Meta material porous/poro-elastic sound absorbers

Abstract

An acoustic metamaterial (AMM) passive impedance matching technique to enhance the acoustic performance of porous/poro-elastic sound absorbing materials is disclosed. An AMM passive matching device is implemented by achieving negative refractive index with double negative parameters, i.e., negative effective mass density and effective bulk modulus scheme, using various acoustic elements. The AMM technique consists of meta material architecture of acoustic inductive open tubes positioned strategically around the outside surfaces of the porous media and perforated screens inserted inside porous media to generate complex acoustic impedance load of the porous media; the inductance defined by a predetermined lengths of the open tubes. The device includes open tubes extending from the porous media to the outside ambient medium generating the desired reactive load over the broadband frequency region of the complex acoustic impedance of the porous media. The AMM open tubes and the resistive perforated screens generate conjugate acoustic impedance that matches the complex acoustic impedance.

Claims

1. An acoustic metamaterial passive impedance matching device for use in porous/poro-elastic materials to match the impedance load of the porous/poro-elastic materials on an ambient medium, comprising: a plurality of open tubes attached to at least two outer surfaces of a block of porous/poro-elastic material, and a resistive element in the form of a plurality of perforated screens positioned inside the block of porous/poro-elastic material, wherein the plurality of open tubes and the plurality of open screens generate an acoustic resistance and reactive impedance that matches the complex acoustic impedance load of the block of porous/poro-elastic material on an ambient medium.

2. The acoustic metamaterial passive impedance matching device of claim 1, wherein the plurality of open tubes are spaced evenly around the outer surfaces of the block of porous/poro-elastic material, the plurality of open tubes partially submerged inside the block of porous/poro-elastic material.

3. The acoustic metamaterial passive impedance matching device of claim 2, wherein the plurality of open tubes and the plurality of perforated screens may alternate in arrangement.

4. The acoustic metamaterial passive impedance matching device of claim 3, wherein each of the plurality of open tubes includes a different extent of its part outside the block of porous/poro-elastic material with respect to one another.

5. The acoustic metamaterial passive impedance matching device of claim 4, wherein the number of open tubes and the number of perforated screens are functions of the reactance and resistance of the block of porous/poro-elastic material.

6. The acoustic metamaterial passive impedance matching device of claim 5, wherein the dimension of the open tubes is a function of the reactance of the block of porous/poro-elastic material, the dimensions of the perforated screen dependent on the resistance of the block of porous/poro-elastic material.

7. The acoustic metamaterial passive impedance matching device of claim 6, wherein the plurality of open tubes increase in diameter from a first end of the block of porous/poro-elastic material to a second end of the block of porous/poro-elastic material, such that the plurality of open tubes taper in diameter from the second end to the first end.

8. The acoustic metamaterial passive impedance matching device of claim 6, wherein the plurality of open tubes are uniform in diameter from a first end of the block of porous/poro-elastic material to a second end of the block of porous/poro-elastic material, such that the plurality of open tubes include substantially equal diameters.

9. The acoustic metamaterial passive impedance matching device of claim 6, wherein the plurality of open tubes further comprises a second set of open tubes, each of the open tubes of the second set of open tubes includes open ends to provide an inductive reactance.

10. The acoustic metamaterial passive impedance matching device of claim 5, wherein the dimension of the plurality of open tubes is a function of the reactance of the block of porous/poro-elastic material.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1A shows a sound absorbing material comprising a fiberglass blanket according to one embodiment of the present disclosed technology.

(2) FIG. 1B shows the fiberglass blanket of FIG. 1A on a microscopic scale according to one embodiment of the present disclosed technology.

(3) FIG. 1C shows a sound absorbing material comprising a melamine foam according to one embodiment of the present disclosed technology.

(4) FIG. 1D shows the melamine foam of FIG. 1C on a microscopic scale according to one embodiment of the present disclosed technology.

(5) FIG. 2 shows sound absorption coefficients of poro-elastic foam samples calculated using numerical Johnson-Champoux-Allard (JCA) model according to one embodiment of the present disclosed technology.

(6) FIG. 3 shows real and imaginary parts of acoustic impedance of poro-elastic foam samples calculated using numerical Johnson-Champoux-Allard (JCA) model according to one embodiment of the present disclosed technology.

(7) FIG. 4A shows a unit cell with a perforated plate within a graph showing the mass density with respect to the geometrical parameters of the unit cell with the perforated plate according to one embodiment of the present disclosed technology.

(8) FIG. 4B shows a graph showing the bulk modulus with respect to the geometrical parameters of the unit cell with the perforated plate according to one embodiment of the present disclosed technology.

(9) FIG. 5A shows a unit cell with a side pipe within a graph showing the mass density with respect to the geometrical parameters of the unit cell with the side pipe according to one embodiment of the present disclosed technology.

(10) FIG. 5B shows a graph showing the bulk modulus with respect to the geometrical parameters of the unit cell with the side pipe according to one embodiment of the present disclosed technology.

(11) FIG. 6A shows equivalent circuits with distributed elements for a cell of Right-Handed (RH)-TL according to one embodiment of the present disclosed technology.

(12) FIG. 6B shows equivalent circuits with distributed elements for a cell of Left-Handed (LH)-TL according to one embodiment of the present disclosed technology.

(13) FIG. 6C shows the distributed equivalent circuit for a cell of CRLH-TL according to one embodiment of the present disclosed technology.

(14) FIG. 7 shows a schematic view of AMM impedance matching device consisting of open tubes and perforated screen in a block of poro-elastic foam backed by a hard wall according to one embodiment of the present disclosed technology.

(15) FIG. 8 shows a schematic view of AMM impedance matching device consisting of open tubes and perforated screen in a block of poro-elastic foam backed by a hard wall according to one embodiment of the present disclosed technology.

(16) FIG. 9 shows a schematic view of AMM impedance matching device consisting of open tubes and perforated screen in a block of poro-elastic foam backed by a hard wall according to one embodiment of the present disclosed technology.

(17) FIG. 10 shows sound absorption coefficient of a 3-inch (7.62 cm) thick AMM poro-elastic foam sample, calculated using the Johnson-Champoux-Allard (JCA) model according to one embodiment of the present disclosed technology.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE DISCLOSED TECHNOLOGY

(18) The main objective of this disclosure is to devise a passive method for management of acoustics and impedance matching for porous/poro-elastic sound absorbing materials with the ambient medium to maximize their sound absorption and enhance their sound absorption performance over a wide frequency range using acoustic metamaterial (AMM) principles.

(19) The present disclosed technology provides an acoustic metamaterial passive impedance matching system for use in porous/poro-elastic sound absorbing materials to match the complex acoustic impedance of the material with the ambient medium. The acoustic impedance device consists of an arrangement of inductive elements of open tubes, which connect poro-elastic medium to the ambient medium to provide passive impedance matching. The device may include a plurality of open tubes extending along the edges/circumference. An open tube of predetermined dimensions representing defines an acoustic inductance and resistance. A system of AMM inductive channels generates complex acoustic impedance that matches the acoustic impedance of the foam.

(20) The energy dissipated within a medium as sound travels through it is analogous to the energy dissipated in electrical resistors or that dissipated in mechanical dampers for mechanical motion transmission systems. All three are equivalent to the resistive part of a system of resistive and reactive elements. The resistive elements dissipate energy (irreversibly into heat) and the reactive elements store and release energy (reversibly, neglecting small losses). The reactive parts of an acoustic medium are determined by its bulk modulus and its density, analogous to respectively an electrical capacitor and an electrical inductor, and analogous to, respectively, a mechanical spring attached to a mass.

(21) Since dissipation solely relies on the resistive element it is independent of frequency. In practice however the resistive element varies with frequency. For instance, vibrations of most materials change their physical structure and so their physical properties; the result is a change in the ‘resistance’ equivalence. Additionally, the cycle of compression and rarefaction exhibits hysteresis of pressure waves in most materials which is a function of frequency, so for every compression there is a rarefaction, and the total amount of energy dissipated due to hysteresis changes with frequency. Furthermore, some materials behave in a non-Newtonian way, which causes their viscosity to change with the rate of shear strain experienced during compression and rarefaction; again, this varies with frequency. Gasses and liquids generally exhibit less hysteresis than solid materials (e.g., sound waves cause adiabatic compression and rarefaction) and behave in a, mostly, Newtonian way.

(22) Combined, the resistive and reactive properties of an acoustic medium form the acoustic impedance. The behavior of sound waves encountering a different medium is dictated by the differing acoustic impedances. As with electrical impedances, there are matches and mismatches and energy will be transferred for certain frequencies (up to nearly 100%) whereas for others it could be mostly reflected (again, up to very large percentages).

(23) Since bulk modulus and density of a medium control propagation of acoustic waves in the medium, it is important to focus on their variability as the wave propagates. These two parameters are analogous to the electromagnetic parameters, permittivity E and permeability μ, as can be seen in the following expression of the refractive index n and the impedance Z.

(24) $n=\sqrt{\frac{\rho }{B}}\left(\mathrm{acoustics}\right),n=\sqrt{\mathrm{\epsilon \mu }}\left(\mathrm{electromagnetism}\right)$$Z=\sqrt{\rho B}\left(\mathrm{acoustics}\right),Z=\sqrt{\mu /\epsilon }\left(\mathrm{electromagnetism}\right)$

(25) The mass density and the bulk modulus are always positive in conventional media and hard to modify because the material properties are directly associated with the chemical composition and bonding structures of the constituted atoms. However, a variety of effective acoustic parameters including negative values which never existed in nature can be obtained by metamaterials whose properties are mainly governed by the meta-atom structures that behaves like a continuous material in the bulk. According to the sign of the mass density and the bulk modulus, acoustic metamaterials can be classified to negative mass density, negative bulk modulus, double negative parameters, near-zero and approaching infinity mass density.

(26) With, either effective mass density or effective bulk modulus of acoustic parameters being negative, a fully opaque acoustic material is possible. However, an inverse effect in which sound wave energy propagates instead of being attenuated, when both of these two parameters are negative simultaneously.

(27) It is the intent of this patent to realize double negative parameters, with negative effective mass density and negative effective bulk modulus scheme, using passive impedance matching approach by combining various acoustic elements.

(28) Acoustic impedance is the opposition of a medium to a longitudinal acoustic wave motion. It characterizes the relationship between the acting sound pressure and the resulting particle velocity. This impedance is called the specific acoustic impedance of the medium because it characterizes the medium itself. When a sound source transfers its energy to a medium, however, the medium opposes the movement of the source with some kind of average impedance that is dependent not only on the medium, but also on the size of the air mass pushed by the sound source.

(29) Energy is dissipated in resistive elements. In a resistor, the current and voltage are always in phase. Inductive impedance stores energy. In inductors the current does not flow immediately upon the application of voltage. The current flow lags the voltage. In a pure acoustic inductance (no resistance), the particle velocity through lags the acoustic pressure across by 90°. Changes in velocity value and direction occur after changes in pressure and there is no dissipation of energy.

(30) In the porous/poro-elastic foam having a complex impedance of Z.sub.s(ω), with a negative inductive impedance X.sub.s(ω) (see Equation 2), it can be deduced that the sound waves entering the foam will be reflected back due to the impedance mismatch and there will be very little or no dissipation of acoustic energy depending on the characteristics of X.sub.s(ω). In fact, there has been no attempt to provide passive impedance matching of foam with the ambient medium (i.e., air/water, etc.) and as a result, some or most of the sound energy is reflected back from the foam due to impedance mismatch, thereby making them quite inefficient in the low frequency region.

(31) The load, i.e., the surface impedance Z.sub.s(ω), that the surrounding medium places on the porous media is an important factor. The knowledge of Z.sub.s(ω) allows us to quantify: (1) power dissipated in the porous media; and (2) the resistive and reactive forces of the medium on the source.

(32) The imaginary part of the porous/poro-elastic media impedance (the reactance, X.sub.s) can be considered as governing the energy stored in the fluid that continually reacts with the ambient medium surface and affects or impedes the energy transfer. This stored energy does not travel away from the ambient medium into the porous media. If efficient and or maximum dissipation of sound, that is sound transmission into porous media from the ambient medium, is desired, then impedance matching between the source (e.g., ambient medium) and the porous media must be considered.

(33) The resistive component is the only part involved in dissipation of real sound energy. Thus, the transmitted sound energy related to the real part of the resistive impedance is useful and represents the power dissipation capacity of the porous media.

(34) The sound power used up by the reactance, on the other hand, “is ‘watt-less’ power, involving energy which comes from the source and then back towards the source, without ever being dissipated as sound waves and that it involves “the mass or inertial property of the air that is involved.” It is “the mass reaction of the porous medium to the ambient medium”, the “additional apparent mass of the porous media.” “The fluid inside the porous media behaves like an effective mass”.

(35) The maximum power transfer theorem is a fundamental rule that can facilitate maximum power transfer between two circuit elements when their impedances are matched. The maximum power transfer theorem, states that a power source with source impedance Z.sub.s will transfer the maximum amount of power to a load impedance Z.sub.s* (e.g., ambient load) which is the complex conjugate of the source impedance. The theorem includes the complex impedance (i.e., reactance), and gives a condition that maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance. If maximum power transfer between the ambient medium and the porous media is facilitated using the impedance matching device proposed in this invention disclosure, sound energy will propagate unimpeded into the porous media and be dissipated.

(36) Referring to FIG. 2 and FIG. 3, simultaneously, FIG. 2 shows sound absorption coefficients of poro-elastic foam samples calculated using numerical Johnson-Champoux-Allard (JCA) model according to one embodiment of the present disclosed technology. FIG. 3 shows real and imaginary parts of acoustic impedance of poro-elastic foam samples calculated using numerical Johnson-Champoux-Allard (JCA) model according to one embodiment of the present disclosed technology. Porous/poro-elastic media can be modeled using the JCA model described earlier. 3D numerical models based on the finite element (FE) approach can be used to accurately describe acoustic behavior of the porous media taking thermo-viscous effects into consideration. FIG. 2 and FIG. 3 show sound absorption coefficient (SAC) and acoustic impedance of porous samples based on such a model using COMSOL software. FIG. 2 and FIG. 3 show the effect of reactive impedance of porous media on the SAC, particularly at low frequencies when the reactive impedance load, X.sub.s, is quite high, and apparently reflecting all the incident sound energy.

(37) The effectively dissipated power W by porous/poro-elastic media is:
W=Q2×Real[Z.sub.s]
where, Q is volume flow (product of velocity) and Re[Z.sub.R] is real (active) part of radiation impedance. The measured absolute impedance Z.sub.s, imaginary part (X.sub.L) and real part (R.sub.L) of the impedance of porous media are shown in FIG. 3. Similarly, calculated impedance curves using analytical models are shown in FIG. 3. The imaginary part, which is the reactive part of the radiation impedance, is more dominant below 1000 Hz, whereas the resistive part is quite robust and rises at lower frequencies, as observed in FIG. 3.

(38) The reactive part, which is inductive, implies that particle velocity lags acoustic pressure in the low frequency region (<1000 Hz). The reactive impedance, jωX.sub.L, below 1000 Hz, of the porous/poro-elastic media is like that of an inductive element. The real part (i.e., the resistive impedance) also increases steadily below 1000 Hz.

(39) Specific acoustic impedance (z) (characteristic impedance, wave impedance) is the opposition of a medium to wave propagation, and it depends on the medium properties and the type of wave propagating through the medium. The specific impedance of a medium opposing the propagation of a plane sound wave is equal to:
z=K×ρ.sub.0  (1)
where K is the stiffness (e.g., bulk modulus) of the medium in N/m2 and ρ.sub.0 is the density of the medium in kg/m3. The acoustic surface impedance Z.sub.s(ω), is not a fundamental acoustical property of porous media because it depends on the dynamic density and complex compressibility through the following equations. where, z.sub.b(ω), is the characteristic impedance of the porous media, and k(w) is the complex wavenumber.

(40) In the low-frequency limit, an open tube is called an acoustic inductance or an inertance and it has a direct analogy to the inductance in electrical circuit analysis or the mass in mechanical system analysis. The acoustic impedance of an open tube of length, L, and area A, is then given by:
Z(ω)={P(ω)}/{U(ω)}=jω(ρ.sub.0L/A),
where, U (ω)=AV (ω) is the acoustic volume velocity of the air mass and P(w) is applied sinusoidal pressure.

(41) Using acoustic metamaterials, acoustic wave propagation can be controlled by appropriate design of the refractive index distribution of the medium. In addition to the refractive index, the acoustic impedance also affects the sound propagation characteristics. For loudspeaker driver in the headphone, the radiation impedance allows the phase relationship between the surface pressure and the object velocity to be quantified. At lower frequencies, these two quantities are generally not in phase, with the velocity lagging behind the surface pressure by 90°.

(42) It is possible to obtain some extraordinary acoustic fluid parameters (ρ.sub.0 and B.sub.0), i.e., density and bulk modulus, by modifying the structural parameters of acoustic metamaterials, that cannot be realized easily using natural materials. These parameters include negative mass density and negative bulk modulus values, anisotropic mass density tensors, and anisotropic elasticity tensors.

(43) Referring now to FIG. 6A, FIG. 6B, and FIG. 6C, simultaneously, FIG. 6A shows equivalent circuits with distributed elements for a cell of Right-Handed (RH)-TL according to one embodiment of the present disclosed technology. FIG. 6B shows equivalent circuits with distributed elements for a cell of Left-Handed (LH)-TL according to one embodiment of the present disclosed technology. FIG. 6C shows the distributed equivalent circuit for a cell of CRLH-TL according to one embodiment of the present disclosed technology. Recently, metamaterials with simultaneously negative permittivity (E) and permeability (μ), more commonly referred to as left-handed (LH) materials, have received substantial attention. In the realm of electromagnetics, there is a common distinction between two types of metamaterials: arrays of resonant inclusions, such as the split-ring resonator and transmission line (TL) based metamaterials. While the materials of the upper kind are inherently narrow band and lossy due to their resonant nature, the latter can exhibit the desired meta-properties, such as negative refraction, over a much larger bandwidth and with lower losses since they do not explicitly rely on resonance.

(44) Most of the acoustic metamaterials reported to date belong to the category of resonant additions, whereas very few works on the acoustic counterparts of TL-based metamaterials have been reported. This requires the realization of acoustic or mechanical elements, which implement shunt “inductances” (i.e., acoustic masses) and series “capacitances” (i.e., acoustic compliances).

(45) Left-handed materials (LHMs), which in a wider sense, are also referred to as negative index materials (NIMs), simultaneously have negative permittivity, E, negative permeability, μ, and negative refractive index, n, over a common frequency band. The term “left-handed material” (LHM) was first introduced by Veselago in 1968, who predicted there exists such a medium in which the electric field, E, the magnetic field, H, and the wave vector, k, form a left-handed orthogonal set. However, left-handed materials do not exist in nature.

(46) Transmission line approach is based on the dual conventional transmission line. Backward wave transmission line (TL) can form a non-resonant LHM. Series capacitance (CO and shunt inductance (L.sub.L) combination supports a fundamental backward wave. Perfect LH TL is not resonant dependent but has a low loss and broad-band performance.

(47) In acoustic circuit modeling, the acoustic pressure p represents the electric voltage, and the volume velocity q flowing through a surface S substitutes for the electric current. Following this convention, an incremental section of a conventional fluid can be described by the model of FIG. 6A, where m.sub.a (i.e., L′.sub.R=ρ/S) is an acoustic mass (or inertance, L′.sub.R) and C′.sub.R (=S/B.sub.0) is an acoustic compliance, and ρ.sub.0 and B.sub.0 are the density and bulk modulus of the medium (e.g., air), respectively. The corresponding wave velocity is given by √ρ.sub.0B.sub.0 340 m/s. n acoustic waveguide, for example, can be described a purely right-handed (PRH) acoustic TL structure and can be represented by the TL circuit of FIG. 6A. It describes the propagation of acoustic waves inside the waveguide with positive index of refraction. The characteristic acoustic impedance of an open-open un-baffled waveguide may be given by: ρ.sub.0c [(ka).sup.2+j(0.6ka)] for (ka<<1) and is of the form: R+jX. The reactive impedance part (X) renders the waveguide as a PRH system with positive refractive index. The porous/poro-elastic media has a similar characteristic impedance as given in Equations (2-2).

(48) The purely Left-Handed (PLH) TL model, shown in FIG. 6B, is the combination of a times-unit length series capacitance C′.sub.L and a times-unit length shunt inductance L′.sub.L and is the dual of the PRH TL. Such a structure is known to exhibit a negative refractive index over an infinite bandwidth. In reality, a PLH structure is not possible because of unavoidable RH parasitic series inductance (L) and shunt capacitance (C) effects (parasitic capacitance is due to development of voltage gradients, and unavoidable parasitic inductance is due to current flow along the metallization).

(49) Considering the natural contribution of the non-vanishing connections between these two PRL and PLH circuits, the resulting periodic structure unit cell is the one shown in FIG. 6C. At low frequencies, the response is dominated by m′.sub.L and C′.sub.R, resulting in a left-handed (LH) behavior (negative refractive index), whereas m′.sub.R and C′.sub.L are predominant at higher frequency, which then results in a right-handed (RH) behavior (positive refractive index). In microwave engineering, interesting applications exist where both of these bands are used, which is why this structure has been named the composite right/left-handed transmission line (CRLH TL).

(50) An acoustic metamaterial that does not cause reflections at boundaries in all frequency regions while exhibiting positive and negative refractive index properties will be preferential. In most of the cases, an anti-reflection property was only achieved at a specific refractive index range or angle of incidence, and there have been no reports to date of an anti-reflection property being achieved for all refractive indices, including positive and negative indices, and regardless of the angle of incidence. In transmission line metamaterials, the impedance of the metamaterial can be matched with that of the air when the balanced condition is satisfied. This condition can be achieved by ensuring that the product of the shunt inductance and the capacitance has the same value as the product of the series inductance and the capacitance (e.g., L′.sub.RC′.sub.L=L′.sub.LC′.sub.R). The lumped series capacitance is indexed, C′.sub.L, and the shunt inductance, L′.sub.L. In such a balanced metamaterial, reflections can be strongly suppressed and the transmission can be maximized over the entire refractive index range.

(51) The equivalent circuits of a cell, for RH-TL and LH-TL are shown in FIG. 6A and FIG. 6B, respectively. In these circuits, L′.sub.R, C′.sub.R and L′.sub.L, C′.sub.L are the distributed inductance and capacitance for RH-TL and LH-TL, respectively. For a balanced CRLH-TL, the impedance matching conditions over a large frequency domain can be easily fulfilled.
Z.sub.CRLH-TL=Z.sub.LH-TL=Z.sub.RH-TL

(52) The equivalent balanced circuit of CRLH-TL is a combination of the equivalent circuits for RH-TL and LH-TL. The equivalent circuit for CRLH-TL is given in FIG. 6C, where, similar to RH-TL and LH-TL, Δl must be small enough compared to the wavelength. In CRLH-TL circuit, LH circuit balances the RH circuit to give a metamaterial impedance matching condition, which is similar to putting a conjugate impedance load on the initial complex impedance load. From the maximum power transfer theorem, thus, the added matching conjugate impedance Z*.sub.L (i.e., R.sub.L+X.sub.L) balances the existing Z.sub.L (i.e., R.sub.L-X.sub.L).

(53) The balanced (CRLH) metamaterial approach can now be seen as an implementation of the maximum power transfer theorem. It also explains how the maximum power transfer really works and can be achieved in nature.

(54) Circuit-theory concepts have been used to conceptualize and design an acoustic non-resonant TL-based metamaterial. Series compliances were implemented using membranes whereas the shunt acoustic masses were realized with transversally connected open channels. Such a metamaterial exhibits a negative refractive index over almost one octave (0.6-1 kHz), which is larger than what can be achieved with locally resonant acoustic metamaterials. However, one-octave coverage is very inadequate for practical applications and must be extended over at least 3 or more octaves.

(55) In the present disclosed technology, an acoustic metamaterial impedance matching device for porous/poro-elastic media, using open-tube inductive and resistive architecture, that is impedance matched for a porous media for all refractive indices including negative indices, is disclosed. This arrangement is highly distinctive and different from previous attempts and is based on the fact that the impedance of the porous media itself, as described earlier, consists mostly of resistive and inductive elements. It is important to note that the resistive and inductive impedance of a porous media needs to be matched with a similar but conjugate environment.

(56) The characteristic impedance of air is specific acoustic impedance (z) (characteristic impedance, wave impedance) is the opposition of a medium to wave propagation, and it depends on the medium properties and the type of wave propagating through the medium. The specific impedance of a medium opposing the propagation of a plane sound wave is equal to: Z=√B.sub.0ρ.sub.0=ρ.sub.0c, where B.sub.0 is the bulk modulus of the medium in N/m2, ρ.sub.0 is the density of the medium in kg/m.sup.3 and c is speed of sound in m/s. Thus, Z depends on both bulk modulus and density of the medium. The pressure in a periodic sound wave can be related to the displacement:
ΔP.sub.max=B.sub.0ks.sup.2.sub.max,
where, B.sub.0 is the bulk modulus of the medium, k (=ω/c) is wavenumber, and s.sub.max is the displacement of sound wave. The average intensity (the rate at which the energy being transported by the wave transfers through a unit area) over one period of the oscillation is:

(57) ${\left(I\right)}_{\mathrm{avg}}=\frac{1}{2}\sqrt{B_0{\rho }_0}{\omega }^2{s}_{\max }^2$
where, ω is the angular frequency. Thus, power or intensity carried by sound wave is proportional to the square root of both bulk modulus and density of air.

(58) An acoustic inductive element is analogous to an open pipe/tube. By combining acoustic inductors and resistors in a series acoustic element, a device with negative refractive index can be achieved. The acoustic mass is equivalent to the mass of the air in the enclosed element divided by the square of the cross-sectional area of the element. Also, since some small volume of the medium on either end of the tube is also entrained with the media inside the tube, the “acoustic” length is usually somewhat larger than the physical length of the tube. For a single open end, the difference between the physical length and the acoustic length is Δ1≈0.8a, also called the end correction. A structure that may be well approximated by an acoustic compliance is an enclosed volume of air with linear dimensions (<0.1λ). The variations in sound pressure within an enclosed air volume generally occur about the steady-state atmospheric pressure, the ground potential in acoustics.

(59) The basic constituent parameters that determine the propagation characteristics of acoustic waves in a medium are the density of the medium ρ.sub.0 and its bulk modulus B.sub.0. The velocity of an acoustic wave in the medium c and the refractive index relative to air n are given by:

(60) $c=\sqrt{\frac{B_0}{{\rho }_0}};n=\sqrt{\frac{{\rho }_r}{B_r}}$
where, B.sub.r=B/B.sub.0 and ρ.sub.r=ρ/ρ.sub.0 are the relative values of the bulk modulus and the mass density of the medium, respectively, with respect to values in air, which are B.sub.0=1.42×105 Pa and ρ.sub.0=1.22 kg/m3.

(61) When open tubes (OTs) are installed periodically as lumped elements in a one-dimensional acoustic waveguide, the pressure amplitude in the waveguide is affected by the dynamic motion of the air column that exists in the OT, and the value of the bulk modulus thus changes. In this case, the bulk modulus of the medium B is given by:
B=B.sub.0/[1−(ω.sup.2.sub.OT/ω.sup.2)], where, the transition frequency of the bulk modulus is given by:

(62) ${\omega }_{OT}=c\sqrt{\frac{S}{l^{\prime }\mathrm{dA}}}$
and, if only OTs have been installed, the mass density of the metamaterial p is equal to that of air ρ.sub.0. Here, c, S, d, and A are the speed of sound in air, the cross-sectional area of the OT, the effective length of the OT, the unit cell length, and the cross-sectional area of the waveguide, respectively.

(63) The two types of unit cells, e.g., open tubes with resistive elements can be combined to obtain a new complex unit cell, as shown in FIG. 7, FIG. 8, and FIG. 9, discussed in more detail below, which can be used to modify the mass density and bulk modulus, needed to modify resistance and reactance, in the porous media simultaneously. Porous media impedance is simulated by appropriate selection of the design parameters (e.g., A, L, d) of the inductance and resistance (i.e., the inductive tube/channel and open holes).

(64) Referring now to FIG. 5A and FIG. 5B, simultaneously, FIG. 5A shows a unit cell with a side pipe within a graph showing the mass density with respect to the geometrical parameters of the unit cell with the side pipe according to one embodiment of the present disclosed technology. FIG. 5B shows a graph showing the bulk modulus with respect to the geometrical parameters of the unit cell with the side pipe according to one embodiment of the present disclosed technology. A side tube in a unit cell could be used to modulate the bulk modulus of the medium by varying the side tube's height. The change in pressure in the main tube is p=−B.sub.0 (ΔV−ΔV.sub.h)/V, and the change in pressure in the side tube is p.sub.h=−B.sub.0DV.sub.h=ΔV.sub.h/V.sub.h. Here, V and V.sub.h represent the volumes of the main tube and the side tube, respectively, while ΔV and ΔV.sub.h are the small changes in the main tube and side tube volumes, respectively. The effective bulk modulus is only dependent on the observable volume change ΔV, and thus, the formula becomes p=−B.sub.eff ΔV/V. Because p=P.sub.h, the effective bulk modulus is given by B.sub.eff=B.sub.0/(1+V.sub.h/V), which means that as the height of the side tube increases, the effective bulk modulus decreases.

(65) Referring now to FIG. 4A and FIG. 4B, simultaneously, FIG. 4A shows a unit cell with a perforated plate within a graph showing the mass density with respect to the geometrical parameters of the unit cell with the perforated plate according to one embodiment of the present disclosed technology. FIG. 4B shows a graph showing the bulk modulus with respect to the geometrical parameters of the unit cell with the perforated plate according to one embodiment of the present disclosed technology. The action of an acoustic resistor is to absorb sound power. The viscous forces within a narrow tube convert the sound power into heat that dissipates away. A narrow tube or radius a (<<0.001λ) can represent an acoustic resistor. Thus, a perforated plate with miniature holes can provide desired resistance. The perforated plate can be regarded as a tiny pipe with an impedance of Z′.sub.0=ρ.sub.0c.sub.0/S. Thus, the variation of the sectional area of the hole is equivalent to the variation of the effective mass density, where a larger radius leads to a smaller effective mass density. A unit cell with a perforated plate, as shown in FIG. 4A, can be used to modulate the mass density of the medium by varying the radius of the hole. The size and shape of the perforation determines the momentum in the rigid plate produced by a wave propagating perpendicular on the plate, and, therefore, can be used to control the corresponding mass density component seen by this wave. This property is used to obtain the higher density component. If, on the other hand, the wave propagates parallel to the plate, it will have a very small influence on it, and consequently the wave will see a density close to that of the background fluid. The compressibility of the cell, quantified by the lower effective parameter, the bulk modulus, is controlled by the fractional volume occupied by the plastic plate.

(66) In the case of an acoustic metamaterial with a composite structure in which perforated plates and open channels, each lumped element affects the constituent parameters of the medium independently. The static density of the medium then becomes ρ′ rather than ρ.sub.0 because of the effect of the perforated plate, and the transition frequency of the bulk modulus should be modified to take the form ω.sub.OT=c√(ρS/ρ.sub.01′dA), which comes from the continuity equation of the medium.

(67) Referring to FIG. 7, FIG. 8, and FIG. 9, simultaneously, FIG. 7 shows a schematic view of AMM impedance matching device consisting of open tubes and perforated screen in a block of poro-elastic foam backed by a hard wall according to one embodiment of the present disclosed technology. FIG. 8 shows a schematic view of AMM impedance matching device consisting of open tubes and perforated screen in a block of poro-elastic foam backed by a hard wall according to one embodiment of the present disclosed technology. FIG. 9 shows a schematic view of AMM impedance matching device consisting of open tubes and perforated screen in a block of poro-elastic foam backed by a hard wall according to one embodiment of the present disclosed technology.

(68) The AMM passive impedance matching device is shown with resistive perforated screen 11 and open tubes 14, 16 integrated with a porous foam block 10. In embodiments, sound waves 13 travel to the AMM passive impedance matching device and through the perforated screen 11. The dimensions of the open tubes 14, 16 depend on the acoustic inductance required. Inductive reactance of the porous media determines the dimensions and number of open tubes 14, 16.

(69) In some embodiments, a plurality of open tubes 14, 16 are spaced around the outside surface 21 to the main foam block 10. In other embodiments, the plurality of open tubes 14, 16 and the perforated screen 11 alternate in arrangement. The plurality of open tubes 14, 16 each include one side of the open ends 14A, 16A to the outside, and the other side of the open ends to the inside of the foam block 10, to provide the desired inductive reactance.

(70) In view of the foregoing, the number of open tubes 14, 16 are functions of the impedance of the foam block 10. Indeed, the quantity of the open tubes 14, 16 and the pattern and the number of the perforated screens 11 are dependent on the impedance of the porous media 10. Further, the dimension of the open tubes 14, 16 is a function of the reactance of the porous media 10. Indeed, the dimensions of the open tubes 14, 16 and perforated screens 11 are dependent on the reactive impedance of the porous media 10.

(71) In embodiments, the AMM impedance matching device consists of open tubes 14, 16, 18, 20 and a perforated screen 11 in a block of poro-elastic foam 10 backed by a hard wall 22. In embodiments, the AMM passive impedance matching device is situated outside a porous foam block 10. The AMM passive impedance matching device is based on resistive and inductive TL elements. The inductive elements are implemented using open tubes 14, 16, which are open at both ends. The open tubes 14, 16 are partially submerged inside the foam block.

(72) In embodiments, the plurality of open tubes 14, 16 further comprises a second set of open tubes 18, 20 in addition to the inductive reactance provided by the first set of open tubes 14, 16, as shown in FIG. 8. These open tubes 14, 16, 18, 20 are partially submerged inside the porous foam block 10 to provide a predetermined amount of reactive impedance.

(73) Referring now to FIG. 10, FIG. 10 shows the predicted sound absorption coefficient of a 3-inch (7.62 cm) thick poro-elastic foam sample with AMM impedance matching device, calculated using the numerical Johnson-Champoux-Allard (JCA) model, compared with that of the baseline foam sample without the AMM device. The predicted curves using the JCA model for both 0 degree and 45 degree (0.784 radians) incidence are shown in FIG. 10. It may be observed that the AMM impedance matching device improves the sound absorption coefficient (SAC) of the porous foam block to almost 0.98-1.0 over the entire frequency range of 10-10000 Hz, whereas the baseline foam block shows a SAC of near 1.0 only over the frequency range of 1100-10000 Hz. There is a small frequency range of 1000-2000 Hz, where the SAC of the AMM foam block is slightly lower (about 0.96-0.99), which can be improved by adjusting the impedance using the resistive elements of the AMM impedance matching device.

(74) The portals of embodiments of the disclosed technology pass all the way through walls to allow dissipation of energy, while at the same time, taking into account reactive impedance of the form/wall materiel. The active impedance of the foam is canceled, at least partially or substantially fully with the tubes which extend, bulge, or exit from the foam/wall. The velocity and pressure of the sound waves simultaneously become in phrase. A screen (rigid or semi-rigid mesh material) is used in embodiments of the disclosed technology over portals/tubes in part or in full to add resistance and dissipation of energy there-into.

(75) For purposes of this disclosure, the term “substantially” is defined as “at least 95% of” the term which it modifies.

(76) Any device or aspect of the technology can “comprise” or “consist of” the item it modifies, whether explicitly written as such or otherwise.

(77) Any device or step to a method described in this disclosure can comprise or consist of that which it is a part of, or the parts which make up the device or step. The term “and/or” is inclusive of the items which it joins linguistically and each item by itself.

(78) When the term “or” is used, it creates a group which has within either term being connected by the conjunction as well as both terms being connected by the conjunction.

(79) While the disclosed technology has been disclosed with specific reference to the above embodiments, a person having ordinary skill in the art will recognize that changes can be made in form and detail without departing from the spirit and the scope of the disclosed technology. The described embodiments are to be considered in all respects only as illustrative and not restrictive. All changes that come within the meaning and range of equivalency of the claims are to be embraced within their scope. Combinations of any of the methods and apparatuses described herein above are also contemplated and within the scope of the invention.