Grating‐like anechoic layer for broadband underwater sound absorption

To address the challenging task of effective sound absorption in the low and broad frequency band for underwater structures, we propose a novel grating‐like anechoic layer by filling rubber blocks and an air backing layer into metallic grating. The metallic gratings are incorporated into the anechoic layer as a skeleton for enhanced viscoelastic dissipation by promoting shear deformation between rubber and metal plates. The introduction of an air backing layer releases the bottom constraint of the rubber, thus intensifying its deformation under acoustic excitation. Based on the homogenization method and the transfer matrix method, a theoretical model is developed to evaluate the sound absorption performance of the proposed anechoic layer, which is validated against finite element simulation results. It is demonstrated that a sound absorption coefficient of the grating‐like anechoic layer of 0.8 can be achieved in the frequency range of 1294–10 000 Hz. Given the importance of sound absorption at varying frequencies, the weighted average method is subsequently used to comprehensively evaluate the performance of the anechoic layer. Then, with structural density taken into consideration, an integrated index is proposed to further evaluate the acoustic properties of the proposed anechoic layer. Finally, the backing conditions and the boundary conditions of finite‐size structures are discussed. The results provide helpful theoretical guidance for designing novel acoustic metamaterials with broadband low‐frequency underwater sound absorption.


| INTRODUCTION
As an effective medium for long-distance transmission of underwater information, sound waves have been widely used in underwater reconnaissance. As a typical antireconnaissance method, a welldesigned anechoic layer works to minimize the intensity of reflected sound waves by absorbing a large portion of the incident sound energy. 1 Viscoelastic materials, such as rubber and polyurethane, are often used as the matrix of the anechoic layer, because their acoustic impedance matches that of water. [2][3][4][5][6] When excited by incident sound waves, the polymer chains inside a viscoelastic material vibrate, which enables intramolecular frictions to consume part of the sound energy. At present, sonar technology has the characteristics of a low detection frequency with a wide band, and it is constantly expanding to low frequency. However, due to the inherently weak dissipation of typical viscoelastic materials within the low-frequency range, 7,8 the absorption of low-frequency sound remains a great challenge. Coupled with the long wavelength of low-frequency sound waves in water, it appears that, at present, the only way to effectively attenuate the sound using a traditional uniform viscoelastic material is to increase its thickness.
To improve the viscoelastic dissipation performance of rubber at low frequencies, while a variety of sound-absorbing structures have been proposed, the more mature one is the cavity resonance structure 9-14 that shows cavity-related absorption peaks at low frequencies. The cavity weakens the stiffness of rubber and reduces the natural frequency of the structure. When the frequency of incident sound is close to this natural frequency, the vibration of polymer chains in rubber is intensified, resulting in significantly increased energy dissipation via intramolecular frictions. In recent years, by periodically arranging metallic spheres or cylinders coated with soft materials in a host polymer matrix, locally resonant anechoic layers have been widely investigated. [15][16][17][18][19][20][21] When the frequency of a sound wave approaches the natural frequency of local resonators, the latter force the polymer chains to vibrate violently, thus achieving strong sound absorption. 16 In recent years, the development of metamaterials/metasurfaces has led to new vitality to the design of underwater anechoic layers. Typically, by periodically embedding air cavities in soft materials, bubble metasurfaces were invented, and relevant acoustic properties including sound scattering, [22][23][24] transmission, [25][26][27][28][29] and absorption [30][31][32][33][34] were extensively studied. High sound absorption was achieved thanks to Fabry-Perot resonance created due to interference between scattered waves from the voids and reflected waves from the steel backing, as well as lumped spring-mass resonance formed with the soft material and steel backing. For enhanced low-frequency sound absorption, metallic structures with different topologies were introduced into rubber to form spring-mass resonant metamaterials. [35][36][37][38] In addition, quasi-Helmholtz resonance metamaterials constructed by incorporating rubber into the Helmholtz structure were exploited for lowfrequency underwater sound absorption. [39][40][41] Generally speaking, however, the resonant anechoic structures proposed hitherto by existing studies show a narrow frequency bandwidth of sound absorption, thus restricting their practical applications. 17,42,43 Therefore, broadening the effective absorption bandwidth of lowfrequency sound waves has become the focus of underwater acoustic research. 17,42 It should be mentioned that, focusing upon underwater low-frequency broadband sound absorption structure, Qu et al. 44 reduced the equivalent longitudinal sound speed by mixing tungsten into polyurethane and realized subwavelength as well as broadband sound absorption by designing the distribution of Fabry-Perot resonance, which has practical significance.
To meet the pressing practical demands, more research is needed in the design of low-frequency broadband absorption structures. In recent years, novel lightweight hybrid structures constructed by combining two or more materials (or structures) in such a way as to achieve attributes not offered by individual constituent have received increasing attention. [45][46][47] Cellular lattice trusses, because of the large number of cavities inside, have been selected for such hybrid design.
For instance, upon filling the interstices of aluminum corrugations with trapezoidal aluminum honeycomb blocks, the new hybrid structure showed considerable enhancement in specific strength and specific energy absorption. 48 In the field of sound absorption, Tang et al. [49][50][51] proposed a hybrid acoustic metamaterial by using a honeycomb-corrugation hybrid as a sandwich core and introducing perforations on both the top face sheet and corrugation. The hybrid idea confers the structure with superior broadband low-frequency sound absorption as well as excellent mechanical stiffness/strength, which confirms the potential of the hybrid idea in structural design.
Previously, inspired by the idea of hybrid design, we proposed an underwater anechoic layer filled with rubber between metal plates and demonstrated that the new structure improved the soundabsorbing performance of rubber by enlarging the shearing strain increment of rubber density as follows. 52 The underlying physical mechanisms are as follows: due to significant impedance mismatch between rubber and metal, vibration between the two is often uncoordinated under acoustic excitation; therefore, by assuming that the rubber block is well bonded to the metallic gratings, molecular chain frictions inside the viscoelastic rubber near each rubber-grating interface are considerably enhanced, thus enabling more acoustic energy to be dissipated. However, since the wavelength of a lowfrequency sound wave is much larger than the thickness of the structure, storage of the energy of sound in rubber is difficult, and only its upper part vibrates at low frequencies. In other words, based on the original design of the grating-like anechoic structure (i.e., without air backing), it is difficult to convert the energy of a lowfrequency sound wave into the kinetic energy of rubber. Thus, increasing the vibration of bottom rubber and converting more energy of low-frequency sound wave into the kinetic energy of rubber are the key for enhanced sound absorption performance.
Therefore, in the present study, we introduce an air layer at the bottom of rubber to address this issue, as shown schematically in

| THEORETICAL MODEL
As shown in Figure 1A, an incident plane sound wave normally impinges on the proposed anechoic layer immersed in water. The anechoic layer is periodic and composed of three main parts: rubber blocks, air layers, and the parallel steel plates connected with rigid steel backing. Each rubber block is closely bonded to adjacent steel plates, with no slip allowed at the interfaces between the steel plates and rubber. The air layer is located between the bottom of the rubber block and the steel backing. As the steel plates are periodically arranged in the x-direction and infinite in the y-direction, the grating-like anechoic layer can be evaluated via a two-dimensional (2D) unit cell; Figure 1B.
The thickness of the rubber block and that of the air layer are h 1 and h 2 , respectively. The thickness of a single steel plate is t and that of the plate spacing is l. Besides, the steel plates are assumed to be rigid because steel is hundreds of times stiffer than rubber.
Typically, viscoelastic material is a kind of material with both elasticity and viscosity. When viscosity is dominant, such as in petroleum, it is commonly characterized using the complex viscosity 53 while when elasticity is dominant, such as in rubber, it is usually characterized using the complex modulus model 54,55 where G c and η s are, respectively, the complex shear modulus and the loss factor, and μ c and γ s are the complex viscosity and the elastic phase, respectively. With small deformation assumed, when rubber is subjected to the excitation of a harmonic wave, the solid constitutive model using complex modulus is completely equivalent to the fluid constituting using complex viscosity and G c and μ c can be related by Equation (1)  homogenization method is usually adopted, that is, the influence of viscosity between the grating and rubber on the vibration of rubber is transformed into a complex increment of rubber density as follows 56 : Schematic of the grating-like anechoic layer and (B) two-dimensional unit cell.
where ρ eq 1 and K eq 1 are, respectively, the equivalent density and equivalent modulus of the first layer, ρ r is the density of rubber, ϕ l t l = /( + ) represents the proportion of rubber in the first layer, j = −1 is the imaginary unit, c l is the compressional wave speed, and η l is the loss factor of the compression modulus, Q a ωρ μ = / / 2 r r denotes the ratio of plate spacing a to the thickness of viscous boundary layer δ μ ω ρ = 2 /( · ) c r , ω πf = 2 is the angular frequency, and f is the frequency of sound. Detailed derivations of these equations can be found in our previous study. 52 According to , where c s is the shear wave speed and η s is the loss factor of the shear modulus. Then, the wavenumber and characteristic impedance of the rubber layer can be expressed, respectively, as k ω ρ K = / eq 1 eq 1 eq For the second layer of air with steel backing, the plate spacing is much larger than the viscous boundary layer of air, and hence viscous dissipation is negligible compared to that of the first layer.
Consequently, for theoretical calculations, the second layer can be regarded as a homogenized air layer, whose equivalent dynamical density and bulk modulus are the same as those of air: ρ ρ = eq 2 air and K ρ c = eq 2 air air 2 , where ρ air and c air are the density and sound speed of air, respectively. Similarly, the wavenumber and characteristic impedance of the air layer are determined by k ω ρ K = / eq 2 eq 2 eq 2 and Upon the above homogenization, the grating-like anechoic layer is simplified to a multilayer sound-absorbing structure, as shown in Figure 2B. When sound propagates in such an anechoic layer, the pressure and velocity of sound satisfy: where p A ( ) and v A ( ) represent, respectively, the sound pressure and the vibration velocity of the incident surface, while p B ( ) and v B ( ) represent the sound pressure and the vibration speed of the transmission surface, respectively. T is the transfer matrix of the structure, given by Equations (4) and (5) present the transfer matrices of the structure. With the back surface of the structure taken as fixed, namely, v B ( ) = 0, the surface acoustic impedance of the front surface can be obtained as To evaluate the sound absorption performance of the proposed anechoic layer, the absorption coefficient α is calculated by where z Z Z = / s s 0 is the relative acoustic surface impedance of the anechoic layer, Z ρ c = 0 0 0 is the characteristic impedance of water, and ρ 0 and c 0 are the density and sound speed of water, respectively.

| NUMERICAL MODEL
To validate the theoretical model, a 2D FE model considering acoustic-structure coupling is developed with COMSOL multiphysics, 55  The perfectly matching layer (PML) is used to simulate the anechoic end, and the incident plane sound wave is applied to the background pressure field. As to the boundary conditions, the lateral boundaries are set as periodic boundaries, while the bottom is fixed according to the assumption of rigid backing. With the FE model, the sound absorption coefficient can be calculated as where p in and p re are the incident and reflected sound pressure, respectively, and    represents the surface average over the inlet interface.

| Sound absorption performance
The relevant physical parameters of rubber are listed in Table 1  energy. This is mainly attributed to increased rubber deformation caused by the introduction of air backing layer. For a more quantitative evaluation, the distribution of energy dissipation density on the rubber-plate interface is presented in Figure 5C. Compared with GAL, although the amount of rubber in GALA is less, the viscous energy dissipation is more than twice that of GAL. In the intermediate-and highfrequency bands (i.e., 6000 and 10 000 Hz), the two structures show similar particle vibration velocities, and viscous energy dissipation remains concentrated near rubber-plate interfaces. The results presented in Figure 4 for GAL (i.e., without air backing) and GALA (i.e., with air backing) reveal that the fundamental reason for the poor lowfrequency sound absorption of the former is its weak ability to store lowfrequency sound energy. The introduction of an air layer as the backing of rubber perfectly solves this problem. As can be seen in Figure 5A, the air layer releases the fixed boundary condition of rubber, thus providing more space for the rubber to vibrate. Further, in sharp contrast to the original design (GAL), the new design (GALA) not only enables the lower portion of rubber to vibrate and hence store sonic energy but also enhances the vibration of the upper portion of rubber. Eventually, the kinetic energy of rubber is converted into heat due to viscous dissipation at rubber-metal interfaces. Therefore, the introduced air layer can significantly improve the acoustic performance of rubber-filled gratings.
From another point of view, the sound absorption capacity of the proposed anechoic layer is mainly determined by two factors: the ability to convert acoustic energy into kinetic energy, that is, the acoustic resistance Z Im( ) s , and the ability to convert kinetic energy into heat, that is, the acoustic reactance Z Re( ) s . To achieve good sound absorption, Z Im( ) → 0 s and Z Re( ) → 1 s need to be satisfied simultaneously, which means that as much incident sound energy as possible can be converted into kinetic energy and then dissipated by the anechoic layer. From the comparison of acoustic resistance between GAL and GALA in Figure 6A, it is clear that the acoustic resistance is not sensitive to the air layer below 2000 Hz, which means that the embedded air layer does not affect the kinetic energy dissipation capacity of the anechoic layer at low frequencies. Besides, the acoustic reactance is considerably improved at low frequencies due to the introduction of an air layer, as shown in Figure 6B. Together with the sound absorption coefficient of Figure 4, these noticeable features of resistance and reactance indicate that the air layer can effectively improve the low-frequency energy conversion capability of the anechoic layer, thereby improving its sound absorption performance.

| Parametric study
To explore the influence of key geometrical parameters on the sound absorption performance of a grating-like anechoic layer, the sound absorption spectrum as a function of upper rubber layer . In Figure 7A, while the thickness of rubber h 1 is varied, the plate spacing is set to l = 19 mm. Decreasing h 1 leads to a reduction in the absorption coefficient within the mid-and high-frequency range. Interestingly, an absorption peak is present in the spectrum at low frequencies, which shifts to ever lower frequencies as h 1 is reduced. The physical mechanism underlying this phenomenon is the resonance of the rubber-grating system, similar to a cantilever beam fixed at both ends. To demonstrate this, for selected values of h 1 , the displacement of the anechoic layer at the frequency corresponding to its  Figure 7B. When the frequency of sound is close to the resonant frequency, the rubber oscillates violently, thus forming a narrow sound absorption band. When the rubber layer becomes thinner, the stiffness of the system decreases, moving the resonance to lower frequencies.
With h = 45 mm 1 and h = 5 mm 2 , the effect of plate spacing l on the sound absorption spectrum is shown in Figure 7C. When the plate spacing is increased, it is not surprising to find an absorption peak at low frequencies, which can be explained by the resonance of the system consisting of rubber and steel grating. As the plate spacing increases, the stiffness of the system formed by the rubber layer and vertical plates decreases, the resonance frequency shifts to low frequencies, the bandwidth of the absorption peak narrows, and the magnitude of the peak becomes smaller, as shown in

| Optimization
To obtain the maximum weighted average absorption coefficient, the anechoic layer is optimized within the key parameter space of (h 1 , l).

| Lightweight evaluation
In practice, as an additional functional structure, the proposed anechoic layer inevitably increases the weight of a submarine and reduces its flexibility. Therefore, it is necessary to evaluate the lightweight design of the anechoic layer. As a preliminary study, an integrated performance index accounting for both (weighted average) sound absorption coefficient and mass density is introduced as follows where ρ is the normalized mass density, defined as the ratio of the mass density of the anechoic layer to the density of water, and is defined as h a t ρ = ( + ) + + ( + )( + ) . , l = 25 mm), which means that the best sound absorption with the least structure mass is achieved in this configuration.

| Evaluation of backing effects
The anechoic layer attached to an underwater vehicle actually works under complex backing conditions. Thus, the effects of backing on the present grating-like anechoic layer should be considered for actual applications. 18,38,57 To this end, we investigated the sound absorption performance of the GALA with a backing of a 10-mmthick steel plate 58 under three common backing conditions: rigid backing, water backing (followed by half-infinite water), and air backing (followed by half-infinite air). The thicknesses of the anechoic layer and the air layer were fixed at 50 and 5 mm, respectively. Based on the FE model detailed in Section 1, we developed FE models with two-sided infinite fluid domains for water and air backing samples, as shown schematically in the insets of Figure 11A. For each FE model, the sound absorption coefficient is calculated using where T p p = / tr in is the transmitted sound pressure coefficient and p tr is the transmitted sound pressure. Figure 11A compares the sound absorption coefficients obtained numerically for the three backing cases. It can be seen that, for the cases considered, the backing effect is mainly manifested in the relatively low-frequency range. The sound absorption coefficient of the GALA with water backing is consistently lower than that with rigid backing, with a shift to higher frequencies. This is because the acoustic impedance of water is close to that of steel, so that the transmission of low-frequency sound waves increases, as shown in Figure 11B. For the air backing case, the anechoic layer and the backing steel plate form a mass-spring resonant system to improve the relative motion between rubber and steel gratings, as shown in the displacement distribution of Figure 11B. Therefore, the energy dissipation caused by the resonance leads to a new absorption peak at 2000 Hz ( Figure 11A). Nonetheless, at frequencies below 1100 Hz, the results of Figure 11A demonstrate that the performance with air backing is inferior to that with rigid backing. This is because air and water backings are both weak constraints, and hence the rubber and steel gratings essentially move in the same direction at sufficiently low acoustic frequencies: the relative movement between the rubber and metal grille is thus weakened, causing inferior absorption performance.

| Evaluation of boundary conditions
The above study is based on wave propagation in an infinite periodic 2D medium. However, in practice, the rubber layer as well as the It is demonstrated that the additional air layer under the rubber block improves low-frequency sound absorption by enhancing the ability of low-frequency energy conversion from sound into kinetic energy of the anechoic layer. Upon reducing the rubber layer thickness or increasing the plate spacing, an absorption peak greater than 0.8 appears at low frequency, depending on the resonance of the rubber layer. However, when the rubber layer thickness is reduced, due to insufficient high-frequency energy dissipation, highfrequency sound absorption becomes inferior, making it difficult to achieve broadband sound absorption. In contrast, increasing the plate spacing can achieve both low-frequency and broadband absorption.