes is 0.20 mm, and they are made of rubber. The wall thickness of the small cylinders is 0.5 mm, and their height is 1.0 mm. The diameter of the platelet is 4 mm, its thickness 0.4 mm. The mass density of the cylinders is 1.0 g/cm 3 , while that of the platelet is 13.6 g/cm 3 . 
         [0062]      FIGS. 11A-D  are diagrams showing eigenmodes of the example alternate structure of  FIG. 9  having the 20 mm large cylinder and the 12 mm smaller cylinders. Numerical simulations show that such structure has two dipole-like eigenmodes and one monopole-like eigenmode. The lowest mode-1 is dipole-like at 227 Hz ( FIG. 11A ), followed by the monopole mode-2 at 341 Hz ( FIG. 11B ). The second dipole-like mode-3 is at 581 Hz ( FIG. 11C ), and the anti-resonance formed by mode-1 and -3 is at 447 Hz ( FIG. 11D ). The sequence of eigenmodes and anti-resonance of this structure is the same as the structure shown in  FIG. 1 . It is therefore expected that a wide frequency band will exist within which both the effective mass and modulus are negative; i.e., there is a double negativity region. 
         [0063]      FIGS. 12 and 13  are graphical depictions of calculated transmission coefficient and effective parameters for the structure of  FIG. 9 . The transmission spectrum ( FIG. 12 ) and band diagram ( FIG. 13 ) of the configuration of  FIG. 9  were obtained by numerical simulations. The shaded area denotes the double negative region. There is such a band between 450 Hz and 620 Hz, as shown by the shaded area in  FIG. 13 . 
         [0064]    In the configuration of  FIG. 10 , the single hollow cylinder is attached onto the middle of the large membrane.  FIGS. 14 and 15  are graphical depictions of calculated transmission coefficient and effective parameters for the structure of  FIG. 10 . The shaded area denotes the double negative region. Numerical simulations show that this structure has two dipole-like eigenmodes and one monopole-like eigenmode, similar to that of the first alternate structure ( FIG. 9 ), and as such, the eigenmodes are as reflected in  FIG. 11A-D . The lowest mode-1 is dipole-like at 299 Hz, as depicted in  FIG. 11A , followed by the monopole mode-2 at 341 Hz, as depicted in  FIG. 11   11 B. The second dipole-like mode-3 is at 662 Hz, as depicted in  FIG. 11C , and the anti-resonance formed by mode-1 and -3 is at 540 Hz, as depicted in  FIG. 11D . The sequence of eigenmodes and anti-resonance of this structure is the same as for the configuration of  FIG. 9 . It is therefore expected that a wide frequency band within which both the effective mass and modulus are negative; i.e., there is a double negativity region. The calculated transmission coefficient and effective parameters are given in  FIGS. 14 and 15 . There is such a band between 450 Hz and 620 Hz, as shown by the shaded area in the  FIG. 15 . 
         [0065]    Hierarchical Self-Similar Architecture 
         [0066]    The structures of  FIGS. 9 and 10  can be made into a hierarchically scaled repeating architectures that can possess a range of interesting acoustic characteristics.  FIG. 16  is a schematic diagram showing a two-level hierarchically scaled repeating architecture based on the configuration of  FIG. 9 , depicting platelets  1631  mounted on membranes  1633 . In this arrangement, the configuration of  FIG. 9  can be viewed as the basic unit of the hierarchically scaled repeating architecture. In the depicted configuration, platelets  1631  are on the outer membranes  1633  but not on inner membranes  1635 . 
         [0067]    Similarly,  FIG. 17  is a schematic diagram showing a two-level hierarchically scaled repeating architecture based on the configuration of  FIG. 10 , with the configuration of  FIG. 10  viewed as the basic unit of the hierarchically scaled repeating architecture. In the depicted configuration, platelets  1731  are on both outer and inner membranes  1733 ,  1735 . 
         [0068]      FIG. 18  is a diagram modeling the hierarchically scaled repeating architecture of  FIG. 17 , and is used in eigenmode representation.  FIGS. 19A-G  are diagrams showing the eigenmodes obtained by numerical simulations. In the examples, platelets  1631 ,  1731  (shown in  FIGS. 16 and 17 ) are positioned on the lowest hierarchical level, but it is also possible to add platelets to other positions. 
         [0069]    In calculating the eigenmodes represented in  FIGS. 19A-G , the following dimensions and materials parameters were used:
       Platelet: Lead, Diameter=5 mm, Thickness=0.5 mm   Membrane-1: Rubber, Diameter=16 mm   Membrane-2: Rubber, Width=6 mm   Membrane-3: Rubber, Width=4 mm, fixed on the outer cylinder with inner diameter=28 mm   Cylinder-1: Acrylonitrile Butadiene Styrene (ABS), Thickness=1 mm, Height=3.2 mm, Inner diameter=16 mm   Cylinder-2: ABS, Thickness=1 mm, Height=6.2 mm, Inner diameter=23 mm       
 
         [0076]    Mode 1 ( FIG. 19A ) at 95.3 Hz and Mode 3 ( FIG. 19C ) at 324 Hz are dipole-like of the first level unit without distortion of the second level unit, while Mode 2 ( FIG. 19B ) at 104.6 Hz is a monopole-like excitation corresponding to the first level unit, but without the distortion of the second level unit. Modes 4 and 5 ( FIGS. 19D  and E) at 458 Hz are degenerate monopole excitations of the second level unit. Mode 6 ( FIG. 19F ) is a hybrid of first level monopole excitation and second level dipole excitation. The contribution of such mode to the effective mass and effective modulus could give rise to new acoustic phenomenon. Mode 7 ( FIG. 19G ) is a collective dipole mode, with both the first and the second level units in dipole excitation. 
       CONCLUSION 
       [0077]    It will be understood that many additional changes in the details, materials, steps and arrangement of parts, which have been herein described and illustrated to explain the nature of the subject matter, may be made by those skilled in the art within the principle and scope of the invention as expressed in the appended claims.