We present a design for a two-dimensional omnidirectional acoustic absorber that can achieve *98.6%* absorption of acoustic waves in water, forming an effective acoustic black hole. This artificial black hole consists of an absorptive core coated with layers of periodically distributed polymer cylinders embedded in water. Effective medium theory describes the response of the coating layers to the acoustic waves. The polymer parameters can be adjusted, allowing practical fabrication of the absorber. Since the proposed structure does not rely on resonances, it is applicable to broad bandwidths. The design might be extended to a variety of applications.

## I. INTRODUCTION

A perfect acoustic absorber can achieve almost unity absorption for broadband incident acoustic waves from all angles, which makes it useful in wide practical applications in sonar, vibration and noise reduction, among others. Although there are many conventional acoustic absorption materials available, few can reach nearly perfect absorption because of the limitations in their primary absorption mechanisms, including the sound attenuation attributed to the material's viscosity, such as soft tissues and polymers, and the structure of the material, such as porous media, panel resonators and Helmholtz resonators.^{1,2} The absorption of acoustic attenuators depends on the intrinsic dissipation property of the material and the inevitable reflection on the boundary of the device, preventing *100%* absorption.^{3} For porous absorbers, such as a tapered foam sound isolated room, porous materials lower sound waves by disseminating energy and turning it into heat, and the tapered structure is used to reduce reflection because the fraction of echoes is directed toward another tapered foam instead of back into the room. But the acoustic absorption of the tapered foam depends on the frequency of the sound. Meanwhile, resonator absorbers reduce waves by transforming energy into the vibration of the resonators themselves. But the frequency band of predominant sound absorption depended on the resonators diameter, thus the most efficient absorption occurs only in a specific range of incidence angles or in a narrow bandwidth.^{3–5}

Recently, with advances in the development of acoustic metamaterials,^{6–11} artificial structures with tailored acoustic responses determined by the building blocks have markedly broadened the realm of possible designs of acoustic absorbers.^{7–9} A dark acoustic metamaterial containing an elastic membrane decorated with asymmetric rigid steel plates was reported to absorb low-frequency airborne sound completely at selective resonance frequencies. It utilized the dissipative property of the membrane to absorb highly concentrated elastic energy when the resonance occurred.^{11} Inspired by achievements in transformation optics,^{12,13} researchers have used coordinate transformation in the study of acoustic waves^{14–16} and proposed an acoustic absorber with omnidirectional high absorption in a broad bandwidth. Although an experimental demonstration of acoustic absorber has been implemented,^{9} the design of a perfect omnidirectional absorber is rarely discussed. Thus, the designing proposal of an acoustic absorber based on transmission acoustics is novel and is highly desirable to foster practical applications.

In this work, we propose a new type of two-dimensional (2D) acoustic metamaterial that can absorb waterborne sound nearly perfectly. The mechanism of the absorber is based on a combination of structure and energy dissipation. The artificial structure comprises circularly distributed polymer cylinders whose radii are designed using effective medium theory^{17–19} to achieve the position-dependent material parameters specified by transformation acoustics. Here, the mass density is close to that of water and the bulk modulus is azimuthally symmetric and radially gradient. Compared with the traditional absorbing materials, such a structure is able to capture *98.6%* incident acoustic waves from all angles and guide them into a lossy core that absorbs the energy without much reflection, leading to an effective, albeit artificial, “acoustic black hole”. Since the composite does not rely on the resonances, it is applicable to a broad bandwidth. The performance of the acoustic absorber is demonstrated by finite-element simulations and compared with the theoretical design.

## II. AN ACOUSTIC BLACK HOLE

For an ideal fluid without viscosity, we can derive, from Newton's second law and continuity, the following linear acoustic wave equation by neglecting the heat transfer between the adjacent media:

where *p* and |$\vec v$|$v\u20d7$ are the pressure and velocity fields, respectively. ρ_{0} denotes the unperturbed mass density and *B* corresponds to the bulk modulus, both of which are position dependent in general. For time harmonic vibrations, Eqs. (1) and (2) can be changed into the acoustic wave equation as follows:

For a 2D problem with *z*-invariance, it is well known that the governing wave equation for acoustic waves is equivalent to that for electromagnetic (EM) waves. If the polarization of the EM wave is transverse magnetic, i.e., the electric field is parallel to the z-axis, the mapping of the variables is *E*_{z}↔*p*, μ↔ρ_{0} and |$\varepsilon \leftrightarrow {1 / B}$|$\u025b\u21941/B$. Here, μ and ε are permeability and permittivity, respectively. Thus, techniques developed in transformation optics^{12,13} can be carried over to transformation acoustics,^{14–16} and the previously proposed optical black hole^{20,21} may be generalized to its acoustic counterpart by introducing a concentric cylinder of radius *R* with an internal core immersed in a fluid host. The bulk moduli of the inner core and the coating layer satisfy the following equation:

where *B*_{0} is the bulk modulus of the host and *R*_{c} is the radius of the core, which is determined by the following equation,

The imaginary part, γ, appearing in the inverse of the bulk modulus of the core, enables its function of absorption, which can be achieved by a vaiety of commonly available materials, such as a sponge or other porous media. We note that the bulk moduli of the coating layers at the outer and inner surfaces match those of the host and the inner core, and the mass densities of both the core and the coating layer are set to be equal to the mass density of the host. Thus, the impedence is matched on the two boundaries of the coating layer, and the buoyancy effect is small.

To demonstrate the absorption property of the acoustic black hole described by Eqs. (4) and (5), we use the acoustic module integrated in COMSOL Multiphysics, a finite-element based solver, to compute the pressure field of a Gaussian beam with frequency of *15kHz* impinging on an acoustic black hole whose inner and outer radii are *0.5m* and *1.0m*, respectively. The results are plotted in Fig. 1 for both on- and off-center cases. It is clear in the figure that it is very difficult for the acoustic fields to escape from the structure and they are almost completely absorbed regardless of the incidence angle.

## III. DESIGNED STRUCTURE AND ARTIFICIAL MATERIALS

It is a bit challenging to realize the proposed black hole by a bulk medium. However, the task may be accomplished by a 2D structure composed of a core cylinder coated with multiple layers of concentric annuli embedded in water. A schematic of the structure is shown in Fig. 2, where *20* annuli of the same thickness are used and each one consists of circularly periodically distributed identical polymer cylinders. These polymer cylinders can be fabricated from organic synthesis, with their bulk moduli smaller than that of water and their mass density comparable to that of water such that any extra mechanical interference from buoyancy can be avoided. Since the velocity contrast between the polymer and the water is high, we can ignore the shear wave inside the polymer cylinder and the use of acoustic wave equations for this system is thereby justified.^{22,23} When the wavelength is much greater than the thickness of each annulus, the properties of such a multilayered structure can be delineated by effective medium theory. To acheive the desired effective bulk modulus, which is azimuthally symmetric but radially varied, we may adjust the radii of the polymer cylinders in each layer. In this case, as will be shown later, decreasing the radii of the polymer cylinders from the inner rings to the outer ones is plausible in realizing the effective bulk modulus defined in Eq. (4).^{19,23}

Let us take the design of the previously mentioned acoustic black hole, i.e., *R*_{c} = *0.5m* and *R* = *1m* as an example. Given the material parameters of the water host (*B*_{0} = 2.19 × 10^{9}*Pa*, ρ_{0} = 998*kg*/*m*^{3}), we can easily obtain from Eq. (5) that the real part of the bulk modulus of the core should be *B*_{c} = 5.475 × 10^{8}*Pa*. Thus, for convenience, the organic polymer can be synthesized to possess the following parameters: *B*′ = 5.475 × 10^{8}*Pa* and ρ′ = 998*kg*/*m*^{3}. The remaining task is to construct the multilayered structure by using periodically distributed polymer cylinders. As the thickness of each layer (*l* = *25mm*) is small compared to both the wavelength (*λ* = *98.7mm*) and the circumference of the smallest ring, we may approximate the unit cell with a polymer cylinder located in the center of a water square with length *l* as shown in Fig. 2. The number of polymer cylinders in each ring is propotional to the radius of the ring. The further out the ring, the greater the number of unit cells (polymer cylinders). The effective medium parameters of such a unit cell can be evaluated by COMSOL Multiphysics and a retrieving method.^{21} The results are plotted in Fig. 3(a), which displays the dependence of various effective medium parameters on the radius of the polymer cylinder, *a*, when the length, *l*, is fixed at *25mm*. To ensure that the description of the effective medium is highly accurate, the size of the unit cell should be much smaller than the wavelength. Another effective medium theory that requires the wave speed inside the inclusion much smaller than the host can be applied to high frequencies beyond the long-wavelength limit.^{24} Here, the acoustic wave frequency is set at *f* = *15kHz* such that the corresponding wavelengh is roughly four times *l.* Fig. 3(a) clearly shows that the relative effective mass density, bulk modulus and impedence vary as the size of the polymer cyliner increases, and the relative effective bulk modulus drops rapidly from *1.0* to *0.2*. In order to meet the requiements of the effective bulk modulus specified by Eq. (4), we carefully chose *20* different radii of the polymer cylinders for the *20* rings and plotted their effective bulk modulus as a function of the radii of the corresponding rings (layers) in Fig. 3(b) (red squares). Also plotted in Fig. 3(b) are the required values of the bulk modulus given by Eq. (4), which coincide with the red squares.

## IV. RESULTS AND DISCUSSION

The acoustic black hole effect of the designed structure can be numerically demonstrated by COMSOL. Fig. 4(a) and 4(b) exhibit the pressure field distributions with the same incident wave configurations as shown in Fig. 1(a) and 1(b), respectively, but the theoretical acoustic black hole is changed into the designed artificial structure with polymer cylinders. Similar results to Fig. 1 are found in Fig. 4, which again shows that almost *100%* of the incident waves are absorbed by the structure without reflection. To further assess the performance of the artificial black hole, the distribution of integrated scattering energy of all solid angles is plotted in Fig. 5(a), where only minor reflection can be detected at the incident angle. The agreements between the results of the artificial struture and its theoretical model provide clear evidence that the designed structure is indeed a good realization of an acoustic black hole. It is worth mentioning that to ensure the quality of the black hole, the layer should be thin enough that the effective medium parameters smoothly change in the neighbouring layers. The impedence is therefore matched between two adjacent layers, significantly reducing the reflection between the layers. Actually, if the shell layer is infinite, this artificial structure can manifest the perfect absorption with *100%* efficiency as the theoracal model does shown in the Fig. 1. However, it is difficult to divide the shell into infinite small layers. Since the performance of *20* layers shell is significant, the fabrications of finite layers with nearly *100%* absorbing efficiency are excellent enough and more approachable in applications.

Since the design does not rely on resonant structures and the effective medium properties do not require resonances, this artificial acoustic black hole is expected to work in a broad bandwidth. Fig. 5(b) shows the performance of the structure under the illumination of acoustic waves at various frequencies from *5 kHz* to *20 kHz*. It indicates that the black hole is highly efficient in absorbing the acoustic wave at frequencies of *6.1 kHz* and *17.7 kHz* as the scattering efficiency is lower than *0.1*. At lower frequencies, the wave is not completely absorbed by the core, which leads to a small leak. At higher frequencies, the performance of artificial black hole deviates from simulation after *16 kHz*. Though the absorption of the core becomes more efficient, the wavelength (*λ* = *62.4mm*) is only about *2.5* times the size of the unit cell, which greatly enhances the backscattering effect of the periodic structures. As frequency increases, the scattering efficiency rises rapidly and is higher than 0.1 beyond 17.7kHz, dooming the breakdown of the acoustic black hole. Clearly, to achieve the best performance of the black hole, the frequency of the incident wave cannot exceed the range where the effective medium holds. Within the frequency regime that the artificial black hole shows great absorption efficiency, the scattering efficiency can arrive at its minimum of *0.014*, indicating a peak absorptivity of *98.6%* at *15kHz*.

We have successfully desmonstrated a realization of an acoustic black hole with *R*_{c} = *0.5m* and *R* = *1m* by using a polymer whose bulk modulus, *B*′, is derived from Eq. (5). In practice, the design is not limited to this configuration. Knowing any two parameters of *R*_{c}, *R*, and *B*′, we can devise an acoustic black hole as long as the following requirements are satisified. First, the bulk *B’*modulus of the core should be smaller than that of the host, which is restricted by Eq. (5); second, the mass density of the core and the cylinders in the coating layers should be close to that of the host to minimize mechanical interferences from buoyancy. While many other materials may also be suitable for constructing an acoustic black hole, polymer is considered to be one of the best candidates because of the flexibility in the fabrication and the wide range of possible values of material parameters.

## V. CONCLUSION

In conclusion, we have designed a 2D artificial structure that can achieve broadband omnidirectional acoustic absorption with *98.6%* efficiency. This artificial structure is composed of *20* equally thick annuli, each of which contains circularly periodically distributed identical polymer cylinders. The sizes of the cylinders are carefully chosen based on effective medium theory. The material of the cylinders, i.e., polymer, can be easily fabricated by organic synthesis at the required bulk modulus and mass density. Since the design does not depend on resonance, it is suitable for a broad bandwidth. Finite-element simulations show excellent agreement between the theoretical acoustic black hole and the designed structure, which undoubtly suggests that the designed structure is indeed a good realization of an acoustic black hole. This work provides a simple but effective approach to sound absorption. The idea might be further extended to many other fields, including sonar, noise reduction and seismology.

## ACKNOWLEDGMENT

The work was jointly supported by the National Basic Research Program of China (Grant nos. 2012CB921503 and no. 2013CB632702) and the National Nature Science Foundation of China (Grant no. 1134006). We also acknowledge the Academic Development Program of Jiangsu Higher Education (PAPD), China Postdoctoral Science Foundation (Grant no. 2012M511249) and KAUST Baseline Research Funds.