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Frontiers in Materials ·

Volume 13 - 2026 | https://doi.org/10.3389/fmats.2026.1929207

A gradient-lattice-constant material for enhanced low-frequency and broadband underwater sound absorption

  • Shibo Wang1
  • Qicheng Zhang2
  • Di Wu1
  • Zhaohua Su1
  • Sen Zhang1*corresponding author
  1. 1Naval University of Engineering, Wuhan, China
  2. 2Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology Wuhan University, Wuhan, China
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Abstract

Acoustic coatings with low-frequency and broadband sound absorption are vital for underwater vehicle stealth. Multi-layer cavity structures are of particular interest, as inter-layer coupling enables broadband absorption, while a gradient in cavity radii facilitates acoustic impedance matching. However, cascading more cavity layers within a given material thickness reduces the elastic medium thickness, thereby weakening sound absorption. To overcome this limitation, we propose and fabricate a multi-layer cavity material with uniform cavity radii but gradient lattice constants, dubbed gradient-lattice-constant material (GLM). The reduced cavity occupancy effectively increases the elastic medium thickness, thereby enhancing the underlying Fabry-Pérot (F-P) resonance and improving sound absorption. Specifically, the F-P resonance theory, displacement field and power dissipation density simulations are employed to evaluate the absorption performance of GLM under varying cavity parameters, including radius, lattice constant, and interlayer distance. The results demonstrate that the advantage of GLM becomes increasingly pronounced with four and more layers, effectively enhancing low-frequency and broadband sound absorption. Furthermore, we experimentally implement a five-layer GLM in water, which achieves effective absorption from 100 Hz to 2000 Hz, agreeing well with theoretical and simulation results. Our findings provide an efficient route to lightweight, high-performance acoustic coatings and absorbers.

Keywords

1 Introduction

In the context of marine science and engineering, effective acoustic absorption is not only a material science problem but also a pivotal technology for naval applications (Akram and Israr, 2025; Chai et al., 2026). Specifically, the survival and operational effectiveness of submarines rely heavily on their ability to minimize target strength in sonar detection bands, which are often dominated by low-frequency noise in ocean environment. Therefore, the development of lightweight (Bezancon et al., 2024), robust (Cai and Xin, 2025; Feng et al., 2025), and broadband acoustic coatings is a persistent research focus within the marine engineering community.

Traditional underwater acoustic coatings use rubber as an elastic matrix, converting sound energy into heat via internal friction (Nori and Venegas, 2017; Venegas et al., 2022; Parra et al., 2026) and elastic relaxation. Rubber materials usually contain periodically distributed cavities, including different cavity shapes such as spheroids (Skvortsov et al., 2019; Thieury et al., 2020), cones (Ye et al., 2018; Wang Z. et al., 2020), ellipsoids (Ivansson, 2008), cylinders (Yang et al., 2022; Zhou and Fang, 2022; Kaganowich et al., 2026), and disks (Sharma et al., 2020; 2023). Among these, transverse cylindrical cavities, whose axes are perpendicular to the direction of incidence, have gradually become a research hotspot for lightweight acoustic coating applications because of their very thin thickness (Liu et al., 2022; Fu et al., 2023). Sharma and Skvortsov et al. developed a theoretical model for cylindrical cavities using effective medium theory and interpreted the monopole and dipole resonance mechanisms (Sharma et al., 2017b; 2017a; 2019b). They also investigated acoustic radiation from an underwater cylindrical shell, attributing significant noise reduction to Fabry-Pérot (F-P) resonance between the cavity layers and shell (Sharma et al., 2019b; Lin et al., 2022; 2024; Modur et al., 2024). This work provided new insights for the subsequent application of materials with cylindrical cavities.

Transverse cylindrical cavities have given rise to many different arrangements of structures. Research has shown that by applying gradient cavity parameter designs (Bian et al., 2021; Shi et al., 2023), broadband sound absorption performance can be achieved (Liu et al., 2023; Mcintosh et al., 2024). Wang et al. (Wang S. et al., 2020; 2022) developed a model of periodic cylindrical cavities in a gradient-parameter soft elastic medium, achieving three distinct high absorption peaks and confirming the tandem capability of F-P resonance. Leveraging this tandem capability, Fang et al. (Fang et al., 2023; Pan et al., 2024) designed 4-layer and 9-layer elastic materials with cylindrical cavities with gradient parameters, generating discrete absorption peaks corresponding to each layer for multiband performance. Subsequent tests under static pressure validated the effectiveness of sustained F-P resonance for sound absorption (Fang et al., 2024). These studies confirm that multilayer cylindrical cavity materials with gradient parameters can achieve broadband absorption through synergistic F-P resonance.

However, the adoption of the design concept involving tandem cylindrical cavities with gradient radii presents two potential challenges as the number of cavity layers increases and the cavity radius gradually expands. First, the implementation of such tandem cavities with gradient radii within a constrained thickness leads to progressive compression of the elastic medium thickness. Since rubber serves to convert sound energy into heat through internal friction, this compression effect results in a degradation of the sound absorption performance. Second, in the absence of thickness constraints, an increase in the number of tandem layers would lead to a substantial increase in the thickness of the acoustic coating, which is detrimental to its practical application and to the achievement of lightweight design requirements. Given the inherent thickness limitation of acoustic coatings for submarines, the present study focuses on addressing the first challenge.

In this study, a gradient-lattice-constant material (GLM) is proposed. This model consists of tandem cylindrical cavities with gradient changes in the lattice constant. Specifically, layers of small-radius, densely arranged cylindrical cavities are employed to replace the previously used layers of large-radius, sparsely arranged cylindrical cavities. The frequency-dependent characteristics of the equivalent impedance of cylindrical cavity layers are given. To this end, we convert the problem with the equivalent impedance as the control variable into one with the effective medium layer thickness as the control variable. In this paper, the geometric model of the proposed GLM is presented first. A theoretical model of multilayer composite homogeneous layers is subsequently established on the basis of the effective medium theory, followed by an introduction to the transfer matrix method for calculating sound absorption coefficients and the F-P resonance theory for analyzing sound absorption mechanisms. Afterward, the sound absorption performance results of the proposed GLM are presented, and through comparisons with the gradient-radius-material (GRM) designed with gradient changes in the radius of the cylindrical cavity, the intrinsic reasons for the GLM outperforming the GRM are comprehensively interpreted. Furthermore, the effects of the cavity radius and distance on F-P resonance are investigated. To confirm these findings, an optimization process is used to verify that the sound absorption performance of the GLM is superior to that of the GRM when the number of cylindrical cavity layers increases to 4 and 5. Finally, acoustic samples are fabricated, and acoustic performance tests are conducted to experimentally validate the proposed GLM.

2 Theory and model

In this section, we will first introduce the structures of the proposed GLM. Then the theoretical model for GLM with two layers of cavity layers will be elucidated, which consists of the effective medium approximation (EMA), transfer matrix method (TMM) and F-P resonance theory. The theoretical method is the effective tool for obtaining acoustic performance and mechanism analysis.

2.1 Geometric structure of GLM

Figure 1a illustrates the proposed GLM is perpendicular to the incident waves and extends infinitely in the x-direction. The schematic shows only a partial view of GLM and does not represent an entire unit cell. From top to bottom, the materials are water, GLM (a soft elastic material embedded with cylindrical cavity layers), steel backing and air. Figure 1b depicts the two-dimensional cross section of the GLM. The thicknesses of the GLM and steel backing are d and s, respectively. GLM contains two cylindrical cavities layers (cavity layers 1 and 2) which consist of periodical cylindrical cavities perpendicular to the incident plane waves and extend infinitely in the z direction. The lattice constants of the two cavity layers are a 1 and a 2, respectively. The distances between the upper boundary of the soft material, the two cavity layers and the steel backing in the y-direction are denoted by d 1, d 2 and d 3, respectively. The thicknesses of the elastic medium above cavity layer 1, between cavity layers 1 and 2, and below cavity layer 2 are l 1, l 2 and l 3, respectively. The radii of the cylindrical cavities embedded in the material are all the same as r. All cylindrical cavities are assumed to be evacuated.

Schematic of the proposed GLM (not a unit cell): **(a)** The model’s spatial arrangement consists of the soft elastic material embedded with multilayers of periodical cavity and steel backing. **(b)** Cross section of the GLM.
Figure 1. Schematic of the proposed GLM (not a unit cell): (a) The model’s spatial arrangement consists of the soft elastic material embedded with multilayers of periodical cavity and steel backing. (b) Cross section of the GLM.

2.2 Theoretical model for GLM

This section begins with the application of the EMA for layer homogenization, yielding the effective material properties, including complex density, sound speed, and acoustic impedance. TMM is then introduced to calculate the acoustic performance of the proposed GLM. F-P resonance theory is employed to investigate the acoustic response of the cavity layers and to provide insight into the physical mechanisms governing sound absorption.

2.2.1 Effective parameters in the homogeneous medium layers

According to the EMA (Sharma et al., 2017a), a cylindrical layer can be homogenized by a layer of homogeneous layer with thickness l *effi* (i = 1,2), as shown in Figure 2.

Layer homogenization applied to the proposed GLM.
Figure 2. Layer homogenization applied to the proposed GLM.

The effective parameters depend mainly on the radius and lattice constant of the cavities in the material (Brooke et al., 2020). l *effi* can be calculated by

l e f f i = 2 a i π ln sec π r a i (1)

In line with effective medium approximation, the effective density ρ e i and effective longitudinal modulus κ e i of Homogenized layer i are given by (Wu et al., 2007):where α e i = l e f f i / a. ρ and κ are the density and longitudinal wave modulus of the soft elastic material. ρ g i and κ g i are global average density and global average longitudinal wave modulus, which are calculated bywhere α i = π r / a i 2 is the filling fraction for cavities in a square array. Q i = δ + 2 / 1 − ω / Ω 0 i 2, where Ω 0 i is the resonant frequency of a layer of cylindrical cavities (Sharma et al., 2017b).

ρ g i = ρ e i α e i + ρ 1 − α e i , 1 κ g i = α e i κ e i + 1 − α e i κ (2)

ρ g i = ρ 1 − α i , κ g i = κ 1 − α i Q i 1 − α i (3)

Substitution of Equation 3 and Q *i* into Equation 2 gives the following expressions for the effective parameters of the homogenized layer:

ρ e i = ρ 1 − π r 2 a l e f f i , κ e i = a κ 1 − α i l e f f i π r 2 1 + Q i + 1 − α i (4)

Finally, the effective acoustic impedance of the homogeneous layer Z e i can be obtained by ρ e i and κ e i in Equation 4 aswhere Z is the acoustic impedance of the soft elastic material.

Z e i = ρ e i κ e i = Z 1 − α i 1 − α i / α e i α i 1 + Q i + 1 − α i / α e i (5)

2.2.2 Transfer matrix method

The reflection and transmission performances of GLM can be calculated by the TMM. One can see on the right side of Figure 2, the homogenized model consists of six homogeneous layers. The thickness d *i* of the six layers from top to bottom (j = 1, 2, 3) is l *1* +r-l *eff1* /2, l *eff1*, l *2* +2r-l *eff1* /2-l *eff2* /2, l *eff2*, l *2* +r-l *eff2* /2 and s, respectively. For the jth homogeneous layer, the relationship between the output sound pressure p *outj* and particle velocity v *outj*, and the input pressure p *inj* and particle velocity v *inj*, is given by (Sharma et al., 2019a).where [T*j*] in Equation 6 is the jth transfer matrix given bywhere k *j* = 2 π f/c *j* is the wavenumber, c *j* is the sound speed of the material in the jth layer, and f is the frequency of the incident wave, and Z *j* can be calculated by Equation 5. The total transfer matrix [T] of the composite homogeneous medium layer shown on the right side of Figure 2 is obtained by multiplying the transfer matrices of each layer in Equation 7, as follows:where T 11, T 12, T 21 and T 22 in Equation 8 are four elements of the total matrix, and the sound pressure (output pressure p *out* and input pressure p *in*) and particle velocity (output particle velocity v *out* and input particle velocity v *in*) on both sides of the soft elastic material can be related by [T] as Equation 9 shows:

p o u t j v o u t j = T j p i n j v i n j (6)

T j = cos ⁡ k j d j i Z ⁡ sin ⁡ k j d j i Z j sin ⁡ k j d j cos ⁡ k j d j (7)

T = T 1 T 2 T 3 T 4 T 5 T 6 = T 11 T 12 T 21 T 22 (8)

p i n v i n = T p o u t v o u t (9)

Then, the reflection coefficient r and the transmission coefficient t can be obtained by Equation 10 (Sharma et al., 2017a)

r = p r p i = T 11 + T 12 Z t r − Z i n c T 21 − Z i n c Z t r T 22 T 11 + T 12 Z t r + Z i n c T 21 + Z i n c Z t r T 22 , t = p t p i = 2 T 11 + T 12 Z t r + Z i n c T 21 + Z i n c Z t r T 22 − 1 (10)

Z *inc* is the acoustic impedance of the fluid on the incident side, which is a function of the density ρ *inc* and the sound speed c *inc* of the water on the incident side. Z *tr* is the acoustic impedance of the fluid on the transmission side, which is a function of the density ρ *tr* and the sound speed c *tr* of the air on the transmitted side. The sound absorption coefficient α is given by α = 1-r 2-[(Z *tr*/Z *inc*)t]2.

2.2.3 F-P resonance sound absorption mechanism

F-P resonance originated as an optical component known as the F-P resonator, forming the design basis for F-P cavities (Yuan, 2002; Zhang et al., 2008). This device comprises two parallel planar reflective mirrors, where light undergoes multiple internal reflections within the cavity, forming oscillating beams that interfere before output. Today, F-P resonance serves as a key tool for investigating cavity–scatterer interactions in acoustic structures (Liu et al., 2019; Xie and Wang, 2021). Thus, for elastic materials with multilayer cylindrical cavities, F-P resonance can be used to explain the sound absorption mechanism of the model. To simplify the analysis, a two-layer cylindrical cavity configuration is chosen as the study case to examine the F-P effect within it.

Figure 3 illustrates the interaction between two cavities or homogenized layers in the soft elastic material. Incident sound waves impinge on the surface of cavity 1; some are reflected (denoted as r *f*), while the remainder are transmitted through the cavity and continue propagating (denoted as t *f*). This transmitted wave travels farther and then reflects off the surface of cavity 2, producing a reflected wave (r *b*). t *f* interacts with r *b*, resulting in F-P resonance. The structure consisting of the two cavities or homogenized layers and the intervening elastic material is denoted as the “F-P resonant structure” here.

Schematic of the F-P resonant structure of the soft elastic material with cavities or homogenized layers.
Figure 3. Schematic of the F-P resonant structure of the soft elastic material with cavities or homogenized layers.

A coefficient denoted as c *b* represents the strength of the interaction between t *f* and r *b*. This coefficient can be expressed as as Equation 11 (Sharma et al., 2017a):where l is the thickness of the elastic material between two layers of cavities or homogenized layers and k *l* is the longitudinal wavenumber in the elastic material. r *f* and r *b* can be obtained by the TMM; thus, the total reflection coefficient of the F-P resonant structure is given by r *FP* = r *f* + c *b*.

c b = r b t f 2 ⁡ exp − 2 i k l l 1 − r b r f ⁡ exp − 2 i k l l (11)

The sound absorption coefficient of the F-P resonant structure can be obtained by α F P = 1 − r F P 2.

3 Results and discussions

3.1 Acoustic performance for GLM

Here, the correctness of the theoretical model is validated initially. A single-cell finite element model (FEM) of GLM is established using the finite element software COMSOL Multiphysics v6.2. The elastic material is submerged in water and subjected to a plane wave. Rubber and steel are modeled using the “solid mechanics module”. Water is modeled using a “pressure acoustics module”. The interior domain of the cavities is assumed to be evacuated. Free boundary conditions are applied on the internal surface of the material and at the interface between the material and embedded cavities. A continuity boundary condition is applied at the interface between the coating and the steel backing. An acoustic-structure boundary condition is applied at the boundary of the coating and water domain to simulate the interaction between the solid and fluid media. A perfectly matched layer boundary condition is applied at the outer boundary of the fluid domain to model the anechoic termination of the outgoing waves. During mesh division, according to finite element theory, the size of all the solid mesh elements is less than 1/4 of the corresponding shear wavelength of the soft material. The material properties of the materials used in FEM and analytical model are given in Table 1.

TABLE 1. Material properties of the elastic material and the surrounding fluid.

SolidsDensity (kg/m3)Longitudinal modulus (GPa)Shear modulus (GPa)
Elastic material1,0001 + 0.01i0.0006 + 0.00018i
Steel7,89017580.7
FluidDensity (kg/m3)Sound speed (m/s)
Water1,0001,500
Air1.2340

To highlight the superior absorption performance of the proposed GLM, we compare it with the model featuring periodic cavities with gradient changes in the cavity radius (gradient-radius-material, GRM) from the previous literature (Wang S. et al., 2020; Pan et al., 2024), as shown on the right side in Figure 4. The structural parameters of the proposed GLM are as follows: d = 60 mm, r = 1.5 mm, a 1 = 10.3 mm, a 2 = 4.5 mm, d 1 = 12 mm, d 2 = 38 mm, d 3 = 10 mm, l 1 = 10.5 mm, l 2 = 35 mm and l 3 = 8.5 mm.

Schematic of the homogenization for GLM and GRM.
Figure 4. Schematic of the homogenization for GLM and GRM.

The GRM contains two layers of cylindrical cavities with the same lattice constant, a = 40 mm. The total thickness is d = 60 mm. The relative positions of the cylindrical cavity in GRM are set the same as those in GLM; thus, l 1 = 9 mm, l 2 = 30 mm and l 3 = 5 mm for GRM can be obtained. To compare the acoustic properties of the two materials under the same condition, the thickness of the homogenized layer l *eff* is set equal. The effective thicknesses l *eff1* = and l *eff2* in GLM are 0.714 mm and 2.016 mm, respectively. According to Equation 1, the radius values of cavities 1 and 2 for GRM should be set as r 1 = 3 mm and r 2 = 5 mm, respectively.

Figure 5 shows a comparison of the sound absorption performance of the GRM obtained theoretically and numerically across the frequency range of 1–2000 Hz, which is denoted by “EMA” and “FEM”, respectively. For the results obtained by FEM, two absorption peaks appear in the given frequency range for both the GLM and GRM. The first absorption peak of the GRM occurs at 181 Hz, with a peak value of 0.87; the first absorption peak of the GLM occurs at 181 Hz, with a peak value of 0.838. Previous studies (Wang S. et al., 2020; Pan et al., 2024; Fang et al., 2023) have shown that the first absorption peak is caused by the F-P resonance of cavity layer 2 and the steel backing. This combination is denoted as F-P resonant structure 1, which is shown in the green fill in Figure 5. The result that the peak value of the GRM is higher than that of the GLM leads to a deduction—the F-P resonance of F-P resonant structure 1 in the GRM is stronger than that in the GLM, which is discussed in the following text. The second absorption peak of the GRM occurs at 841 Hz, with a peak value of 0.566; the second absorption peak of the GLM occurs at 881 Hz, with a peak value of 0.664. The second peak value of the GLM is 9.8% greater than that of the GRM. Because the second absorption peak is attributed to the F-P resonance of cavity layers 1 and 2, this combination is denoted as F-P resonant structure 2, which is shown in the yellow fill in Figure 5. Thus, it can be inferred that the F-P resonance between the F-P resonant structures 2 in the GLM is stronger than that in the GRM. The above results indicate that under the same homogenized layer thickness conditions, the peak values of the GLM are greater than those of the GRM. The average sound absorption coefficient α ¯ across the frequency range under 2 kHz is calculated, indicating that the GLM (α ¯ G L M T H = 0.221) is 2.8% greater than the GRM (α ¯ G R M T H = 0.193). Compared with the GRM, the GLM has higher absorption coefficients around 1,000 Hz, and although its absorption coefficients in the low-frequency range (approximately 200 Hz) are lower, its average sound absorption performance across the entire frequency range is better than that of the GRM.

Comparison of the sound absorption performance between the GLM and GRM obtained theoretically and numerically.
Figure 5. Comparison of the sound absorption performance between the GLM and GRM obtained theoretically and numerically.

In addition, Figure 5 also show that the second absorption peak frequency predicted by FEM is higher than that from EMA, with a shift of approximately 140 Hz. This deviation is primarily attributed to two factors: Firstly, the EMA model is based on a fluid-equivalent assumption that neglects shear wave effects in the soft elastic medium. In contrast, FEM fully accounts for both compressional and shear wave coupling. At higher frequencies, shear waves significantly increase the effective dynamic stiffness of the elastic layer between the two cavity layers, thereby shifting the F-P resonance toward higher frequencies. Secondly, at higher frequencies, transverse coupling between adjacent cavities and mode conversion become more pronounced, which EMA inherently fails to capture. At low frequencies (the first absorption peak), shear and coupling effects are weaker, and EMA agrees well with FEM. Although quantitative deviations exist, EMA reliably captures the relative trend. Thus, the limitations of EMA do not compromise the core conclusions of this work.

3.2 Mechanism of F-P resonance in GLM

To explore the physical mechanisms underlying the superior absorption performance of the proposed GLM, Figure 6 presents a comparison of α F P of F-P resonant structures 1 and 2 in GLM and GRM. For F-P resonant structure 2 (Figure 6a), the peak values of α F P for the GLM and GRM are0.814 and 0.736, respectively. The result shows that the F-P resonance between cavity layers in the GRM is stronger than that in the GLM, causing the second absorption peak of GLM to be higher than that of GRM. However, the results of α F P in Figure 6b verify that the resonance effect of F-P resonant structure 1 in GRM (0. 971) is stronger than that in GLM (0. 967). One can see that the F-P resonance of F-P resonant structure 1 is stronger than that of F-P resonant structure 1, resulting in that the first absorption peak value is larger than 0.8.

Comparison results of α F P of F-P resonant structures 1 and 2 in GLM and GRM. **(a)** F-P resonant structure 2. **(b)** F-P resonant structure 1.
Figure 6. Comparison results of α F P of F-P resonant structures 1 and 2 in GLM and GRM. (a) F-P resonant structure 2. (b) F-P resonant structure 1.

To provide an intuitive explanation of the absorption mechanism, Figure 7 presents the displacement field of the GLM and GRM at their second absorption peaks, i.e., at 881 Hz and 841 Hz, respectively.

Displacement field at the second absorption peak frequency of GLM and GRM (unit: m). **(a)** Displacement field. **(b)** Power dissipation density.
Figure 7. Displacement field at the second absorption peak frequency of GLM and GRM (unit: m). (a) Displacement field. (b) Power dissipation density.

As shown in Figure 7a, regions with larger displacement magnitudes consistently appear between the two cavity layers in the GLM and GRM. This occurs because sound waves primarily propagate through the soft elastic medium between the cavity layers: after passing through the first cavity, the incident sound waves reach the second cavity, reflect off its surface, undergo attenuation between the two layers, and ultimately contributing to sound absorption. Comparing the high-value displacement regions of the GLM with the case GRM reveals that, on the one hand, the maximum displacement between the two cavity layers in the GLM is 4.6310<sup>−10</sup> m, while in the GRM is 4.5610−10 m; on the other hand, the average displacement field magnitude between the two cavity layers is 22×10−10 m for the GRM, while that of the GLM is 25 × 10−10 m. Furthermore, the area of the regions with larger displacement magnitudes in GLM is larger than GRM.

To further investigate the coupling between F-P resonance and local resonances in multi-layer structures, the power dissipation density at the second absorption peak frequencies for both GLM and GRM are calculated. The results are presented in Figure 7b, where the left and right panels correspond to GLM (at 881 Hz) and GRM (at 841 Hz), respectively.

As shown in Figure 7b, the dissipation patterns of the two configurations differ significantly. In the GRM (right panel), the power dissipation density is primarily concentrated around the surfaces of the cylindrical cavities, indicating that the energy attenuation is dominated by local monopole and dipole resonances of individual cavities. In contrast, the GLM (left panel) exhibits a much broader dissipation distribution: high-dissipation regions not only appear around the cavity surfaces but also extend throughout the elastic medium between the cavity layers. The coexistence of dissipation both at the cavity surfaces and within the inter-layer elastic medium reveals a synergistic coupling between local resonances and F-P resonance in the GLM. Furthermore, the expanded dissipation zone in GLM is attributed to its thicker elastic interlayers (l 2 = 35 mm vs. 30 mm in GRM), which provide a longer propagation path for acoustic waves, thereby enhancing F-P resonance-induced dissipation. The increased interlayer thickness also promotes larger local deformation around individual cavities, leading to higher overall dissipation. In contrast, the GRM employs large-radius, sparsely distributed cavities with thinner elastic interlayers, resulting in weaker F-P resonance and less dissipation in the elastic medium. This dual mechanism explains why the second absorption peak of GLM is significantly higher than that of GRM.

3.3 Parameter influences

The effects of different geometric parameters, including the cavity arrangement and the distance between cavity layers, on the acoustic performance of the proposed model are now examined.

3.3.1 Influence of the cavity radius r

According to Equation 1, various cavity arrangements combining different radii and lattice constants under a fixed l *eff* can be obtained. The influence of different cavity arrangements on the sound absorption performance of materials with two cavity layers is studied below. The effective thicknesses of cavity layers 1 and 2 are l *eff1* = 0.714 mm and l *eff2* = 2.016 mm, respectively. Three cavity arrangement cases are considered: Case (i): r = 1 mm, a 1 = 4.8 mm, a 2 = 2.4 mm, l *1* and l *2* are 9 mm and 36 mm, respectively; Case (ii): r = 1.5 mm, a 1 = 10.3 mm, a 2 = 4.5 mm, l *1* and l *2* are 8.5 mm and 35 mm, respectively; Case (iii): r = 2 mm, a 1 = 18 mm, a 2 = 7.2 mm, l *1* and l *2* are 8 mm and 34 mm, respectively. The total thickness of the GLM for the three cases is the same, with d = 60 mm. The relative positions of the cylindrical cavities in each layer are the same: d 1 = 12 mm, d 2 = 38 mm, d 3 = 10 mm, s = 20 mm.

Figure 8a shows a comparison of the total absorption coefficients of the GLM for the three cases. The trends of the absorption curve are consistent with each other. Differences occur at the second absorption peak. Among them, the peak of Case (i) is the highest, followed by Case (ii), and Case (iii) has the lowest peak. The peak value of Case (i) is 0.0228 higher than that of Case (ii), and the peak value of Case (ii) is 0.0233 higher than that of Case (iii). For every 1 mm increase in l, the peak value increases by approximately 0.023, which is an approximately linear change. The first absorption peak values for the three cases are almost identical. Figure 8b presents the α F P values of F-P resonant structure 2 in the GLM for three cases. The α F P value of the F-P peak for Case (i) is the highest, followed by that for Case (ii), and that for Case (iii) is the lowest. The results show that the smaller and denser the cavities are arranged, the stronger the F-P resonance effect between the cavity layers, and the higher the sound absorption capacity of the resonant structure. This improvement is primarily attributed to the increased thickness of the elastic medium between cavity layers, which promotes sound attenuation. Therefore, when the GLM is designed, if the process technology permits, the radius and lattice constant can be appropriately reduced to achieve better sound absorption.

Comparison of total absorption and α F P between three cases. **(a)** Total sound absorption coefficients of three cases. **(b)** α F P of F-P resonant structure 2.
Figure 8. Comparison of total absorption and α F P between three cases. (a) Total sound absorption coefficients of three cases. (b) α F P of F-P resonant structure 2.

3.3.2 Influence of distance between cavity layers

Next, the sound absorption performance of the GLM with two cavity layers at different distances between cavity layers is investigated, where l changes from 0.5 mm to 40mm, as shown in Figure 9a. The total thickness of the GLM for the three cases is the same, with d = 60 mm. The cavity radius is r = 1.5 mm. The lattice constants of the two cavity layers are a 1 = 10.3 mm and a 2 = 4.5 mm, respectively. d 3 = 10 mm, and s = 20 mm.

Comparison of total absorption and α F P for three cases. **(a)** Diagram showing the change in the distance between the cavities. **(b)** Total sound absorption coefficients GLM for three cases. **(c)** α F P of F-P resonant structure 2 for three cases.
Figure 9. Comparison of total absorption and α F P for three cases. (a) Diagram showing the change in the distance between the cavities. (b) Total sound absorption coefficients GLM for three cases. (c) α F P of F-P resonant structure 2 for three cases.

One can see that the second absorption peak value increases as l increases in Figure 9b. The greater the distance is, the smaller the increase in the peak value. This is because the attenuation of the sound wave becomes less pronounced as the propagation distance increases. Additionally, as l increases, the second absorption peak moves toward lower frequencies. Cavity layers 1 and 2 and the elastic medium can be modeled as a spring-mass-spring system, as described by Sharma et al. (Sharma et al., 2017b; 2017a; 2019a). Cavities 1 and 2 act as springs, and the mass is provided by the elastic medium. Thus, increasing l leads to an increase in the mass of the elastic medium layer, which decreases the resonance frequency of the system because the frequency of the resonance is inversely proportional to the mass.

The results of α F P for three cases are given in Figure 9c. The F-P peak value also increases with increasing l, which verifies the aforementioned conclusion that an increase in the thickness of the elastic medium enhances the F-P resonance, resulting in an increase in the sound absorption capacity of the material.

3.4 Optimized design of GLM4 and GLM5

The tunable broadband acoustic performance at low frequencies is a critical concern for improving the sound stealth performance of underwater vehicles. The above analysis indicates that the GLM exhibits superior sound absorption coefficients. However, this is achieved only by embedding two layers of cylindrical cavities within the material. Inspired by the broadband absorption metagrating, which consists of tandem meta-units proposed in waterborne environments (Pan et al., 2024), GLMs with more cavity layers (4 and 5) (denoted as GLM4 and GLM5, respectively) are presented in this paper to expand the number of F-P resonant modes. Through this approach, a broader absorption frequency range can be obtained, and the high sound absorption advantage of the GLM can also be highlighted. To obtain satisfactory absorption performance, an optimization for GLM and GRM is carried out as follows. In addition, a negative gradient change in cylindrical cavity distances is added to the existing distribution. That is, comparing the absorption performance of materials with cavities with “gradient lattice constant + gradient distance” and with “gradient radius + gradient distance”.

3.4.1 Optimization configuration

The optimization of the multi-layer cavity structures is performed using the Nelder-Mead simplex algorithm implemented in COMSOL Multiphysics. This derivative-free, gradient-free optimization method is well-suited for continuous multi-parameter acoustic design problems, as the objective function (average sound absorption coefficient) varies smoothly with the geometric parameters. The key parameters of the Nelder-Mead algorithm are configured as follows. The optimization tolerance is set to 0.01, serving as the stopping criterion based on the change in the objective function value, and the maximum number of model evaluations is limited to 50 iterations per optimization run. For constraint handling, the augmented Lagrangian method is employed as a penalty strategy to enforce geometric constraints, with the penalty parameter initialized to 10, the penalty tolerance factor set to 0.5, the penalty multiplication factor set to 10, the Lagrange multiplier upper bound defined as 1.7977 × 10308, the maximum number of augmented iterations set to 10, and the constraint tolerance set to 0.01. The sub-solver tolerance adopts the default Dynamic I setting in COMSOL, which is automatically adjusted by the software. These parameter settings were systematically tested and determined through preliminary numerical experiments, ensuring robust convergence of the optimization process with acceptable computational cost and yielding physically reasonable optimized designs.

Taking GRM4 (a GRM with 4 cavity layers) as an example, the cavity radius (r 2, r 3, and r 4) and the distances between the boundary cavity layers (l 2, l 3, l 4, and l 5) serve as optimization variables. The radius of cavity layer 1 is fixed to 1.5 mm, and the lattice constant of each layer is fixed to 40 mm. The objective is to determine the optimal solution for the average sound absorption coefficient over the frequency range of 1–2000 Hz. After obtaining the optimal GRM4, l *eff* of each cavity layer is computed and used as constraints for GLM4. With these l *eff* values fixed, GLM4 was then optimized to maximize the same average absorption coefficient, allowing its radii, lattice constants, and distance distributions to vary. For GLM, the radius of cavity layer 1 is fixed to 1.5 mm, and the lattice constant of each layer is fixed to 40 mm. The optimization formulas of GRM4 are listed in Table 2.

TABLE 2. Optimization formulas for GRM4 and GLM4.

MaterialGRM4GLM4
Optimization configurationF i n d : X = r 2 , r 3 , r 4 , l 2 , l 3 , l 4 , l 5 max ⁡ : F X = 1 N ∑ i = 1 N α f i 1 Hz ≤ f i ≤ 2000 Hz s . t . : 2 mm ≤ r 2 ≤ 6 mm , 3 mm ≤ r 3 ≤ 8 mm 4 mm ≤ r 4 ≤ 10 mm 7 mm ≤ l 2 ≤ 20 mm , 5 mm ≤ l 3 ≤ 15 mm , 3 mm ≤ l 4 ≤ 12 mm , 1 mm ≤ l 5 ≤ 10 mm contraint : r 1 ≤ r 2 , r 2 ≤ r 3 , r 3 ≤ r 4 l 2 ≥ l 3 , l 3 ≥ l 4 , l 4 ≥ l 5F i n d : X = a 1 , a 2 , a 3 , a 4 , l 2 , l 3 , l 4 , l 5 max ⁡ : F X = 1 N ∑ i = 1 N α f i 1 Hz ≤ f i ≤ 2000 Hz s . t . : 3.1 mm ≤ a 2 ≤ 25 mm , 3.1 mm ≤ a 3 ≤ 15 mm 3.1 mm ≤ a 4 ≤ 10 mm 4 mm ≤ l 2 ≤ 20 mm , 3 mm ≤ l 3 ≤ 15 mm , 2 mm ≤ l 4 ≤ 10 mm , 1 mm ≤ l 5 ≤ 5 mm contraint : l e f f i G L M = l e f f i G R M i = 1 , 2 , 3 , 4 a 2 ≤ a 1 , a 3 ≤ a 2 , a 4 ≤ a 3 l 2 ≥ l 3 , l 3 ≥ l 4 , l 4 ≥ l 5

X is the optimization variable parameter of the four-layer cylindrical cavity, **F(X)** is the optimization index within the required frequency band, that is, to maximize the average sound absorption coefficient within the frequency band, and N is the number of frequency sweeps within the frequency band, with a frequency interval of 20 Hz. The subscripts “min” and “max” represent the lower and upper bounds, respectively, of the variables. l e f f i G L M and l e f f i G R M are the ith effective thickness of the homogenized layer in GLM and GRM, respectively.

3.4.2 Absorption performance of GLM4 and GLM5

In this section, we first investigate the absorption performance of the optimized GLM4 and GRM4. To mitigate the risk of converging to local optima, we performed five independent optimization runs for each configuration, each starting from a different initial simplex (randomly generated within the variable bounds). The best result among the five runs, i.e., the one with the highest average absorption coefficient, is selected as the final design. All optimized designs are subsequently validated using full-wave COMSOL simulations and experimental measurements to ensure their reliability. The final optimal variables and values are detailed in Table 3.

TABLE 3. Geometric parameters of GRM4, GLM4, GRM5 and GLM5 (Unit: mm).

ParametersGRM4GLM4GRM5GLM5
r 11.51.51.51.5
r 22.41.52.61.5
r 34.11.53.31.5
r 45.61.54.51.5
r 5——5.51.5
a 140404040
a 27.910.2
a 34.48.1
a 44.14.2
a 5——3.3
l 225.235.1 (9.9 ↑)20.129.9 (9.8 ↑)
l 38.812.9 (4.1 ↑)6.29.1 (2.9 ↑)
l 43.97.1 (3.2 ↑)3.86.8 (3.0 ↑)
l 52.12.2 (0.1 ↑)2.26.1 (3.9 ↑)
l 6——1.11.1 (0)
l 2 +l 3 +l 4 +l 5 +l 640.057.2 (43.0%)33.453.0 (58.7%)

Figure 10a presents a comparison of the sound absorption performance between GRM4 and GLM4. When GLM4 is applied, four absorption peaks are observed (Values in brackets represent the peak values): 120 Hz (0.77), 893 Hz (0.712), 1,392 Hz (0.728) and 1,655 Hz (0.652). Four absorption peaks are also observed when GRM4 is applied: 152 Hz (0.705), 922 Hz (0.648), 1,323 Hz (0.632) and 1709 Hz (0.636). The coefficients are about 6.4% and 9.6% greater at approximately 900 Hz and 1,300 Hz, respectively. The half-absorption bandwidth (the frequency range where the absorption coefficient is equal to or greater than 0.5 (α ≥ 0.5)) of GLM4 is 996 Hz (108–132Hz, 784–1056Hz, and 1,159–1859 Hz), which is 6 Hz wider than that of GRM4 (138–166Hz, 841–1056Hz, and 1,159–1906 Hz). In terms of average performance, compared with GRM4, GLM4 has an average sound absorption coefficient α ¯ of 0.39, which is 1% greater. While the difference in half-absorption bandwidth and average absorption coefficients between the two materials is minimal, GLM4 delivers the higher second and third absorption peaks, as shown in the pink-filled areas in Figure 10a. Below 200 Hz, the peak value of GLM4 is 0.07 larger than that of GRM4. These results indicate that for four-layer cylindrical cavity configurations, GLM4 exhibits superior absorption peaks. Figure 10b shows GLM5 exhibits significantly higher absorption peaks than GRM5 when the number of cavity layers increases to five. The absorption peak is 11.8% greater at 800 Hz and 15.1% greater at 1,300 Hz. Notably, GLM5 outperforms GRM5 across the frequency range over 1,250 Hz (495–1763 Hz), as shown in the pink-filled areas in Figure 10b. The half-absorption bandwidth of GLM5 is 1,300 Hz (120–144Hz, 687–848Hz, and 1,140–1876 Hz), which is 135 Hz wider than that of GRM5 (85–115Hz and 1,233–1965 Hz). GLM5 has an average sound absorption coefficient α ¯ of 0.39, which is 3% greater than that of GRM5 (0.36).

Comparison of the acoustic performance between GRM and GLM. **(a)** Optimized GRM4 and GLM4. **(b)** Optimized GRM5 and GLM5.
Figure 10. Comparison of the acoustic performance between GRM and GLM. (a) Optimized GRM4 and GLM4. (b) Optimized GRM5 and GLM5.

Results compared by Figures 10a,b show that the half-absorption bandwidth and average absorption coefficient of GLM are enhanced as the number of cavity layers increases. The advantage of the GLM design over GRM one becomes increasingly evident with increasing layer count. Regardless of whether four or five cavity layers are employed, the improved absorption capability of the GLM design is consistently attributable to the considerable thickening of the elastic medium, which facilitates superior absorption coefficients. As summarized in Table 3, the elastic medium thickness of GLM4 exceeds that of GRM4 by 43% (reaching 17.2 mm), and that of GLM5 surpasses its GRM counterpart by more than 58% (reaching 19.6 mm).

3.5 Experimental validation

In this section, we conduct an experiment in an impedance tube filled with water to validate the correctness of the simulation results. The experimental setup for underwater absorption measurement is shown in the “Instruments and samples” column in Figure 11 and consists of an impedance tube, transducer, power amplifier, signal conditioner, and data acquisition device. The inner diameter of the tube is 208 mm, corresponding to the reasonable measuring frequency band of 100 Hz∼10 kHz. The sample preparation procedure is shown in the “Sample preparation” column in Figure 11. Soft elastic materials are prepared through sulfurization and molding using liquid rubber (type Heisen Yoko 3,400), with a ρ of 1,100 kg/m3 and a Young’s modulus (1.62 + 0.49i) MPa. The properties of the elastic medium are assumed to be independent of the frequency. Two samples with different radii distributions, lattice constants and distances between cylindrical cavities are fabricated by the same procedure, which correspond to GRM5 and GLM5, labeled S1 and S2, respectively. All these samples have the same thickness 70 mm. The radius, lattice constant, and distances of the cavities in the samples coincide with those in Table 3. The acoustic performance test procedure of the samples is shown in the “Acoustic performance test” column in Figure 11. The tested sample is the prepared sample with a steel backing fed with sound waves generated by the transducer controlled by the power amplifier. Pressure signals are collected using hydrophones, amplified, and sent to the data acquisition device. Afterward, we can obtain the complex reflection coefficient from the sample surface, where the sound absorption coefficient can be calculated.

Sample preparation and acoustic performance test procedure and measurement instruments.
Figure 11. Sample preparation and acoustic performance test procedure and measurement instruments.

Figure 12 shows the absorption coefficients compared with the measured results. Five absorption peaks are observed in the experimental results for both samples. For GRM5, absorption peaks are located at approximately 100, 700, 1,200, 1,300 and 1,600 Hz, whose peak values are 0.57, 0.55, 0.62, 0.65 and 0.67, respectively. For GLM5, absorption peaks are located at 80, 700, 1,200, 1,400 and 1,600 Hz, whose peak values are 0.58, 0.75, 0.72, 0.71 and 0.74, respectively. The maximum peak value of GLM5 reaches 20% higher than that of GRM5. The average absorption coefficient for GLM5 is 0.49, which is 5% higher than that of GRM5 (0.44). One can see that the experimental absorption performance of GLM5 at peak frequencies and over the whole frequency range are superior to that of GRM5, which validates the simulation results.

Comparison of the absorption performance of the samples from the simulation and experimental results.
Figure 12. Comparison of the absorption performance of the samples from the simulation and experimental results.

As shown in Figure 12, the experimentally measured absorption coefficients exhibit a trend that closely follows the simulation results, confirming the validity of the numerical model. While noticeable deviations exist in the absolute values of the absorption peaks, the corresponding peak frequencies are found to be in excellent agreement between the two approaches. First, the deviation in the amplitudes of the peaks is due to the nonuniformity of the damping material in the sample, which contains small irregular holes caused by insufficient vacuum defoamation processes, as well as the nonflat surface of the sample. These irregular scatterers in the medium enhance multiple scattering, resulting in a higher sound absorption coefficient in the experimental results. Furthermore, manufacturing errors results in the structural dimensions of the samples being slightly different from those of the simulation model. Second, the difference may result from the simplified modulus of the soft rubber in the finite element model, which is assumed to be frequency independent. However, the real part and the loss factor of the complex modulus vary with the frequency in practice. In addition, the outer layer of the sample is wrapped with a layer of rubber for water tightness, which increases the content of soft elastic materials in the sample. Thus, the material’s loss of sound waves is enhanced, resulting in an increase in the sound absorption coefficient.

4 Conclusion

In summary, a GLM is proposed to achieve underwater low-frequency, broadband and high sound absorption. The proposed design is composed of multilayer cavities with gradient changes in lattice constants embedded in a soft elastic material. The mechanism is analyzed on the basis of F-P resonance theory, the displacement and power dissipation density field. This enhancement is primarily attributed to the fact that cavities with gradient lattice constants substantially reduce the thickness of the cavity layer within the GLM, thereby yielding a thicker elastic medium. This, in turn, strengthens the F-P resonance and promotes greater sound attenuation. The sound absorption performance and F-P resonance for cylindrical cavities with different radii, lattice constants, and distance arrangements are investigated. The results show that the greater the distance between the F-P resonant structures is, the stronger the F-P resonance and the greater the degree of sound absorption. Optimizing the GLM with 4 and 5 cavity layers has demonstrated that more layers of cylindrical cavities with gradient changes in lattice constants can be used to obtain a greater number of absorption peaks, resulting in high low-frequency multiband absorption performance. The peak values of GLM4 are more than 6.4% higher than those of GRM4, and the peak values of GLM5 are at least 11.8% higher than those of GRM5, which indicates that the more layers of the cavity there are, the greater the advantages of the GLM. Acoustic samples with 4 and 5 cavity layers are fabricated and tested in an impedance tube. The theoretical and experimental results are consistent, thus verifying the validity of the proposed design.

Nevertheless, the elastic modulus of the host medium is assumed to be constant within the frequency range for simplicity. The spectrum of the elastic modulus requires quantification in subsequent research. On the other hand, the structural design of the proposed model still relies mainly on experience and involves adjustments in various structural parameters and many repeated iterations, which is inefficient. Applying a neural network (He et al., 2024; Li et al., 2025; Xu et al., 2025) training method to optimize the design process of acoustic coating and reverse design can greatly reduce design costs. In addition, the sound absorption performance under hydrostatic pressure is not considered. However, the geometric deformation and modification of mechanical parameters under hydrostatic pressure affect the prediction of absorption performance in deep-water environments. Therefore, future studies should focus on practical submarine applications.

Although the GLM proposed in this paper addresses only the issue of the decline in sound absorption performance caused by a tandem cylindrical cavity with gradient changes in radius, it provides a new method for high-absorption acoustic coating design and has broad potential for use in underwater applications.

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Footnotes

Edited by: Shuqing Ma, National University of Defense Technology, China

Reviewed by: Wei Guo, National University of Defense Technology, China

Kunsheng Xing, National University of Defense Technology, China

Statements

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This work was funded by the Postdoctoral Fellowship Program of CPSF, (Grant No. GZC20252802) and National Key Research and Development Program of China (Grant No. JMRH-2024-0007-00), the Research Project of the Education Department of Hubei Province (Grant No. Q202513509), and the China Postdoctoral Science Foundation (Grant No. 2026M794865 and No. 2025M784483).

Data availability

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions. Requests to access the datasets should be directed to SW, liandanshibo@163.com.

Author contributions

SW: Formal Analysis, Writing – original draft, Data curation, Funding acquisition, Conceptualization, Writing – review and editing. QZ: Writing – original draft, Validation, Conceptualization. DW: Writing – review and editing, Methodology, Data curation, Investigation. ZS: Investigation, Writing – review and editing, Visualization. SZ: Writing – review and editing, Supervision.

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Citation

Wang, S., et al. (2026). A gradient-lattice-constant material for enhanced low-frequency and broadband underwater sound absorption. Frontiers in Materials, Volume 13. https://doi.org/
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13 - 2026

Copyright

Copyright © 2026 Wang, Zhang, Wu, Su and Zhang.

License: https://creativecommons.org/licenses/by/4.0/

*Correspondence: Sen Zhang

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