WO2018107098A1

WO2018107098A1 - Double-zero-index structural waveguide

Info

Publication number: WO2018107098A1
Application number: PCT/US2017/065423
Authority: WO
Inventors: Fabio Semperlotti; Hongfei Zhu
Original assignee: University Of Notre Dame; Purdue Research Foundation
Priority date: 2016-12-09
Filing date: 2017-12-08
Publication date: 2018-06-14

Abstract

An acoustic waveguide includes a plate having a first surface and an opposing second surface. The plate includes an incident portion and a transmission portion. A plurality of unit cells is arranged in an array having a predetermined shape. Each cell of the plurality of unit cells includes a mass located at a center of the unit cell and having a central axis perpendicular to the first surface. A channel is disposed in the first surface and surrounds the mass and a radius of revolution of the channel is a predetermined distance from the central axis. The mass is integrally formed as one piece with the plate. The array is configured such that an acoustic wave incident on the array from the incident portion emerges in the transmission portion.

Description

DOUBLE-ZERO-INDEX STRUCTURAL WAVEGUIDE

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Application No. 62/431,880, filed with the United States Patent and Trademark Office on December 9, 2016; and U.S. Provisional Application No. 62/508,561, filed with the United States Patent and Trademark Office on May 19, 2017.

STATEMENT REGARDING FEDERALLY-SPONSORED RESEARCH

[0002] This application was made with government support under grant YIP FA9550-15-1-0133 awarded by the Air Force Office of Scientific Research. The government has certain rights in the application.

BACKGROUND

[0003] This application relates generally to acoustic waveguides. More specifically, this application relates to a double-zero-index elastic waveguide and corresponding effects produced thereby.

[0004] The concept of "metamaterials" is an approach to designing materials and structures able to manipulate wavelike fields in a manner typically not achievable in naturally-occurring materials. Metamaterials were initially studied in optics and implemented via optical antennas or microwave metamaterials in the context of light propagation across phase discontinuities.

[0005] Despite progress in the development of subwavelength designs in optics and attempts to apply these to acoustics, the extension of this concept to the control of elastic waves has not yet progressed much beyond a few preliminary studies on elastic waveguides with metamaterial inserts. There exists a need to extend the concept of metamaterials to a solid, so as to provide novel and important functionalities for structural acoustic waveguides while drastically expanding their range of application. [0006] One such application is the possibility of designing materials having near-zero effective parameters, such as refractive index. This class of materials was first formulated for

electromagnetic waves where epsilon-near-zero (ENZ), mu-near-zero (MNZ), and epsilon-and- mu-near-zero (EMNZ) properties were first obtained. Among the peculiar characteristics of these materials is the spatial independence of the phase from the propagation distance. This means that a wave entering an ideal double-zero material emerges on the other side having the exact same phase as the input. Double-zero materials are further characterized by a high level of

transmissibility, ideally acting as a non-reflective waveguide even in the presence of sharp discontinuities.

[0007] While some materials with near-zero permittivity may be found in nature, in acoustics applications near-zero density and elastic compliance must be achieved as effective quantities by leveraging the local dynamic response of the medium. Recently, progress has been made in designing acoustic metamaterials exhibiting single-zero effective parameters, such as near-zero density, analogous to single-zero electromagnetic materials such as ENZ. It has been difficult to design acoustic media with double-zero effective properties given that, for example, these materials are not readily available in nature.

SUMMARY

[0008] Theoretical studies on photonic and phononic crystals revealed that when a Dirac-like Cone (DC) can be obtained at the center of the Brillouin zone, such lattice can be mapped into a double-zero refractive index material. While different applications of this basic concept have been explored in photonics and phononics, there has not yet been successful implementation of these effective material properties in solids.

[0009] Various aspects of the present disclosure provide for the theoretical, numerical, and experimental realization of a structural phononic waveguide exhibiting double-zero-index- material (DZIM) behavior and capable of achieving novel behavior, including acoustic cloaking, supercoupling, and energy squeezing.

[0010] In one aspect of the present disclosure, an acoustic waveguide includes a plate having a first surface and an opposing second surface. The plate includes an incident portion and a transmission portion. A plurality of unit cells is arranged in an array having a predetermined shape. Each cell of the plurality of unit cells includes a mass located at a center of the unit cell and having a central axis perpendicular to the first surface. A channel is disposed in the first surface and surrounds the mass and a radius of revolution of the channel is a predetermined distance from the central axis. The mass is integrally formed as one piece with the plate. The array is configured such that an acoustic wave incident on the array from the incident portion emerges in the transmission portion.

[0011] In another aspect of the present disclosure, a waveguide unit cell includes a plate having a first surface and an opposing second surface. The plate includes a mass located at a center of the plate and having a central axis perpendicular to the first surface and a channel disposed in the first surface and surrounding the mass. A radius of revolution of the channel is a predetermined distance from the central axis. The mass is integrally formed as one piece with the plate.

[0012] According to various aspects of the present disclosure, an acoustic waveguide exhibiting double-zero-index behavior is provided. That is, acoustic waveguides according to the present disclosure exhibit zero effective mass density and zero reciprocal effective shear modulus at a predetermined wavelength.

[0013] The foregoing summary is intended merely to provide a general overview of various aspects of the present disclosure, and is not intended to limit the scope of this application in any way.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] These and other more detailed and specific features of various aspects of the present disclosure are more fully described in the following description, reference being had to the accompanying drawings, in which:

[0015] FIG. 1 A illustrates an isometric view of an exemplary unit cell according to various aspects of the present disclosure;

[0016] FIG. IB illustrates a cross section of the unit cell of FIG. 1A;

[0017] FIG. 2A illustrates an isometric view of another exemplary unit cell according to various aspects of the present disclosure;

[0018] FIG. 2B illustrates a cross section of the unit cell of FIG. 2A;

[0019] FIG. 3 illustrates a chart showing the dispersion relations for an exemplary unit cell according to various aspects of the present disclosure;

[0020] FIG. 4 illustrates an Equi -Frequency Surface plot for the dispersion relations of FIG. 3; [0021] FIG. 5 illustrates the dispersion relations for a modification to the exemplary unit cell of FIG. 3;

[0022] FIG. 6A illustrates a chart showing wave transmission behavior for an exemplary waveguide according to various aspects of the present disclosure;

[0023] FIG. 6B illustrates another chart showing wave transmission behavior for an exemplary waveguide according to various aspects of the present disclosure;

[0024] FIG. 6C illustrates another chart showing wave transmission behavior for an exemplary waveguide according to various aspects of the present disclosure;

[0025] FIG. 7 illustrates a waveguide according to various aspects of the present disclosure;

[0026] FIG. 8 illustrates another waveguide according to various aspects of the present disclosure;

[0027] FIG. 9 illustrates another waveguide according to various aspects of the present disclosure;

[0028] FIG. 10 illustrates another waveguide according to various aspects of the present disclosure;

[0029] FIG. 11 illustrates another waveguide according to various aspects of the present disclosure;

[0030] FIG. 12 illustrates an exemplary fuselage implementing various aspects of the present disclosure; and

[0031] FIG. 13 illustrates an exemplary airfoil-shaped body implementing various aspects of the present disclosure.

DETAILED DESCRIPTION OF THE INVENTION

[0032] In the following description, numerous details are set forth, such as flowcharts, data tables, and system configurations. It will be apparent to one skilled in the art that these specific details are merely exemplary and explanatory, and are not intended to limit the scope of this application.

[0033] [Unit Cell - Structure]

[0034] The present waveguide employs a tapered unit cell in an array configuration. The unit cell consists of a waveguide plate having opposing first and second surfaces, an embedded channel ("taper," "groove," or "grooved taper"), and a (resonating) center mass. In one aspect, the taper is a toroidal taper; that is, a groove defined by an elliptical surface of revolution about a center point coincident with a center of the unit cell. While the exemplary aspect shows in elliptical surface of revolution, the present disclosure is not so limited. In practice, the surface of revolution may take any shape that defines a groove, including an ellipse or a polygon.

Moreover, while the exemplary taper is circular in a plan view, the present disclosure is not so limited. In practice, the taper may be an ellipse or a polygon such as a rectangle or hexagon. Additionally, while the exemplary unit cell shows the mass located at the center of the unit cell and centered within the taper, the present disclosure is not so limited. In practice, the mass may be located anywhere within the taper; for example, if the taper is an ellipse in the plan view, the mass may be centered at a focus of the ellipse or at another interior point. Furthermore, the mass and the taper may be centered around a point that is not at the center of the unit cell. The unit cell may be one-sided or two-sided; that is, extending in one or both perpendicular directions with respect to the plate mid-plane.

[0035] FIGS. 1 A-B illustrate a one-sided tapered unit cell 100. FIG. 1 A illustrates an isometric view of the one-sided tapered unit cell 100, whereas FIG. IB illustrates an xz plane cross section of the one-sided tapered unit cell 100.

[0036] The one-sided tapered unit cell 100 includes a center mass 101 surrounded by a grooved taper 102. The center mass 101 and the grooved taper 102 are embedded within a waveguide plate 103. The center mass 101 extends in the +z direction, such that a bottom surface 104 of the waveguide 103 is substantially planar. While FIGS. 1A-B show the center mass 101 extending beyond a virtual plane coincident with a top surface 105 of the waveguide plate 103, this is merely exemplary and the center mass 101 may in fact extend to a point on or beneath the virtual plane.

[0037] FIG. IB also illustrates the geometric parameters of the one-sided tapered unit cell 100. Specifically, L represents the lattice constant (that is, a side length of the square unit); t represents the thickness of the waveguide plate 103; a represents a first half-axis of the ellipse defining the grooved taper 102 (either the semi-major axis or semi-minor axis, depending on the particular value of a); b represents a second axis of the ellipse defining the grooved taper 102 (either the minor or major axis, depending on the particular value of b); r represents a radius of revolution of the toroid measured from the center of the unit cell; and h is the height of the center mass 101 measured from the virtual plane coincident with the top surface 105 of waveguide plate 103. The particular values of the geometric parameters directly affect the band structure of the waveguide plate 103. For example, varying h affects the mass of the center mass 101, whereas varying a and/or b affects the resonant behavior of the grooved taper 102.

[0038] FIGS. 2A-B illustrate a two-sided tapered unit cell 200. FIG. 2A illustrates an isometric view of the two-sided tapered unit cell 200, whereas FIG. 2B illustrates an xz plane cross section of the two-sided tapered unit cell 200.

[0039] The two-sided tapered unit cell 200 includes a center mass 201 surrounded by a grooved taper 202. The center mass 201 includes a first center mass 201a located above a mid-plane 204 of the waveguide and a second center mass located below the mid-plane 204. The grooved taper 202 includes a first grooved taper 202a located above the mid-plane 204 and a second grooved taper 202b located below the mid-plane 204. The center mass 201 and the grooved taper 202 are embedded within a waveguide plate 203. Because two-sided tapered unit cell 200 is symmetric with respect to the mid-plane 204, the first center mass 201a extends in the +z direction and the second center mass 201b extends in the -z direction. While FIGS. 2A-B show the first and second center masses 201a-b extending beyond virtual planes coincident with a top surface 205 and a bottom surface 206, respectively, of the waveguide plate 203, this is merely exemplary and the first and second center masses 201a-b may in fact extend to a point on or beneath the virtual planes.

[0040] FIG. 2B also illustrates the geometric parameters of the two-sided tapered unit cell 200. Specifically, L represents the lattice constant (that is, a side length of the square unit); t represents the thickness of the waveguide plate 203; a represents a first half-axis of the ellipse defining the grooved taper 202 (either the semi-major axis or semi-minor axis, depending on the particular value of a); b represents a second axis of the ellipse defining the grooved taper 202 (either the minor or major axis, depending on the particular value of b); r represents a radius of revolution of the toroid measured from the center of the unit cell; hi is the height of the first center mass 201a measured from the virtual plane coincident with the top surface 205 of waveguide plate 203; and h₂ is the height of the second center mass 201b measured from the virtual plane coincident with the bottom surface 206 of the waveguide plate 203. While FIGS. 2A-B illustrate the two-sided tapered unit cell 200 as being symmetric with respect to the mid- plane 204 such that hi = h₂, the present disclosure is not so limited. The two-sided tapered unit cell 200 may instead be asymmetric with respect to the mid-plane 204. As above, the particular values of the geometric parameters directly affect the band structure of the waveguide plate 203. For example, varying i and/or h₂ affects the mass of the center mass 201, whereas varying a and/or b affects the resonant behavior of the grooved taper 202.

[0041] The waveguide plate 203 may be formed of any linear elastic material. For example, the waveguide plate 203 may be formed of metals, polymers, plastics, foams, wood, or composites (such as short-fiber composites). The particular material is dependent on the end application; for example, if the unit cell is intended for aerospace applications the material selected may be aluminum or an aluminum alloy for structural panels or a composite for internal panels.

[0042] [Unit Cell - Example]

[0043] In order to verify the effectiveness of the above geometrically-tapered metamaterials, an exemplary unit cell was numerically tested. For purposes of testing, it was assumed that the unit cell was made out of aluminum with mass density p = 2700 kg/m³, Young's modulus E = 70 GPa, and Poisson's ratio v = 0.33.

[0044] The dispersion relations for a two-sided tapered unit cell were calculated using a finite element solver. Given the finite dimension of the unit cell in the z direction the dispersion curves are composed by symmetric (S), anti-symmetric (A), and shear horizontal (SH) guided Lamb modes. Given the geometric parameters L = 0.04 m, t = 0.008 m, a = 0.0039 m, b = 0.007442 m, r = 0.012 m, and h = 0.013 m, the band structure of the waveguide was tuned to exhibit a threefold degenerate point (the Dirac-like point) at /= 27.04 kHz and k = 0. This is illustrated in FIGS. 3-5.

[0045] Specifically, FIG. 3 illustrates the dispersion relations around /= 27.04 kHz for the above example two-sided unit cell. FIG. 3 shows the existence of a triple-degenerate point 301 and a Dirac-like cone 302 for the given selection of geometric parameters. FIG. 4 is an Equi-Frequency Surface plot corresponding to the frequency range around the Dirac-like point and shows the formation of the Dirac-like cones 302. As illustrated in FIG. 4, the branches emanating from the triple-degenerate point 301 are isotropic and linear, and form two cones 401 touching at their respective vertices at the triple-degenerate point 301. The cones 401 are made of A₀ modes having non-zero but constant group velocity and are intersected by an A₀ flat band 402 at the triple-degenerate point 301. [0046] That the Dirac-like cone is the result of an accidental degeneracy can be confirmed by slightly perturbing the geometric parameters, as illustrated in FIG. 5. In FIG. 5, the semi-major (or semi-minor) axis a of the unit cell was slightly perturbed from a = 0.0039 m to a = 0.0035 m. As a result of this perturbation, the Dirac-like cone "opens up" (that is, cones 401 separate) and the triple-degenerate point 301 splits into a non-degenerate and a double-degenerate band. The corresponding eigenstates of the three degenerate modes are illustrated in FIG. 5 which shows, from top to bottom, a lower cone 501, a flat band 502, and an upper cone 503. This indicates that the formation of the linear dispersions in FIG. 3 was due to an accidental degeneracy.

[0047] To validate the concept of a double-zero index structural waveguide, shown numerically above, an exemplary physical two-sided unit cell was fabricated and tested. This additional unit cell was identical to the above-described exemplary unit cell, except that the geometric parameters were reduced to half the previous size, including the plate thickness t, which was reduced to 0.004 m. The rescaled waveguide resulted in a Dirac-like point at 54.08 kHz, measured experimentally.

[0048] The waveguide design is highly scalable and allows for operation over a wide range of frequencies. One manner in which to rescale the design in order to provide a Dirac-like point at a new frequency is to isotropically rescale the unit cell according to the relation IL = C, where λ is the wavelength at the desired frequency L is the lattice constant, and C is a constant value. Thus, to modify the wavelength by a particular scaling factor n, it is sufficient to scale all of the geometric parameters (in the example defined above, L, t, a, b, r, h) by a factor of \ln. For example, in order to double the Dirac-like point frequency, it is sufficient to divide all of the geometric parameters by a factor of two.

[0049] Similarly, if it is necessary to rescale the thickness of the supporting plate (for example, if a particular application requires a particular thickness for structural or other reasons), then the target A₀ flexural mode will shift to a new wavenumber according to the following equation (1):

[0050] Above, k and k' are the old and new wavenumber, respectively, and t and V are the old and new thickness, respectively. Alternatively, it is possible to consult tabulated dispersion curves (in the form of / vs. k h) for a thin plate of the selected material to obtain the recalled value of the frequency of the A₀ flexural mode.

[0051] It is also possible to perform a more general rescale involving changes to the thickness, the selected material, and the cell dimensions. Such a general rescale may be performed in a similar manner to the above analysis; that is: (1) perform a numerical dispersion analysis of the new design and plot the curves along the boundary of the first Brillouin zone; (2) identify the location of degenerate points at the Γ point and select only those associated with linear dispersion curves (that is, only those points associated with a conical Dirac dispersion); and (3) subject the geometric parameters of the unit cell to the required theoretical criteria.

[0052] Specifically, the lattice J should meet the criterion J » \ .5λ and the parameters a, b, and r should meet the criteria r + b < J/2, r > b, and a < t/2. A parameter optimization simulation may be required to tune the unit cell and obtain optimal performance.

[0053] Moreover, while the above validating examples are of unit cells having a centimeter scale, this disclosure is not so limited. Depending on the particular frequency at issue, unit cells according to this disclosure may be scaled larger (for example, to a meter scale) or smaller (for example, to a micrometer / nanometer scale).

[0054] [Dirac-like Dispersion]

[0055] Geometrically-tapered metamaterials, such as those described above, exhibit Dirac-like Cones at the center of the Brillouin zone (Γ point) as the result of "accidental" degeneracies. In other words, the degeneracy is induced by the specific combination of geometric parameters of the tapers, and is not protected by the underlying lattice structure as would be the case in, for example, graphene. In the neighborhood of this degenerate point, a waveguide including a particular array of geometrically-tapered metamaterials exhibits simultaneous zero mass density and zero reciprocal shear modulus (or, equivalently, infinite shear modulus).

[0056] The origin of this Dirac-like dispersion may be understood by an extension of the k p method from electronics applications to a phononic system. The system may be described by the Navier's equations with traction-free boundary conditions on the top and bottom surfaces of the waveguide. While imposing these boundary conditions in the presence of geometric tapers introduces complexities due to the changing direction of the unit vector normal to the tapered surface, this may be overcome by applying the three-dimensional plane wave expansion method to 2D phononic waveguides. [0057] By this method, the 2D waveguide is considered part of a layered three-dimensional periodic system that is constructed by alternating the waveguide and vacuum layers along the thickness direction. The vacuum layers have negligible mass density and modulus, so as to allow for the traction-free boundary conditions on the surface of the waveguide to be automatically satisfied. The resulting three-dimensional periodic medium can then be modeled as a layered bulk material according to the Navier's equations. The dispersions for the medium are provided by the solution of the homogeneous problem.

[0058] The general form of the Navier's equations for an inhomogeneous bulk medium can then be expressed according to the following equation (2):

-ρω² ΰ = (λ + μ)ν(ν^■ ϋ) + μν² ϋ + VAV^■ U + ν X V X U + 2(ν ^■ V)U (2)

[0059] Above, U(r) is the particle displacement vector, p(r) is the local density, and A(r) and/(r) are the local Lame constants, all of which are functions of the spatial variables. These values are material dependent. For example, in aluminum, p = 2700 kg/m³, λ = 5.1 GPa, and μ = 2.6 GPa. To apply the k p method to equation (2), the Bloch functions in the vicinity of the Γ point are written as linear combinations of the three degenerate states. Substituting these functions into equation (2), applying periodic boundary conditions, and collecting only the linear order terms in k results in the following equation (3): det| ½ - 6>/₀)/ + P(¾ | = 0 (3)

[0060] Above, n denotes the band index at the Bloch wave vector k, and P is the reduced Hamiltonian matrix with elements Ρ_β = /^'k p,_/, which represent the coupling strength between the degenerate states j and / at the Γ point. The matrix element p₇ can be calculated according to the following equation (4):

(2π ³ f ( → T → → → _{→ →} (4)

Pji W = ^γ- {( + )(νί/_;0) ^■ ί/fo + (λ + ) [ν^■ U_jQ{r)]U_l ^* _Q + 2//Vt/,-₀ ^■ t/f₀

Junitcell

+ [ I^■ ¾] ¾ + 2(¾o^■ ¾)V// + [ ^■ ¾J ¾ - [¾^■ ΰ^μ] [0061] Evaluating equation (4) numerically provides the reduced Hamiltonian matrix. Substituting this matrix into equation (3) provides the dispersion relations of the modes contributing to the Dirac-like cone, as shown in the following equations (5a) and (5b): f ₌ (5a) Ak

[0062] Equation (5a) corresponds to a flat band while each individual solution of equation (5b) (+ and -) corresponds to a linear dispersion associated with cones. These results are independent of the wave vector k, and thus the linear dispersion is isotropic.

[0063] [Effective Medium]

[0064] One property of the Dirac-like cone is that, under certain conditions, it can be mapped to an effective medium. By reducing the dynamic properties to effective medium properties, it is possible to clearly characterize the double zero properties. Boundary effective medium theory may be used to obtain the effective material parameters. For periodic media, the effective medium theory is valid at k = 0 around the standing wave frequency, even if the frequency belongs to the short wavelength regime. Empirically, it is also possible to show that the use of the effective medium description matches the finite element model predictions described above.

[0065] As applied to a phononic waveguide, the effective medium method considers the unit cell as a black box that responds to an incoming wave. The method calculates the eigenstates and then evaluates the effective forces, displacements, strains, and stresses induced by the eigenstate itself along the boundaries of the unit cell. It is then possible to extract the effective density and modulus by using Newton's second law and the constitutive relations.

[0066] For example, for eigenstates along the ΓΧ direction in wave space (corresponding to +x in physical space), the effective mass density can be obtained from Newton's second law according to the following equation (6):

[0067] Above, p is the mass density, F_z is the net force exerted on the unit cell in the out-of- plane z direction, and u_z is the displacement of the unit cell in the z direction resulting from the three degenerate states contributing to the Dirac-like Cone. The superscript eff indicates that these quantities are effective quantities. The effective force and displacement are obtained by integrating and evaluating the stress tensor and displacement in various planes and the constitutive relations according to the following equations (7a) and (7b):

[0068] Above, G^eJJ is the effective shear modulus and S is the strain tensor. Using the above methods, it is possible to obtain the effective shear modulus and the effective mass density as a function of frequency in the range around the Dirac-like point. For the material evaluated using the finite element method above, FIGS. 6A-C illustrate wave transmission behavior thereof. Specifically, FIG. 6A illustrates the effective shear modulus, FIG. 6B illustrates the reciprocal effective shear modulus, and FIG. 6C illustrates the effective mass density near the Dirac-like point. As seen in FIGS. 6A-C, the material exhibits double-zero properties at the Dirac-like point; here, = 27.04 kHz. Therefore, acoustic waves traveling through the medium at the Dirac- like point frequency experience no spatial phase change.

[0069] [Elastic Waveguide]

[0070] A waveguide including a particular array of geometrically-tapered metamaterials can provide a high level of control of the propagation parameters for acoustic waves in the waveguide. FIG. 7 illustrates a basic example of one such waveguide as seen in a plan view.

[0071] Specifically, FIG. 7 illustrates a waveguide 700 formed of a flat plate 701 with a metamaterial array 702 of geometric tapers 703 formed therein. Incident waves are represented by an arrow 704, whereas outgoing waves are represented by an arrow 705. The flat plate 701 includes an incident portion 711 on which acoustic waves are incident and a transmission portion 712 on which outgoing waves emerge. The metamaterial array 702 is disposed between the incident portion 711 and the transmission portion 712.

[0072] The flat plate 701 may be formed of any material that transmits acoustic waves, although the particular geometric parameters may be dictated by the desired material as described above. In one embodiment, the flat plate 701 is formed of aluminum. The metamaterial array 702 is an mxn lattice of unit cells, such as the unit cell 100 illustrated in FIG. 1 A or the unit cell 200 illustrated in FIG. 2 A. While FIG. 7 illustrates the metamaterial array 702 as a 16x8 lattice for ease of explanation, the particular values of m and n are not so limited. The metamaterial array 702 illustrated in FIG. 7 is a rectangular lattice; however, a waveguide according to the present disclosure is not so limited, as will be discussed in more detail below. In order to exhibit sufficient wave-shaping properties, the wave must penetrate a sufficient distance into the metamaterial array 702. Specifically, the wave should penetrate several wavelengths into the metamaterial array 702. Because the geometric parameters of the individual geometric tapers 703 are sub- , features, the metamaterial array 702 is preferably more than 5λ in length along the direction in which the waves travel.

[0073] The particular type of unit cell may be application-specific. For example, one-sided unit cell 100 may allow for a plate that has a substantially flat and uniform opposing surface which exhibits superior aerodynamic properties. Furthermore, the two-sided unit cell 200 may allow for easier modeling, testing, and/or manufacture in some circumstances.

[0074] The waveguide 700 may be formed by any suitable process. The particular process may depend on the material, the scale of the geometric parameters, and/or the intended application, and may be either an additive or subtractive process. An example of additive processes is 3D printing. An example of subtractive processes is (Computer Numerical Control) machining. At small scales, it is also possible to produce waveguide 700 via etching and similar subtractive processes.

[0075] The waveguide 700 may also be formed by a combination of additive and subtractive processes. In one example, the torus-like portions of the geometric tapers 703 may be CNC machined from an initially-flat aluminum plate, while the center-mass portions of the geometric tapers 703 may be successively welded onto the center of the torus-like portions.

[0076] A wave in the A₀ mode incident from the left is illustrated by the arrow 704 in FIG. 7. If the wave has a frequency matching the Dirac-like point of the geometric tapers 703 and impinges with normal incidence on the metamaterial array 702, the resulting wave field (that is, the wave emerging on the opposite side of the metamaterial array 702) exhibits substantially no phase change (for example, less than 10%) as a result of the metamaterial array 702. In this example, the resulting wave field also exhibits substantially no amplitude change (for example, less than 10%) as a result of the metamaterial array 702, such that transmitted amplitude is nearly equal to the incident amplitude due to the double-zero-index nature of the waveguide 700. However, depending on the shape of the metamaterial array 702, the transmitted amplitude may be larger than the incident amplitude. These transmission properties may be utilized to achieve structural waveguides exhibiting behavior such as cloaking, supercoupling, and energy squeezing for acoustic waves.

[0077] FIG. 8 illustrates a waveguide 800 that exhibits cloaking behavior. The waveguide 800 is formed of a flat plate 801 with a metamaterial array 802 of geometric tapers 803 formed therein. Incident waves are represented by an arrow 804, whereas outgoing waves are represented by an arrow 805. The flat plate 801 includes an incident portion 81 1 on which acoustic waves are incident and a transmission portion 812 on which outgoing waves emerge. The metamaterial array 802 is disposed between the incident portion 81 1 and the transmission portion 812. To illustrate the cloaking capability, FIG. 8 shows an embedded object represented by a through- hole 806 within the metamaterial array 802.

[0078] The flat plate 801 may be formed of any material that transmits acoustic waves. In one embodiment, the flat plate 801 is formed of aluminum. The metamaterial array 802 is an mxn lattice of unit cells, such as the unit cell 100 illustrated in FIG. 1 A or the unit cell 200 illustrated in FIG. 2A. As noted above the wave should penetrate several wavelengths into the metamaterial array 802. Therefore, while FIG. 8 illustrates the metamaterial array 802 as a 16x8 lattice and the through-hole 806 having an area of 10x2 unit cells for ease of explanation, the particular values of m and n are not so limited.

[0079] Numerical tests in which the through-hole 806 has clamped boundary conditions show that the incident wave emerges on the opposite side of the metamaterial array 802 being unaffected by the through-hole 802. In other words, the far-field effect of the waveguide 800 including the metamaterial array 802 is identical to the far-field effect of the waveguide 700 including the metamaterial array 701. Therefore, the acoustic field downstream of the through- hole 805 does not carry any information about the embedded object itself, confirming the cloaking capability of the waveguide 800.

[0080] The numerical tests utilizing the through-hole 806 correspond to the physical situation in which an embedded object experiences substantially no vibration (for example, less than 1%) as a result of the incident wave. This capability may thus be harnessed to, for example, vibrationally isolate sensitive components.

[0081] FIG. 9 illustrates a waveguide 900 that exhibits supercoupling behavior. The waveguide 900 is formed of a flat plate 901 with a metamaterial array 902 of geometric tapers 903 formed therein. Incident waves are represented by an arrow 904, whereas outgoing waves are represented by an arrow 905. The flat plate 901 includes an incident portion 91 1 on which acoustic waves are incident and a transmission portion 912 on which outgoing waves emerge. The metamaterial array 902 is disposed between the incident portion 91 1 and the transmission portion 912. To illustrate the supercoupling capability, FIG. 9 shows a discontinuity represented by a notch 906 within the metamaterial array 902.

[0082] The flat plate 901 may be formed of any material that transmits acoustic waves. In one embodiment, the flat plate 901 is formed of aluminum. The metamaterial array 902 is an mxn lattice of unit cells, such as the unit cell 100 illustrated in FIG. 1 A or the unit cell 200 illustrated in FIG. 2A. As noted above the wave should penetrate several wavelengths into the metamaterial array 902. Therefore, while FIG. 9 illustrates the metamaterial array 902 as a 16x8 lattice and the notch 906 having an area of 8x4 unit cells for ease of explanation, the particular values of m and n are not so limited.

[0083] Numerical tests in which the notch 906 has clamped boundary conditions along all walls thereof show that the incident wave propagates through the U-shaped channel of the

metamaterial array 902 unaffected except for a minor phase distortion (for example, less than 10%). In other words, the far-field effect of the waveguide 900 including the metamaterial array 902 is nearly identical to the far-field effect of the waveguide 700 including the metamaterial array 701.

[0084] The numerical tests utilizing the notch 906 correspond to the physical situation in which a wave propagates efficiently across discontinuities. This capability may be harnessed to, for example, efficiently transmit energy across joints and consequently reduce localization and dynamic amplification. [0085] FIG. 10 illustrates a waveguide 1000 that exhibits energy squeezing behavior. The waveguide 1000 is formed of a flat plate 1001 with a metamaterial array 1002 of geometric tapers 1003 formed therein. Incident waves are represented by an arrow 1004, whereas outgoing waves are represented by an arrow 1005. The flat plate 1001 includes an incident portion 1011 on which acoustic waves are incident and a transmission portion 1012 on which outgoing waves emerge. The metamaterial array 1002 is disposed between the incident portion 1011 and the transmission portion 1012. To illustrate the energy squeezing capability, FIG. 10 shows that the incident portion 1011 is wider than the transmission portion 1012, represented by a notch 1006 within the metamaterial array 1002.

[0086] The flat plate 1001 may be formed of any material that transmits acoustic waves. In a preferred embodiment, the flat plate 1001 is formed of aluminum. The metamaterial array 1002 is an mxn lattice of unit cells, such as the unit cell 100 illustrated in FIG. 1 A or the unit cell 200 illustrated in FIG. 2A. As noted above the wave should penetrate several wavelengths into the metamaterial array 1002. Therefore, while FIG. 10 illustrates the metamaterial array 1002 as a 16x8 lattice and the notch 1006 having an area of 8x4 unit cells for ease of explanation, the particular values of m and n are not so limited.

[0087] Numerical tests in which the notch 1006 has clamped boundary conditions along all walls thereof show that the incident wave propagates the metamaterial array 1002 with minimal distortion but increased amplitude. This capability may be harnessed to, for example, efficiently extract energy for dissipation and/or harvesting regardless of the location of the source.

[0088] FIG. 11 illustrates a waveguide 1100 that exhibits wave tunneling behavior. The waveguide 1100 is formed of a flat plate 1101 with a metamaterial array 1102 of geometric tapers 1103 formed therein. Incident waves are represented by an arrow 1104, while outgoing waves are represented by an arrow 1105. The flat plate 1101 includes an incident portion 1111 on which acoustic waves are incident and a transmission portion 1112. The incident waves travel from the incident portion 1111 into the metamaterial array 1102 and emerge from the

metamaterial array 1102 into the transmission portion 1112.

[0089] The flat plate 1101 may be formed of any material that transmits acoustic waves. In one embodiment, the flat plate 1 101 is formed of aluminum. The metamaterial array 1102 is an mxn lattice of unit cells, such as the unit cell 100 illustrated in FIG. 1 A or the unit cell 200 illustrated in FIG. 2A. As noted above the wave should penetrate several wavelengths into the metamaterial array 1102. Therefore, while FIG. 11 illustrates the metamaterial array 1102 as a 16x8 lattice for ease of explanation, the particular values of m and n are not so limited.

[0090] Moreover, while FIG. 11 illustrates the incident waves incident from the incident portion 1111 traveling in the -x direction and outgoing waves in the transmission portion traveling in the +x direction, the wave tunneling behavior is not so limited. For example, the shape of the metamaterial array 1102 may be selected such that the outgoing waves emerge in an arbitrary outgoing direction, depending on the particular application. Moreover, it is possible to pre- process incident waves in the incident portion 1111 or prior to their entry in the incident portion 1111 such that the angle of incidence on the metamaterial 1102 is within a particular acceptance range.

[0091] Thus, the waveguides illustrated in FIGS. 7-11 are double-zero index waveguides. That is, they exhibit zero effective mass density and zero reciprocal effective shear modulus at a wavelength corresponding to the Dirac-like point.

[0092] The above waveguides allow for the control of the dynamics of a host structure while simultaneously yielding a lighter and load-bearing system. In this manner, the above waveguides may be utilized as structural panels or as acoustic insulating panels attached or mounted thereto.

[0093] [Applications]

[0094] The double-zero-index material represented by the waveguides described above may be used in any application where lightweight thin-walled structures are used. For example, the disclosed double-zero-index material has applications in aerospace, commercial and business aircraft, automotive, and locomotive fields, among others.

[0095] FIG. 12 illustrates a section of an exemplary fuselage 1200 implementing acoustic waveguides according to the present disclosure. The fuselage 1200 includes a generally cylindrical main body 1201 (also known as a "skin") and at least one interior baffle 1202 (also known as a "bulkhead"). As illustrated in FIG. 12, the interior baffle 1202 is an acoustic waveguide including a plurality of metamaterial arrays 1203a-c, wherein each of the

metamaterial arrays 1203a-c includes a respective plurality of unit cells 1204a-c. The unit cells 1204a-c within a particular metamaterial array 1203a-c all include the same geometric parameters, but the geometric parameters of unit cells 1204a are different from the geometric parameters of unit cells 1204b which are in turn different from the geometric parameters 1204c. Additionally, each metamaterial array 1203a-c is illustrated with a different shape; however, the present disclosure is not so limited.

[0096] The exemplary fuselage 1200 may be used in a commercial aircraft, and thus may be formed of aluminum. Because the interior baffle 1202 is not exposed to the outside of the aircraft and is thus not required to be an aerodynamic surface, the unit cells 1204a-c may be the unit cell 100 illustrated in FIG. 1 A or the unit cell 200 illustrated in FIG. 2 A.

[0097] While FIG. 12 illustrates the metamaterial arrays 1203a-c in the interior baffle 1202 but not in the main body 1201, the present disclosure is not so limited. Metamaterial arrays may be present in the main body 1201 instead of or in addition to the metamaterial arrays 1203a-c. In a case where metamaterial arrays are present in the main body 1201, it is preferable that the unit cells therein are the unit cell 100 illustrated in FIG. 1 A to preserve the aerodynamic properties of the exterior of the main body 1201.

[0098] FIG. 13 illustrates a section of an exemplary airfoil-shaped body 1300 implementing acoustic waveguides according to the present disclosure. The airfoil-shaped body 1300 includes a main body 1301, a plurality of interior reinforcing beams 1302 (also known as "ribs") which extend from a front of the airfoil-shaped body 1300 to a rear of the airfoil-shaped body 1300, and a plurality of crosswise reinforcing beams 1303 (also known as "spars") which extend from a proximate end of the airfoil-shaped body 1300 to a distal end of the airfoil-shaped body 1300. As illustrated in FIG. 13, the main body 1301 includes a metamaterial array 1304a including a plurality of unit cells 1305a (only one of which is illustrated) and the interior reinforcing beams 1302 includes a metamaterial array 1304b including a plurality of unit cells 1305b. While not shown in FIG. 13, metamaterial arrays may be present in crosswise reinforcing beams 1303 instead of or in additional to one or both of the metamaterial arrays 1304a-b.

[0099] FIGS. 12-13 illustrate commercial-aircraft applications for double-zero-index materials according to various aspects of the present disclosure; however, it will be appreciated that these applications are illustrative and non-limiting. The double-zero-index materials disclosed herein may also be implemented in other aerospace applications and, more generally, any application where thin-walled structures are utilized and/or passive adaptive vibration and noise control capabilities are desired.

[0100] For example, the above acoustic waveguides may be implemented internally or externally in vehicles and other transportation systems, such as aerospace systems (e.g., hypersonic flight and reusable launch vehicles, satellites, helicopters, drones, personal aircraft, research aircraft, and the like), personal transportation vehicles (e.g., automobiles, autonomous vehicles, personal rapid transit, motorcycles, scooters, jet-skis, motorboats, and the like), mass transportation vehicles (e.g., freight trains, passenger trains, bullet trains, trams, light rail stock, streetcars, subway cars, buses, ferries, water taxis, ships, and the like), and military vehicles (e.g., submarines, ships, jets, personnel carriers, and the like).

[0101] The above acoustic waveguides may also be implemented in stationary systems which are negatively affected by noise and/or vibration. For example, the above acoustic waveguides may be implemented in scientific research equipment (e.g., optical telescopes, radio telescopes, microscopes, antennas, particle detectors, electromagnets, photomultipliers, and the like), medical equipment (e.g., MRI machines, MR spectroscopy devices, PCR devices, X-ray machines, nucleic acid hybridization assays, nucleic acid chips, and the like), household goods (e.g., dishwashers, washing machines, clothes dryers, and the like), and structural materials (e.g., seismic damping structures, soundproofing, bridge and roadway materials, and the like).

[0102] It is to be understood that the above description is intended to be illustrative and not restrictive or exhaustive. Although various structures have been described in detail with reference to certain aspects thereof, the disclosure may be variously embodied without departing from the spirit or the scope of the disclosure. Therefore, many aspects and applications other than the specific examples provided herein would be apparent upon a reading of the above description. It is anticipated and intended that future developments will occur in the technologies discussed herein and manufacturing methods thereof, and that the disclosed systems and methods will be incorporated into such future technologies. In other words, it should be understood that the application is capable of modification and variation.

[0103] A double-zero-index material according to the present disclosure may take one or more of the following illustrative configurations:

[0104] (1) An acoustic waveguide comprising: a plate having a first surface and an opposing second surface, the plate including an incident portion and a transmission portion; and a plurality of unit cells arranged in an array having a predetermined shape, each cell of the plurality of unit cells including a mass located at a center of the unit cell and having a central axis perpendicular to the first surface, and a channel disposed in the first surface and surrounding the mass, a radius of revolution of the channel being a predetermined distance from the central axis, wherein the mass is integrally formed as one piece with the plate, and wherein the array is configured such that an acoustic wave incident on the array from the incident portion emerges in the transmission portion.

[0105] (2) The acoustic waveguide according to (1), wherein the channel is a groove defined by an elliptical surface of revolution.

[0106] (3) The acoustic waveguide according to (1) or (2), wherein the second surface defines a plane.

[0107] (4) The acoustic waveguide according to any one of (1) to (3), wherein the mass extends in a direction of the central axis to a predetermined height relative to a virtual plane coincident with the first surface.

[0108] (5) The acoustic waveguide according to any one of (1) to (4), wherein the channel is a first channel, and each unit cell respectively further includes a second channel disposed in the second surface, a radius of revolution of the second channel being equal to the radius of revolution of the first channel.

[0109] (6) The acoustic waveguide according to any one of (1) to (5), wherein the mass is a first mass, each unit cell respectively further includes a second mass located at the center of the unit cell and aligned with the first mass along the central axis, the first mass extends in a first direction of the central axis a predetermined first height relative to a first virtual plane coincident with the first surface, and the second mass extends in a second direction antiparallel to the first direction by a predetermined second height relative to a second virtual plane coincident with the second surface.

[0110] (7) The acoustic waveguide according to any one of (1) to (6), wherein the predetermined shape is configured such that an incident acoustic wave having a predetermined wavelength, incident on the array from the incident portion, emerges in the transmission portion with a substantially unchanged phase from that within the incident portion.

[0111] (8) The acoustic waveguide according to any one of (1) to (7), wherein the predetermined shape is configured such that the incident acoustic wave emerges in the transmission portion with a substantially unchanged amplitude from that within the incident portion.

[0112] (9) The acoustic waveguide according to any one of (1) to (7), wherein the predetermined shape is configured such that the incident acoustic wave emerges in the transmission portion with an increased amplitude from that within the incident portion. [0113] (10) The acoustic waveguide according to any one of (1) to (8), wherein the predetermined shape is configured such that an object embedded in a center of the array experiences substantially no vibration at the predetermined wavelength as a result of the incident acoustic wave.

[0114] (11) The acoustic waveguide according to any one of (1) to (10), wherein a length of the array in a traveling direction of the incident acoustic wave is greater than or equal to five times the predetermined wavelength.

[0115] (12) The acoustic waveguide according to any one of (1) to (11), wherein the waveguide plate is constructed of aluminum or an aluminum alloy.

[0116] (13) A waveguide unit cell comprising: a plate having a first surface and an opposing second surface, the plate including a mass located at a center of the plate and having a central axis perpendicular to the first surface, and a channel disposed in the first surface and surrounding the mass, a radius of revolution of the channel being a predetermined distance from the central axis, wherein the mass is integrally formed as one piece with the plate.

[0117] (14) The waveguide unit cell according to (13), wherein the channel is a groove defined by an elliptical surface of revolution.

[0118] (15) The waveguide unit cell according to (13) or (14), wherein the second surface defines a plane.

[0119] (16) The waveguide unit cell according to any one of (13) to (15), wherein the mass extends in a direction of the central axis to a predetermined height relative to a virtual plane coincident with the first surface.

[0120] (17) The waveguide unit cell according to claim any one of (13) to (16), wherein the channel is a first channel, and the plate further includes a second channel disposed in the second surface, a radius of revolution of the second channel being equal to the radius of revolution of the first channel.

[0121] (18) The waveguide unit cell according to any one of (13) to (17), wherein the mass is a first mass, the plate further includes a second mass located at the center of the plate and aligned with the first mass along the central axis, the first mass extends in a first direction of the central axis by a predetermined first height relative to a first virtual plane coincident with the first surface, and the second mass extends in a second direction antiparallel to the first direction by a predetermined second height relative to a second virtual plane coincident with the second surface.

[0122] (19) The waveguide unit cell according to any one of (13) to (18), wherein the waveguide plate is constructed of aluminum or an aluminum alloy.

[0123] (20) An acoustic waveguide having zero effective mass density and zero reciprocal effective shear modulus at a predetermined wavelength.

Claims

CLAIMS What is claimed is:

1. An acoustic waveguide comprising:

a plate having a first surface and an opposing second surface, the plate including an incident portion and a transmission portion; and

a plurality of unit cells arranged in an array having a predetermined shape, each cell of the plurality of unit cells including

a mass located at a center of the unit cell and having a central axis perpendicular to the first surface, and

a channel disposed in the first surface and surrounding the mass, a radius of revolution of the channel being a predetermined distance from the central axis,

wherein the mass is integrally formed as one piece with the plate, and

wherein the array is configured such that an acoustic wave incident on the array from the incident portion emerges in the transmission portion.

2. The acoustic waveguide according to claim 1, wherein the channel is a groove defined by an elliptical surface of revolution.

3. The acoustic waveguide according to claim 1, wherein the second surface defines a plane.

4. The acoustic waveguide according to claim 3, wherein

the mass extends in a direction of the central axis to a predetermined height relative to a virtual plane coincident with the first surface.

5. The acoustic waveguide according to claim 3 wherein

the channel is a first channel, and

each unit cell further includes a second channel disposed in the second surface, a radius of revolution of the second channel being equal to the radius of revolution of the first channel.

6. The acoustic waveguide according to claim 5, wherein

the mass is a first mass,

each unit cell further includes a second mass located at the center of the unit cell and aligned with the first mass along the central axis,

the first mass extends in a first direction of the central axis a predetermined first height relative to a first virtual plane coincident with the first surface, and

the second mass extends in a second direction antiparallel to the first direction by a predetermined second height relative to a second virtual plane coincident with the second surface.

7. The acoustic waveguide according to claim 1, wherein the predetermined shape is configured such that an incident acoustic wave having a predetermined wavelength, incident on the array from the incident portion, emerges in the transmission portion with a substantially unchanged phase from that within the incident portion.

8. The acoustic waveguide according to claim 7, wherein the predetermined shape is configured such that the incident acoustic wave emerges in the transmission portion with a substantially unchanged amplitude from that within the incident portion.

9. The acoustic waveguide according to claim 7, wherein the predetermined shape is configured such that the incident acoustic wave emerges in the transmission portion with an increased amplitude from that within the incident portion.

10. The acoustic waveguide according to claim 7, wherein the predetermined shape is configured such that an object embedded in a center of the array experiences substantially no vibration at the predetermined wavelength as a result of the incident acoustic wave.

11. The acoustic waveguide according to claim 7, wherein a length of the array in a traveling direction of the incident acoustic wave is greater than or equal to five times the predetermined wavelength.

12. The acoustic waveguide according to claim 1, wherein the waveguide plate is constructed of aluminum or an aluminum alloy.

13. A waveguide unit cell comprising:

a plate having a first surface and an opposing second surface, the plate including:

a mass located at a center of the plate and having a central axis perpendicular to the first surface, and

wherein the mass is integrally formed as one piece with the plate.

14. The waveguide unit cell according to claim 13, wherein the channel is a groove defined by an elliptical surface of revolution.

15. The waveguide unit cell according to claim 13, wherein the second surface defines a plane.

16. The waveguide unit cell according to claim 15, wherein

17. The waveguide unit cell according to claim 13, wherein

the channel is a first channel, and

the plate further includes a second channel disposed in the second surface, a radius of revolution of the second channel being equal to the radius of revolution of the first channel.

18. The waveguide unit cell according to claim 17, wherein

the mass is a first mass,

the plate further includes a second mass located at the center of the plate and aligned with the first mass along the central axis,

the first mass extends in a first direction of the central axis by a predetermined first height relative to a first virtual plane coincident with the first surface, and

19. The waveguide unit cell according to claim 13, wherein the waveguide plate is constructed of aluminum or an aluminum alloy.

20. An acoustic waveguide having zero effective mass density and zero reciprocal effective shear modulus at a predetermined wavelength.