Broadband Low-Frequency Sound Absorption Enabled by a Rubber-Based Ni₅₀Ti₅₀ Alloy Multilayer Acoustic Coating

Abstract

Acoustic coatings play a vital role in enhancing the acoustic stealth of underwater structures across the full depth range and, especially, in the low-frequency band. However, existing small-scale acoustic coatings struggle to achieve low-frequency broadband sound absorption, which limits further performance improvements. Ni₅₀Ti₅₀ alloy, with their shape memory effect, hyper elasticity, and high damping properties, offer promising applications in vibration and noise control. In this study, a rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating is proposed, based on the sound absorption mechanism of rubber and the vibration and noise reduction mechanism of Ni₅₀Ti₅₀ alloy. The sound absorption characteristics of the proposed composite coating were obtained through analytical derivations, numerical simulations, and experimental investigations. The objective was to combine the high-frequency absorption capability of rubber and the low-frequency absorption characteristics of Ni₅₀Ti₅₀ alloy without increasing material dimensions, thereby introducing a novel approach for the design of the next generation of underwater acoustic coatings.

1. Introduction

In underwater acoustic engineering, acoustic stealth performance is a critical indicator of the acoustic characteristics of underwater structures [1]. As a key component, acoustic coatings must achieve stable absorption of underwater sound waves across the entire depth range [2]. However, traditional sound-absorbing structures face challenges in achieving low-frequency broadband absorption at small scales due to limitations in material size and resonance mechanisms. This shortcoming has driven the continuous development of multilayer sound-absorbing materials.

Currently, research on acoustic coatings encompass theoretical analysis, numerical simulation, and experimental studies. With regard to analytical investigations, Cervenka et al. [3] derived the reflection and transmission coefficients of fluid–solid interphase structures based on the attenuation wave theory, revealing the influence of inner material properties and geometric parameters on acoustic performance. Hladky-Hennion et al. [4] developed a periodic cylindrical structure model using the finite element method and analyzed the modulation effect of the periodic structure on sound wave scattering with the ATILA software. Ye et al. [5] employed the transfer matrix method to establish a simplified hull model, demonstrating the critical impact of a silencing layer’s structural parameters on insertion loss and noise reduction performance. Zhang et al. [6] proposed a semi-active coating containing a piezoelectric subwavelength array, revealing its low-frequency broadband sound absorption mechanism. Li et al. [7] embedded microperforated panels into a high-viscosity medium and validated the enhancement of sound absorption performance at the 115 kHz frequency band through the transfer matrix theory and experimental analysis. Li et al. [8] proposed a periodic multilayer solid inclusion structure and optimized its parameters using a genetic algorithm, achieving broadband sound absorption from 1780 to 8890 Hz, thereby providing an analytical foundation for novel coating design.

In the field of numerical simulation, research has increasingly focused on the systematic analysis of multiple structural parameters using the finite element method, aiming to uncover the physical mechanisms underlying acoustic behavior. Yao et al. [9] constructed a double-layer cylindrical hull model based on a three-dimensional finite element analysis, validating the noise reduction effectiveness of a composite cavity structure in double-hull sections. Zhong et al. [10] conducted finite element simulations to investigate the impact of arranging non-coaxial locally resonant scatterers in a viscoelastic plate on low-frequency sound absorption. The results showed that increasing the core offset leads to a broader effective absorption bandwidth. Liu et al. [11] established a multilayer material ellipsoidal cavity model using the COMSOL simulation platform and verified, through the transfer matrix method, that its low-frequency sound absorption performance outperformed that of single-layer structures. Zhong et al. [12] conducted simulation evaluations on the applicability of axisymmetric simplified models, indicating that hexagonal structures offer greater adaptability. A high loss factor was found to mitigate the impact of perforation rate on sound absorption, and cavity shapes were optimized using a genetic algorithm to enhance low-frequency absorption. Ke et al. [13] constructed a combined cavity model using COMSOL to simulate the effects of geometric parameters and micropore structures on sound absorption performance. The study indicated that micropores can enhance low-frequency sound insulation, with simulation results showing high consistencies with experimental data.

With regard to experimental studies, Cheng et al. [14] investigated the sound absorption performance of porous aluminum and found that pore size and thickness significantly affected its underwater acoustic response. Humphrey et al. [15] developed an acoustic pressure vessel system capable of simulating deep-sea environments at a depth of 85 m, enabling the acoustic testing of materials under varying temperature and pressure conditions. Cederholm [16] measured the acoustic performance of Alberich rubber in the 1530 kHz frequency band using a water-filled anechoic tank. Jiang [17] tested interpenetrating network polymer composite coatings under hydrostatic pressure conditions, achieving sound absorption coefficients exceeding 0.9. Fu et al. [18] established an underwater acoustic testing platform to evaluate the broadband sound absorption characteristics of carbon nanotube composites at 1.57 kHz.

To date, research on sound-absorbing materials has focused on single rubber-based and composite systems. Within single rubber materials, Wang [19] proposed the “free volume sound absorption mechanism” and investigated the sound absorption characteristics of modified silicone rubber and polyurethane under microstructural regulation. Li et al. [20] achieved multilayer broadband sound absorption by constructing microparticle impedance-matching structures. In the composite systems field, Garu and Chaki [21] studied the effect of carbon black content on the properties of chloroprene rubber, finding excellent insertion loss and echo reduction performance in the high-frequency range. Wang [22] explored the low-frequency sound absorption mechanisms of porous metal materials coupled with fluids. Gu et al. [23] expanded the sound absorption bandwidth by using a rubber-coupled steel block structure that relied on strong coupling and multiple scattering. Experimental results from these studies [19,20,21,22,23] showed that, over the frequency range of 600–2000 Hz, the mean sound absorption coefficient was 0.78. Although composite structures effectively enhance mid-to-high frequency performance, achieving broadband low-frequency sound absorption without increasing thickness remains a significant challenge.

In recent years, research achievements in the application of acoustic metamaterials and functionally graded materials in the field of underwater acoustic coatings have been remarkable. Zhou et al. [24] proposed an ultra-thin acoustic metamaterial with a rubber-coated spatial spiral channel. This metamaterial can achieve perfect sound absorption at 181 Hz, and its thickness is much smaller than the wavelength. The theoretical results are consistent with the simulation results. The rubber coating can significantly reduce the effective sound speed of the channel. It generates slow sound propagation and internal friction to meet impedance-matching requirements. By adjusting parameters, they also developed an ultra-broadband sound absorber with a thickness of 33 mm and a sound absorption frequency range of 365–900 Hz. This provides a new path for the design of such materials. Ranjbar and Bayer [25] designed an underwater acoustic metamaterial using polyurethane as the matrix. They conducted parameter optimization on conical, new gong-shaped, and hourglass-shaped air cavity structures through COMSOL simulation and mesh analysis. They also compared the multi-frequency sound absorption performance of these structures. The results showed that air cavity parameters have a significant impact on performance. Among them, the gong-shaped structure is more suitable for deep-water environments due to its small volume and broadband sound absorption characteristics. Croenme et al. [26] reviewed micro-inclusion and macro-inclusion coating technologies. They analyzed the acoustic performance and environmental adaptability of materials in combination with new design trends. They concluded that macro-inclusion coatings had higher efficiencies, new designs such as acoustic stealth cloaks had potential, and this technology could be extended to civil noise reduction. Jia et al. [27] conducted a review on underwater sound absorption coating technology under hydrostatic pressure. They pointed out that acoustic metamaterials such as local resonance and five-element metamaterials can be used in underwater structures to improve low-frequency sound absorption. Functionally graded materials can enhance the sound absorption and pressure resistance of coatings through impedance gradient design or combination with other structures. Both provide new ideas for related research. However, they also pointed out that there were few reviews from the perspective of hydrostatic pressure, and the low-frequency sound absorption performance of some structures still needed to be improved. Shi et al. [28] investigated the underwater sound absorption performance of acoustic metamaterials with multilayered locally resonant scatterers (M-LRAMs). Aiming at the problem that conventional locally resonant acoustic metamaterials (LRAMs) had a relatively narrow effective frequency range of sound absorption in the low-frequency band, they broadened and enhanced the sound absorption performance through coupled resonance generated by embedding multilayered locally resonant scatterers. They also analyzed the effects of various parameters on sound absorption performance, which showed that M-LRAMs had significant advantages over conventional LRAMs. In addition, Jia et al. [29] proposed an underwater sound absorption structure that embedded local resonators (LRs) into functionally graded materials (FGMs). They established an acoustic calculation model using the gradient finite element method (G-FEM) and verified the correctness of the model through the transfer matrix method (TMM). The analysis showed that this structure could improve low-frequency sound absorption performance and optimizing parameters could also enhance mid-high frequency sound absorption effects. The performance comparison of References [2,6,8,28] is shown in

Table 1. Comparison of coatings between the literature [2,6,8,28].

Type of Coatings	Thickness (mm)	First Sound Absorption Peak Frequency (Hz)	Bandwidth with α > 0.7 (Hz)	Manufacturability	Pretreatment
Traditional acoustic coating [2]	95	2610	2190~15,620	Easy	\
Piezoelectric shunted patches [6]	83.5	1520	500~3720 4240~10,000	Difficult	External circuit
Periodic multilayer inclusions [8]	80	1960	1780~8890	Difficult	\
Locally resonant acoustic metamaterials [28]	\	3800	3120~4360 8250~18,120	Difficult	\

.

One class of shape memory alloys (SMAs), such as Ni-Ti alloys, exhibit excellent energy dissipation and damping properties due to their unique martensite–austenite phase transformation behavior [30]. The damping performance primarily arises from energy dissipation caused by internal friction, thermally induced phase transformations, and stress-induced hysteresis loops [31]. Pan et al. [32] indicated that Ni₅₀Ti₅₀ alloys exhibit high loss factors under tensile and compressive loads. The dissipation capacity is nearly independent of temperature but closely related to strain rate. Heller et al. [33] identified vibration amplitude, pre-strain, and temperature as key parameters influencing SMA damping capacity. Biffi et al. [34], through numerical simulations and experiments, confirmed that CuZnAl alloy composite structures achieved a damping factor of up to 0.8 by leveraging phase transformation hysteresis and interfacial energy dissipation. Ahmadi and co-workers [35,36] validated, through thermo-mechanical coupled modeling and experimental work, that SMA thin films adjusted structural stiffness via austenite fraction variation, enhancing damping efficiency from 36% to 82%. Liao et al. [37] investigated the effects of annealing behavior on the martensitic transformation and damping peaks of ferromagnetic SMA, enhancing the material’s low-temperature damping capacity. Wang et al. [38] modeled the vibration behavior of SMA microstructures using the finite element method, demonstrating that their porous structures exhibited significant damping advantages due to localized phase transformations. Abirami et al. [39] developed an acoustic stress wave propagation model and experimentally confirmed that twin boundary motion in Ni₄₈Mn₃₂Ga₂₀ single crystals effectively suppressed vibrations. El Khatib et al. [40] introduced a thermo-mechanically coupled phase-field model that accurately predicted hysteresis behavior under various loading conditions, suitable for quasi-static SMA behavior modeling. Saedi et al. [41] reviewed the damping characteristics, energy dissipation mechanisms, and characterization methods of SMAs, emphasizing the decisive role of stress–strain hysteresis loops in energy dissipation.

In view of the low-frequency absorption limitations of traditional coatings and the damping potential of SMAs, in the present work, we propose a rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating. The absorption characteristics of this coating, within the 10–20,000 Hz frequency range, were investigated using theoretical analysis, numerical simulation, and underwater acoustic impedance tube experiments. By combining the low-frequency acoustic energy dissipation of the Ni₅₀Ti₅₀ alloy through phase transformation damping with the high-frequency sound absorption advantage of the rubber matrix, we propose that this coating has high potential to achieve low-frequency broadband sound absorption without an increase in size.

2. Theoretical Analysis

2.1. Sound Velocity of Rubber-Based Ni₅₀Ti₅₀ Alloy Multilayer Acoustic Coating

To theoretically analyze and calculate the sound absorption coefficient of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating, it is first necessary to obtain the longitudinal wave velocity of the material. In view of this, this section investigates the relationship between the longitudinal wave velocity of sound waves and the mechanical parameters of the material under the condition of the normal incidence of sound waves. Under normal incidence conditions, normal stress components exist inside the solid, while shear stress components are zero. To this end, a small portion of the periodic unit is selected for stress analysis [42], and its stress state is shown in Figure 1.

Let the displacement of the periodic unit along the x-axis be $\xi$, where T_xx represents the normal stress on the x-face, and T_yy and T_zz represent the normal stresses on the y-face and z-face, respectively. Since the incident acoustic pressure depends only on the x-coordinate and is uniformly distributed in the y–z plane, the stress distribution within the solid on the y–z plane is also uniform. This causes the periodic unit to reach stress equilibrium in the y and z directions, resulting in motion only along the x direction. Accordingly, the force expression on the periodic unit in the x direction can be written as

$$F_x=\left(T_{xx}+{\displaystyle \frac{\partial T_{xx}}{\partial x}}dx-T_{xx}\right)dydz={\displaystyle \frac{\partial T_{xx}}{\partial x}}dxdydz$$

(1)

According to Newton’s second law of motion, we have

$$\rho dxdydz{\displaystyle \frac{{\partial }^2\xi }{\partial t^2}}={\displaystyle \frac{\partial T_{xx}}{\partial x}}dxdydz$$

(2)

According to Hooke’s law, we have

$$T_{xx}=\left(\lambda +2\mu \right){\epsilon }_{xx}=\left(\lambda +2\mu \right){\displaystyle \frac{\partial \xi }{\partial x}}$$

(3)

Substituting Equation (3) into Equation (2) yields

$$\rho {\displaystyle \frac{{\partial }^2\xi }{\partial t^2}}=\left(\lambda +2\mu \right){\displaystyle \frac{{\partial }^2\xi }{\partial x^2}}$$

(4)

$${\displaystyle \frac{{\partial }^2\xi }{\partial t^2}}={\displaystyle \frac{\left(\lambda +2\mu \right)}{\rho }}{\displaystyle \frac{{\partial }^2\xi }{\partial x^2}}=c^2{\displaystyle \frac{{\partial }^2\xi }{\partial x^2}}$$

(5)

The longitudinal wave velocity of sound in the solid can be expressed as

$$c=\sqrt{{\displaystyle \frac{\lambda +2\mu }{\rho }}}$$

(6)

Next, we conduct an in-depth analysis of the intrinsic relationships among the elastic modulus, Lamé constants, and Poisson’s ratio.

When the periodic unit is subjected to stresses in the x, y, and z directions, these stresses cause deformation of the periodic unit along the x direction. The corresponding relationships are

$$\left\{\begin{array}{l}{{\epsilon }^{\prime }}_{xx}={\displaystyle \frac{T_{xx}}{E}}\\ {{\epsilon }^{″}}_{xx}=-{\displaystyle \frac{v{T}_{yy}}{E}}\\ {{\epsilon }^{‴}}_{xx}=-{\displaystyle \frac{v{T}_{zz}}{E}}\end{array}\right.$$

(7)

The deformation of the periodic unit in the y and z directions is

$$\left\{\begin{array}{l}{{\epsilon }^{\prime }}_{yy}={\displaystyle \frac{T_{yy}}{E}}\\ {{\epsilon }^{″}}_{yy}=-{\displaystyle \frac{v{T}_{xx}}{E}}\\ {{\epsilon }^{‴}}_{yy}=-{\displaystyle \frac{v{T}_{zz}}{E}}\end{array}\right.$$

(8)

$$\left\{\begin{array}{l}{{\epsilon }^{\prime }}_{zz}={\displaystyle \frac{T_{zz}}{E}}\\ {{\epsilon }^{″}}_{zz}=-{\displaystyle \frac{v{T}_{xx}}{E}}\\ {{\epsilon }^{‴}}_{zz}=-{\displaystyle \frac{v{T}_{yy}}{E}}\end{array}\right.$$

(9)

Then, the relative elongations in the x, y, and z directions are

$$\left\{\begin{array}{l}{\epsilon }_{xx}={{\epsilon }^{\prime }}_{xx}+{{\epsilon }^{″}}_{xx}+{{\epsilon }^{‴}}_{xx}={\displaystyle \frac{T_{xx}-v\left(T_{yy}+T_{zz}\right)}{E}}\\ {\epsilon }_{yy}={{\epsilon }^{\prime }}_{yy}+{{\epsilon }^{″}}_{yy}+{{\epsilon }^{‴}}_{yy}={\displaystyle \frac{T_{yy}-v\left(T_{xx}+T_{zz}\right)}{E}}\\ {\epsilon }_{zz}={{\epsilon }^{\prime }}_{zz}+{{\epsilon }^{″}}_{zz}+{{\epsilon }^{‴}}_{zz}={\displaystyle \frac{T_{zz}-v\left(T_{xx}+T_{yy}\right)}{E}}\end{array}\right.$$

(10)

Substituting Equation (10) into Equation (3) rewrites it as

$$\left\{\begin{array}{l}{T}_{xx}=\lambda \left({\epsilon }_{xx}+{\epsilon }_{yy}+{\epsilon }_{zz}\right)+2\mu {\epsilon }_{xx}\\ T_{yy}=\lambda \left({\epsilon }_{xx}+{\epsilon }_{yy}+{\epsilon }_{zz}\right)+2\mu {\epsilon }_{yy}\\ T_{zz}=\lambda \left({\epsilon }_{xx}+{\epsilon }_{yy}+{\epsilon }_{zz}\right)+2\mu {\epsilon }_{zz}\end{array}\right.$$

(11)

It can be further solved as

$$\left\{\begin{array}{l}\lambda ={\displaystyle \frac{Ev}{\left(1+v\right)\left(1-2v\right)}}\\ \mu ={\displaystyle \frac{E}{2\left(1+v\right)}}\end{array}\right.$$

(12)

Substituting Equation (12) into Equation (6) yields

$$c=\sqrt{{\displaystyle \frac{\frac{Ev}{\left(1+v\right)\left(1-2v\right)}+\frac{E}{\left(1+v\right)}}{\rho }}}=\sqrt{{\displaystyle \frac{E\left(1-v\right)}{\rho \left(1+v\right)\left(1-2v\right)}}}$$

(13)

Equation (13) shows that the longitudinal wave velocity in the material can be calculated using the material’s elastic modulus, density, and Poisson’s ratio. In the sections that follow, the longitudinal wave velocity through the material was calculated using Equation (13).

2.2. Four Terminal Grid Theory Modeling

A schematic drawing of the multilayer acoustic coating proposed in this study is given in Figure 2. The coating consists of five layers, labeled from top to bottom with thickness h_i (i = 1, 2, 3, 4, 5). For a given layer, its elastic modulus, Poisson’s ratio, loss factor, and density are Eᵢ, νᵢ, ηᵢ, and ρᵢ, respectively. Among these, the first and fifth layers are homogeneous; the second layer is a non-homogeneous layer incorporating periodically arranged truncated-cone cavities, with upper and lower radii denoted as r₁ and r₂, respectively. And the fourth layer is a non-homogeneous layer containing periodically distributed cylindrical cavities of radius r₃. Of particular importance, the third layer corresponds to the introduced Ni₅₀Ti₅₀ alloy sheet. In the design of the periodic structure, a square with side length L is used as the basic unit, whose outline is indicated by the blue dashed line. In the figure, P_i, P_r, and P_t represent the incident sound pressure, the reflected sound pressure, and the transmitted sound pressure, respectively. To facilitate spatial positioning, a global Cartesian coordinate system is established at the top of the acoustic coating. This coordinate system is based on the arrangement direction of the periodic cavity array, where the x-axis and y-axis are parallel to the two orthogonal directions of the array, and the z-axis is perpendicular to the xy-plane, pointing downward. When a plane wave with a sound pressure amplitude of p_i impinges normally on the upper surface of the acoustic coating, the derivation of the corresponding reflection and transmission coefficients for the reflected and transmitted waves with amplitudes Pᵢ and Pₜ, respectively, is presented in the two sub-sub-sections that follow.

2.2.1. Uniform Layer Transfer Matrix

From Figure 2, considering layers 1, 3, and 5 as single homogeneous layers without cavities, the transmission relationship between the sound pressure p and particle velocity v at the adjacent front and back interfaces is given by

$$\left[\begin{array}{l}{p}_{1,i}\\ v_{1,i}\end{array}\right]=\left[\begin{array}{cc}\cos \left(k_i{h}_i\right)& j{\rho }_i{c}_i\sin \left(k_i{h}_i\right)\\ {\displaystyle \frac{j\sin \left(k_i{h}_i\right)}{{\rho }_i{c}_i}}& \cos \left(k_i{h}_i\right)\end{array}\right]\left[\begin{array}{l}{p}_{2,i}\\ v_{2,i}\end{array}\right]=A_i^{pre}\left(h_i\right)\left[\begin{array}{l}{p}_{2,i}\\ v_{2,i}\end{array}\right]$$

(14)

In the formula, the subscript i denotes the i-th layer of the acoustic covering layer; subscripts 1 and 2 represent the front and back ends of the single homogeneous layer, respectively. The symbols ρ, c, k, h, and A_i^pre represent the density, longitudinal wave speed, longitudinal wave number, layer thickness, and transmission relation of the homogeneous layer, respectively. The symbol j is the imaginary unit. The characteristic impedance ρc of the medium in the homogeneous layer.

Longitudinal wave speed can be expressed as

$$c=\sqrt{{\displaystyle \frac{E\left(1-v\right)}{\rho \left(1+v\right)\left(1-2v\right)}}}$$

(15)

$$E=\overline{E}\left(1+j\eta \right)$$

(16)

In the formula, E, v, and η represent the material’s complex elastic modulus, Poisson’s ratio, and damping loss factor, respectively.

Therefore, under the action of force, the transfer matrix of the homogeneous layer is

$$A_i=S_e\otimes A_i^{pre}\left(h\right)$$

(17)

$$S_e=\left[\begin{array}{cc}1& s_b\\ 1/s_b& 1\end{array}\right]$$

(18)

In the equation, the symbol $\otimes$ represents the Hadamard product between matrices, and ${\mathrm{s}}_{\mathrm{b}} = {\mathrm{L}}^2$ denotes the cross-sectional area of a single periodic unit cell.

2.2.2. Non-Uniform Layer Transfer Matrix

In a non-homogeneous medium layer with cavity distribution, the acoustic properties at the interfaces exhibit specific boundary constraints. Specifically, the acoustic pressure field p is discontinuous across the interface, while the acting force F remains continuously distributed. This behavior can be mathematically described by the following boundary conditions:

(1)

The particle vibration velocity v in the medium remains continuous at the interface;

(2)

The product of the stress tensor component and the acting area, s∙p, exhibits continuity across the interface. Propagation characteristics of the acoustic wave are similar to those of a high-viscosity fluid propagating in a waveguide with a non-uniform cross-section. The corresponding waveguide equation can thus be expressed as

$${\displaystyle \frac{{\partial }^2\xi }{\partial z^2}}+{\displaystyle \frac{1}{s}}{\displaystyle \frac{\partial s}{\partial z}}{\displaystyle \frac{\partial \xi }{\partial z}}+k^2\xi =0$$

(19)

In this equation, $\xi$ and s represent the particle displacement and cross-sectional area of the non-homogeneous layer, respectively, both of which are functions of z. As a result, the expression is a nonlinear equation. To convert it into a linear equation with an analytical solution, the cross-sectional area s(z) of the non-homogeneous layer must satisfy the following condition [43]:

$${\displaystyle \frac{{\left(\sqrt{s}\right)}^{″}}{\sqrt{s}}}={\mu }^2\left(\mu \mathrm{is} \mathrm{a} \mathrm{constant}\right)$$

(20)

In the non-homogeneous layer containing truncated cone-shaped cavities, a conical waveguide $s\left(z\right) = \sqrt{az+b}$ can be used to approximate the cross-sectional area of the cavity. The transfer relationship between F₁, F₂, v₁, and v₂ is given by

$$\left[\begin{array}{l}{F}_1\\ v_1\end{array}\right]=\left[\begin{array}{cc}{b}_{2,11}& b_{2,12}\\ b_{2,21}& b_{2,22}\end{array}\right]=B_2\left[\begin{array}{l}{F}_2\\ v_2\end{array}\right]$$

(21)

The specific expressions for b_2,11, b_2,12, b_2,21, and b_2,22 are as follows:

$$b_{2,11}=\sqrt{{\displaystyle \frac{s_1}{s_2}}}\left[\cos \left(K_2{h}_2\right)+{\displaystyle \frac{1}{K_2{h}_2}}\left(\sqrt{{\displaystyle \frac{s_2}{s_1}}}-1\right)\sin \left(K_2{h}_2\right)\right]$$

(22)

$$b_{2,12}={\displaystyle \frac{j{\rho }_2{c}_2\sqrt{s_1{s}_2}}{k_2}}\left\{\begin{array}{l}{\displaystyle \frac{\cos \left(K_2{h}_2\right)}{h_2}}\left(2-\sqrt{{\displaystyle \frac{s_1}{s_2}}}-\sqrt{{\displaystyle \frac{s_2}{s_1}}}\right)\\ +K_2\sin \left(K_2{h}_2\right)\left[1-{\left({\displaystyle \frac{1}{K_2{h}_2}}\right)}^2\left(1-\sqrt{{\displaystyle \frac{s_1}{s_2}}}\right)\right]\end{array}\right\}$$

(23)

$$b_{2,21}={\displaystyle \frac{j{k}_2}{K_2{\rho }_2{c}_2\sqrt{s_1{s}_2}}}\sin \left(K_2{h}_2\right)$$

(24)

$$b_{2,22}=\sqrt{{\displaystyle \frac{s_2}{s_1}}}\left[\cos \left(K_2{h}_2\right)-{\displaystyle \frac{1}{K_2{h}_2}}\left(1-\sqrt{{\displaystyle \frac{s_1}{s_2}}}\right)\sin \left(K_2{h}_2\right)\right]$$

(25)

In this equation, B₂ represents the transfer matrix of the second non-uniform layer under force conditions and $K_2 = \sqrt{k_2^2-{\mu }^2}$; $s_1 ={ L}^2-\pi {r_1}^2$ and $s_2 = L^2-\pi {r_2}^2$ denotes the cross-sectional areas at the interfaces of the non-uniform layer, where r₁ and r₂ are the smaller and larger diameters of the frustum-shaped cavity, respectively.

The fourth layer, which contains cylindrical cavities, is a special case of the frustum-shaped cavity non-uniform layer, where $s_1 = s_2 = s_3 = L^2-\pi r_3^2$ and K = k. Under the action of force, the corresponding transfer matrix is

$$B_4=\left[\begin{array}{cc}\cos \left(k_4{h}_4\right)& j{\rho }_4{c}_4{s}_3\sin \left(k_4{h}_4\right)\\ {\displaystyle \frac{j\sin \left(k_4{h}_4\right)}{{\rho }_4{c}_4{s}_3}}& \cos \left(k_4{h}_4\right)\end{array}\right]$$

(26)

Then, the global transfer matrix (2 × 2) of the acoustic coating under force conditions can be recursively expressed as

$$T=A_1\left(h_1\right)\times B_2\left(h_2\right)\times A_3\left(h_3\right)\times B_4\left(h_4\right)\times A_5\left(h_5\right)$$

(27)

Therefore, the reflection coefficient and the sound absorption coefficient of the coating are given by

$$r={\displaystyle \frac{Z_{in}-Z_w}{Z_{in}+Z_w}}$$

(28)

$$\alpha =1-r\cdot r^{\ast }$$

(29)

$$Z_{in}={\displaystyle \frac{T_{11}{F}_6+T_{12}{v}_6}{s_b\left(T_{21}{F}_6+T_{22}{v}_6\right)}}$$

(30)

In the formula, the superscript * denotes the complex conjugate of the reflection coefficient, and Z_w = ρ_wc_w is the characteristic impedance of the water medium in layer 0. Therefore, according to Equation (29), the theoretical sound absorption coefficient of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating can be calculated.

3. Sound Absorption Characteristics of Rubber-Based Ni₅₀Ti₅₀ Alloy Multilayer Acoustic Coatings

3.1. Verification of Theoretical Analysis and Structural Design Effectiveness

To verify the accuracy of the results obtained from the theoretical analysis, the finite element simulation software COMSOL Multiphysics 6.1 was used to obtain the sound absorption coefficient. According to the structure described above, a unit cell with a length L = 30 mm was designed. The simulation results were compared with those obtained from the theoretical analysis. The values of the material parameters used are given in

Table 2. Material parameters of rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating model [6].

Layer Number	Material Abbreviation	Layer Thickness (mm)		Elastic Modulus (Pa)		Poisson’s Ratio		Loss Factor		Density (kg/m3)
1	IIR	h1	5	E1	4.24 × 107	v1	0.47	η1	1.7	ρ1	1005
2	NBR	h2	30	E2	8.79 × 108	v2	0.37	η2	0.5	ρ2	1600
3	Ni50Ti50 Alloy	h3	5	E3	9 × 1010	v3	0.33	η3	0.17	ρ3	6450
4	NBR	h4	10	E4	8.79 × 108	v4	0.37	η4	0.5	ρ4	1600
5	SBR	h5	25	E5	9.60 × 107	v5	0.48	η5	1.0	ρ5	1039

.

The material composition of this multilayer acoustic coating is as follows: layer 1 is made of IIR (butyl rubber), layers 2 and 4 use NBR (nitrile rubber), and layer 5 is composed of SBR (styrene–butadiene rubber). Notably, the middle layer 3 is made of Ni₅₀Ti₅₀ alloy. The model established in COMSOL Multiphysics 6.1 finite element simulation software is shown in Figure 3.

During the finite element simulation process, the model was configured as follows:

(1)

In the finite element simulation model, water and the cavities are defined as pressure acoustics domains, while the other parts are set as solid mechanics domains. Considering that the energy loss of low-frequency sound waves below 3 kHz propagating in the water medium can be neglected, a perfectly matched layer (PML) is applied at the outer boundary of the water domain to simulate an infinitely large incident field. The PML truncates the computational domain and allows the simulation of infinite incident conditions within a finite region. It also suppresses boundary reflections, reduces computational cost, eliminates boundary effects, and improves accuracy.

(2)

Floquet periodic boundary conditions are applied to all boundaries of the unit cell in the x and y directions, and the periodic vectors are defined. This introduces a phase-shifted periodic constraint at the boundaries, allowing the response of an infinite periodic array under any incident wave vector to be simulated using only a single unit cell. In this way, an infinitely large acoustic covering layer structure can be modeled, greatly reducing the computational cost. To simulate a rigid backing condition, fixed constraints are imposed on the bottom boundary of the solid mechanics domain.

(3)

The model is discretized using free tetrahedral elements. The mesh size is determined based on the one-sixth wavelength criterion [4]; mesh elements consist of nodes. For linear elements, the nodes are located at the vertices. This model has a total of 87,295 mesh elements, 93,482 nodes, and 213,708 degrees of freedom. In COMSOL, the wave equation uses second-order polynomial interpolation by default. Quadratic elements have additional nodes along the edges, which allow more accurate wave resolution. For free-field wave problems, about 10~12 nodes per wavelength are generally required. Therefore, when using quadratic elements, roughly 5~6 elements per wavelength are needed, meaning that the maximum element size in each region does not exceed one-sixth of the acoustic wavelength in that region.

The acoustic–structure coupling boundary equation is given as

$$-n\cdot (-{\displaystyle \frac{1}{{\rho }_c}}(\nabla {\rho }_t-q_d))=-n\cdot u_{tt},F_A=p_{t}n$$

(31)

In the equation, u_tt denotes the structural acceleration, n is the surface normal vector, p_t is the total acoustic pressure, q_d is the dipole domain acoustic source, and F_A is the load acting on the structure.

The discretized form of the coupled acoustic–structure governing equations can be expressed in the following matrix form:

$$\left[\begin{array}{cc}{M}_s& 0\\ {\rho }_{f}R& M_f\end{array}\right]\left\{\begin{array}{l}{\ddot{U}}^e\\ {\ddot{P}}^e\end{array}\right\}+\left[\begin{array}{cc}{C}_s& 0\\ 0& C_f\end{array}\right]\left\{\begin{array}{l}{\dot{U}}^e\\ {\dot{P}}^e\end{array}\right\}+\left[\begin{array}{cc}{K}_s& -R\\ 0& K_f\end{array}\right]\left\{\begin{array}{l}{U}^e\\ P^e\end{array}\right\}=\left\{\begin{array}{l}{F}_s\\ F_f\end{array}\right\}$$

(32)

In the equation, M, C, and K represent the element mass matrix, stiffness matrix, and fluid–structure coupling matrix of the fluid domain, respectively; the subscripts s and f denote the solid and fluid domains; U^e is the nodal displacement vector in the solid domain, P^e is the nodal sound pressure vector, and F_s and F_f are the external loads applied at the nodes of the solid and fluid domains, respectively; R stands for the coupling matrix at the fluid–structure interface; and ρ_f is the density of the fluid medium.

Let S denote the fluid medium and the acoustic coating layer’s fluid–structure coupling boundary. When a plane acoustic wave with an amplitude of P_in is incident, the scattered sound pressure can be calculated using Equation (32), and the absorption coefficient of the structure can be determined using Equation (33).

$$P_s={\displaystyle \frac{{\displaystyle {\int }_s{P}_{s}ds}}{{\displaystyle {\int }_{s}1ds}}}$$

(33)

$$\alpha =1-{\left|{\displaystyle \frac{P_s}{P_{in}}}\right|}^2$$

(34)

Through the above steps, numerical simulation was conducted on the sound absorption coefficient curve within the frequency range of [10, 20,000] Hz, and the results were compared with those obtained using the theoretical analysis. As shown in Figure 4, there is an excellent agreement between the two sets of results.

After verification of the results obtained from the theoretical analysis, to further confirm the role of the Ni₅₀Ti₅₀ layer in the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating, the influence of the Ni₅₀Ti₅₀ alloy layer on the sound absorption characteristics of the composite system was analyzed by comparing the sound absorption performance between the structure with the Ni₅₀Ti₅₀ alloy layer (original structure) and that without the Ni₅₀Ti₅₀ alloy layer (control group). The results are presented in Figure 5.

As shown in Figure 5, there is a significant difference between the sound absorption curve of the rubber-based multilayer acoustic coating with the Ni₅₀Ti₅₀ alloy layer and that of the control group without the Ni₅₀Ti₅₀ alloy layer. The specific manifestations are as follows: The first sound absorption peak of the original structure with the Ni₅₀Ti₅₀ alloy layer is significantly higher than that of the control group, indicating that the introduction of the Ni₅₀Ti₅₀ alloy layer enhances the sound energy dissipation efficiency in specific frequency bands and improves the peak sound absorption capacity through its unique phase transition damping characteristics. The overall sound absorption curve of the original structure shifts to the low-frequency direction compared with that of the control group, suggesting that the addition of the Ni₅₀Ti₅₀ alloy layer effectively expands the sound absorption coverage of the multilayer structure in the low-frequency band, which is directly related to the absorption characteristics of Ni₅₀Ti₅₀ alloy layer for low-frequency sound waves. When a sound absorption coefficient of 0.8 was used as a benchmark to calculate the sound absorption bandwidth, it was found that, in the 10–20,000 Hz frequency range, the effective bandwidth of the original structure with the Ni₅₀Ti₅₀ alloy layer (where the sound absorption coefficient is greater than 0.8) covered 1040 Hz to 20,000 Hz, with a width of 18,960 Hz. In contrast, the sound absorption coefficient of the control group in the 2420–4200 Hz frequency band is significantly lower than 0.8, showing an obvious sound absorption valley. These results fully demonstrate that the introduction of the Ni₅₀Ti₅₀ alloy layer not only makes up for the deficiency in traditional rubber-based structures in low-frequency sound absorption but also achieves efficient sound absorption in a wide frequency band through the synergistic effect with the rubber matrix, thus verifying the rationality of this structure.

3.2. Experiment Validation

The absorption coefficient measurement of the acoustic coating samples fabricated in this study was conducted in accordance with the specific requirements of the national standard GBJ88-85 “Measurement Range of Absorption Coefficient and Acoustic Impedance Ratio by Standing Wave Tube Method” [44]. The detailed testing system setup is shown in Figure 6.

The testing principle of the system shown in Figure 6 is as follows: First, the operator selects the appropriate acoustic excitation mode through the computer’s human–machine interface. Then, the signal generator in the B&K front-end equipment produces the corresponding electrical signal. This electrical signal is modulated and amplified by a power amplifier, which drives the piezoelectric transducer to generate a plane-wave sound field. In terms of acoustic field measurement, the system uses a hydrophone array to simultaneously collect sound pressure information of the incident and reflected waves. It employs multi-channel multiplexing technology to convert analog signals into digital signals, which are then transmitted to the data acquisition module. Finally, the acquisition module calculates the absorption coefficient and sends the results back to the computer for display and storage. It should be noted that the thick steel plate is used to simulate the rigid backing of the acoustic coating. To minimize amplitude and phase mismatch errors between the sound pressure signals of the two hydrophones, the plane-wave calibration method is employed to perform the consistency calibration of the measurement system, ultimately achieving a matching accuracy with an amplitude deviation of less than 3% and a phase deviation of less than 1°.

When a plane-wave sound field is incident on the surface of the sound-absorbing material inside the underwater acoustic impedance tube, the sound pressure signals detected by the two hydrophones positioned at the measurement locations, denoted as P₁ and P₂, can be expressed, respectively, as

$$\left\{\begin{array}{l}{P}_1=P_i{e}^{-jk{x}_1}+P_r{e}^{jk{x}_1}\\ P_2=P_i{e}^{-jk{x}_2}+P_r{e}^{jk{x}_2}\end{array}\right.$$

(35)

In the formula, (P₁, x₁) and (P₂, x₂) represent the sound pressure values and position coordinates at hydrophone 1 and hydrophone 2, respectively. According to the theory of sound wave reflection, the sound reflection coefficient can be expressed as

$$r={\displaystyle \frac{P_r}{P_i}}={\displaystyle \frac{H_{12}-H_1}{H_R-H_{12}}}{e}^{-2jk{x}_1}$$

(36)

where ${\mathrm{H}}_1 = {\mathrm{e}}^{-\mathrm{jk}({\mathrm{x}}_2-{\mathrm{x}}_1)}$ and ${\mathrm{H}}_{\mathrm{R}} ={ \mathrm{e}}^{-\mathrm{jk}({\mathrm{x}}_2-{\mathrm{x}}_1)}$.

Thus, the sound absorption coefficient can be expressed as

$$\alpha =1-r\cdot r^{\ast }$$

(37)

According to the previously described model, the test specimens were prepared and assembled, with the specific assembly process shown in Figure 7. During the assembly process, since epoxy resin has excellent interfacial wettability with both the Ni₅₀Ti₅₀ alloy and the base materials, and its modulus matches well with them, which can reduce the risk of interfacial debonding, the components were bonded and combined with resin in the following order: uniform layer 1 (IIR), frustum cavity layer 2 (NBR), Ni₅₀Ti₅₀ alloy layer 3, cylindrical cavity layer 4 (NBR), and rubber layer 5 (SBR), ensuring a tight connection. The rubber materials were supplied and processed by Wenji Hardware & Plastic Products Co., Ltd., Shenzhen, China, while the Ni₅₀Ti₅₀ alloy was provided by Gangda Supply Chain Group Co., Ltd. Shanghai, China. It is particularly important to note that during the assembly process of the multilayer rubber-based Ni₅₀Ti₅₀ alloy acoustic coating samples, a specialized sleeve can be used for positioning and fixation to ensure the coaxial accuracy of each layer’s structure. The inner diameter of the sleeve should be designed to match the outer diameter of the acoustic coating, while its outer diameter must be compatible with the inner cavity dimensions of the testing apparatus (underwater acoustic impedance tube). To clearly observe the cavities of the experimental samples, the surfaces of the assembled components were also processed to expose the cavities.

After the assembly of the acoustic coating layer specimen, the experimental tests were conducted at the China Ship Scientific Research Center (702 Research Institute). The main equipment used to measure the sound absorption coefficient during the experiment was an underwater acoustic impedance tube (as shown in Figure 8), with the corresponding instrument parameters detailed in

Table 3. Information table of main instruments and equipment for underwater impedance tube.

Equipment Name and Type	Company	Location
2692-A Measuring Amplifier	Spectris Instrumentation and Systems Shanghai Ltd.	Shanghai China
3560D Signal Analyzer	Brüel & Kjær	Wuxi China
MFT-120 Underwater Intermediate Frequency Tube	Spectris Instrumentation and Systems Shanghai Ltd.	Shanghai China
YB-150A Pressure Gauge	Shanghai Automation Instrument Co., Ltd.	Shanghai China

. It should be noted that, as shown in Figure 5, the three sound absorption peaks of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating all lie within the frequency range of [10, 20,000] Hz. However, considering that the maximum testing frequency of the available equipment is 10,000 Hz and measurements beyond 8000 Hz may suffer from distortion, the underwater acoustic impedance tube tests were conducted under standard environmental conditions (normal temperature and pressure) over the frequency range of [10, 8000] Hz. A comparative analysis between the experimental and numerical results as shown in Figure 9.

In Figure 9, two distinct absorption peaks are observed within the full [10, 8000] Hz frequency range under atmospheric pressure. Specifically, the effective bandwidth with an acoustic absorption coefficient greater than 0.8 covers 786–7152 Hz and 7483–8000 Hz. The two sound absorption peaks of the experimental curve and the simulation curve at [1006, 1400] Hz show a slight mutual shift, mainly due to the fact that the experimental material adopts a finite-sized structure. Restricted by the finite-scale effect, the corresponding surface input impedance changes slightly. Secondly, the experimental test data shows a higher sound absorption performance below 2000 Hz (low frequency), which is mainly attributed to the acoustic leakage phenomenon under real conditions. The experimental absorption coefficient curve of the rubber-based SMA composite acoustic coating fluctuates closely around the simulated curve, indicating that the designed acoustic coating is reasonable and effective.

3.3. Prediction of Sound Absorption Law of Rubber-Based Ni₅₀Ti₅₀ Alloy Multilayer Acoustic Coating

For the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating shown in Figure 3, this section systematically analyzes its sound absorption performance using the method of controlling variables. By sequentially varying the geometric and material parameters of each sublayer and recording the trends in the sound absorption coefficient, the energy dissipation mechanisms and their interactions within the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating are further revealed.

3.3.1. Influence of Unit Cell Length

To investigate the influence of unit cell length on the sound absorption performance of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating, a single-variable approach is adopted. Using an increment of 10 mm as the basis, the effect of cell lengths L = (30, 40, 50) mm on the global sound absorption characteristics is obtained, as shown in Figure 10.

As shown in Figure 10, with the increase in unit cell length, the first sound absorption peak gradually shifts toward higher frequencies. This phenomenon can mainly be attributed to two factors: On one hand, as the unit cell length increases, the overall stiffness of the acoustic cladding is enhanced. A longer unit cell forms a more rigid structure, which in turn affects the interaction mode between sound waves and the material, leading to an increase in the system’s resonance frequency. Since the resonance frequency determines the position of the sound absorption peak, this pushes the sound absorption peak to shift toward higher frequencies. On the other hand, an increase in unit cell length results in a reduction in the relative volume of the cavity, weakening its resonance dissipation capacity for low-frequency sound waves, which also promotes the shift in the sound absorption peak toward higher frequencies.

3.3.2. Influence of Layer Thickness

The rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating is composed of multiple distinct layers, each of which contributes uniquely to the overall sound absorption performance. In the process of achieving global sound absorption, each layer exhibits its own inherent acoustic characteristics and damping behavior. To analyze the influence mechanism of structural layer thickness on sound absorption characteristics, based on the research principle of the control variable method, a thickness variation interval of 10 mm was selected. The thicknesses of layers 1, 2, 3, 4, and 5 were chosen as h₁ = (5, 15, 25) mm, h₂ = (30, 40, 50) mm, h₃ = (5, 15, 25) mm, h₄ = (10, 20, 30) mm and h₅ = (25, 35, 45) mm, respectively. Based on the material parameters provided in Table 1, the variation patterns of the sound absorption coefficient of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating was obtained through theoretical analysis. The results are shown in Figure 11.

Figure 11 shows that the variation trend of the sound absorption coefficient curves of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating is similar as the thickness of each layer increases, with the first sound absorption peak exhibiting a shift toward lower frequencies. This phenomenon can be attributed to the following two factors: (1) the increase in the thickness of the acoustic covering layer leads to a corresponding extension of the propagation distance of sound waves within the medium, thereby enhancing the dissipation effect of the sound waves; and (2) the overall resonant energy-dissipating sound absorption frequency of the acoustic coating decreases with the increase in its thickness. However, in practical underwater structures, an excessively large thickness of the acoustic coating not only leads to increases in the weight and volume of the underwater structure, but also the density of nickel–titanium alloy itself further amplifies the weight effect. This causes a rise in the total weight of the underwater structure and impairs its flexibility and maneuverability. Therefore, in engineering design, the thickness of the acoustic coating must be reasonably designed to balance the relationship between sound absorption performance and the overall performance of the underwater structure.

3.3.3. Influence of Cavity Radius

The geometric parameters of the cavity structure have a dual effect on the acoustic performance of the coating: on the one hand, the geometric characteristics of the cavity directly influence the sound pressure and vibration velocity at the interfaces between adjacent material layers. By regulating the phase coupling relationship between the interface sound pressure field and the particle vibration velocity field, the reflection and transmission behavior of sound waves are significantly affected. On the other hand, the cavity structure alters the propagation path and energy distribution of sound waves within the heterogeneous medium, achieving selective attenuation of sound waves in specific frequency bands through the induction of local resonance effects and multiple scattering phenomena. To investigate the influence of combined cavity radii on the sound absorption characteristics of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating, we used various values of the small radius r₁ = (6, 8, 10) mm, the large radius r₂ = (10, 12, 14) mm, and the cylindrical radius r₃ = (8, 10, 12) mm with an interval of 2 mm. The results are shown in Figure 12.

As shown in Figure 12, increasing the cavity radius results in a generally similar trend in the sound absorption coefficient curves, although the underlying mechanisms differ. Specifically, as r₂ and r₃ increase, the first sound absorption peak shows a slight shift toward lower frequencies. This is because the sound wave propagation path inside the cavity becomes longer, increasing the number of reflections and interferences within the cavity, which leads to a decrease in the resonance frequency. Conversely, as r₁ increases, the peak sound absorption coefficient decreases due to a weakened ability of the cavity to facilitate waveform conversion of the sound waves.

3.3.4. Influence of Elastic Modulus

The elastic modulus of the material is one of the key parameters affecting sound absorption performance, and its variation can significantly alter the acoustic impedance-matching characteristics. By designing composite structures with materials of different elastic moduli, the synergistic effect of multiple energy dissipation mechanisms can be achieved. To analyze the influence of elastic modulus on the sound absorption characteristics of the rubber–Ni₅₀Ti₅₀ alloy multilayer acoustic coating, the elastic moduli of E₁ = (42.4, 92.4, 142.4) MPa, E₂ = (479, 679, 879) MPa, E₃ = (10, 90, 170) GPa, E₄ = (679, 879, 1079) MPa, and E₅ = (96, 196, 296) MPa were selected. The results are shown in Figure 13.

According to Figure 13, it can be observed that as the elastic modulus gradually increases, the first sound absorption peak of the acoustic coating shifts toward higher frequencies. This phenomenon mainly arises because the increase in the material’s elastic modulus enhances the overall structural stiffness of the acoustic covering layer, thereby raising the overall resonant absorption frequency. Generally speaking, using materials with a lower elastic modulus helps achieve impedance matching between the acoustic coating and the seawater medium. However, considering that underwater structures are subjected to large hydrostatic pressure in deep-water environments, a very low elastic modulus will markedly reduce the compressive strength of the acoustic coating. Therefore, in practical engineering design, it is essential to consider both the acoustic and mechanical performances of the coating in a bid to obtain the optimal balance between these two criteria.

3.3.5. Influence of Poisson’s Ratio

In the multilayer acoustic coating, the Poisson’s ratios of materials play a significant role in their acoustic performance. As one of the fundamental mechanical parameters, Poisson’s ratio not only determines the distribution characteristics of vibration modes within the structure but also significantly affects the propagation characteristics of elastic waves. To investigate the effect of changes in Poisson’s ratio on the sound absorption performance of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating, based on the single-variable principle, the Poisson’s ratios were set as v₁ = (0.27, 0.37, 0.47), v₂ = (0.37, 0.47, 0.57), v₃ = (0.23, 0.33, 0.43), v₄ = (0.37, 0.47, 0.57), and v₅ = (0.28, 0.38, 0.48). The results are shown in Figure 14.

Based on the data analysis in Figure 14, a regular pattern can be observed in the sound absorption performance as the Poisson’s ratios of the materials varies. Specifically, with the increase in Poisson’s ratio in each structural layer, the first absorption peak in the sound absorption coefficient curve tends to shift toward higher frequency regions. The mechanism behind this phenomenon mainly involves two aspects: on the one hand, the increase in the material’s Poisson’s ratio significantly enhances the dynamic bending stiffness of the rubber layer; on the other hand, the geometric characteristics of the cavity units modulate the system’s resonant properties, thereby further raising the structure’s resonant frequency.

3.3.6. Influence of Loss Factor

In the design of acoustic materials, the damping characteristics of rubber play a crucial role in its performance. By increasing the material’s loss factor, its ability to absorb sound waves can be significantly enhanced, thereby optimizing the overall acoustic performance. This improvement mechanism mainly originates from the increased internal frictional losses within the material, which convert more acoustic energy into heat and dissipate it. To investigate the effect of variations in the loss factor on the sound absorption performance of the multilayer acoustic coating based on the single-variable principle, the values were set as η₁ = (1.7, 2.7, 3.7), η₂ = (0.5, 1.0, 1.5), η₃ = (0.17, 1.17, 2.17), η₄ = (0.5, 1.0, 1.5), and η₅ = (1.0, 1.5, 2.0). The results are shown in Figure 15.

As shown in Figure 15, the peak frequency of the sound absorption curve of the acoustic coating layer is minimally affected by the material’s damping loss factor; its primary influence is on the amplitude of the absorption peaks. As the damping loss factor increases, the sound absorption performance of the acoustic coating layer before the second absorption peak initially improves and then gradually decreases. This phenomenon is mainly attributed to the increased damping loss factor causing a longer strain lag behind stress. Under the same vibration amplitude, more energy is required to overcome resistance, and ultimately, the acoustic energy is dissipated in the form of heat. However, this also enhances the acoustic coating layer’s ability to reflect sound waves. Based on this characteristic, in engineering, it is necessary to consider the material’s damping loss factor and optimize its value range to achieve an overall improvement in acoustic performance.

3.3.7. Influence of Density

When designing underwater acoustic coatings, impedance-matching characteristics are one of the key factors determining the quality of the structural design. Whether for single-layer or multilayer acoustic coatings, it is essential to ensure good acoustic impedance matching between the coating and the surrounding water medium. To achieve this goal, functional materials with densities close to that of water are typically selected as the matrix materials for acoustic coatings in engineering practice. To investigate the effect of the densities of each layer on the sound absorption coefficient of the rubber-based Ni₅₀Ti₅₀ alloy multilayer acoustic coating, the densities were set as ρ₁ = (1005, 1305, 1605) kg/m³, ρ₂ = (1000, 1300, 1600) kg/m³, ρ₃ = (6150, 6450, 6750) kg/m³, ρ₄ = (1300, 1600, 1900) kg/m³, and ρ₅ = (1039, 1339, 1639) kg/m³. The influence of density on the overall sound absorption characteristics was obtained as shown in Figure 16.

From Figure 16, it is seen that the sound absorption coefficient curves under different densities basically overlap within the frequency range of [10, 20,000] Hz, with only a slight tendency for the first absorption valley to shift toward lower frequencies. This phenomenon mainly originates from the increase in areal density causing changes in the structural vibration characteristics, specifically manifested as a shift in the system’s modal frequencies toward the low-frequency range. Therefore, in engineering practice, in addition to considering the impedance-matching characteristics between the acoustic coating and the water medium, the influence of frequency modal shifts must also be considered.

3.3.8. Influence of Phase Transformation on Ni₅₀Ti₅₀ Alloy

Ni₅₀Ti₅₀ alloy undergoes a martensite–austenite phase transformation under thermally induced conditions, which alters its material parameters and consequently affects the sound absorption performance of multilayer acoustic coatings. To further elucidate the mechanism by which the phase transformation characteristics of Ni₅₀Ti₅₀ alloy influence the acoustic behavior of multilayer coatings, the effects of phase transformation on the global sound absorption properties were evaluated at T = (20, 40, 80) °C, as illustrated in Figure 17.

As shown in Figure 17, the overall sound absorption coefficient in the frequency range of [10, 20,000] Hz is minimally affected by temperature. However, the enlarged view indicates that the sound absorption performance of the multilayer coating improves with increasing temperature. This behavior is primarily attributed to the phase transformation characteristics of the Ni₅₀Ti₅₀ alloy: elevated temperatures promote the stabilization of the austenite phase, enabling a more complete stress-induced martensite transformation and reverse transformation, accompanied by enhanced hysteretic energy dissipation and interfacial friction, thereby increasing the coating’s capacity to dissipate acoustic energy.

4. Conclusions

The present study focused on the theoretical analysis, numerical simulation, and experimental study of the sound absorption mechanism and characteristics of a novel rubber–Ni₅₀Ti₅₀ alloy multilayer acoustic coating, under atmospheric pressure. The following are the main conclusions:

(1)

The design of the coating utilizes the significant high-frequency sound absorption capability of rubber and the vibration damping and noise reduction ability of Ni₅₀Ti₅₀ alloy. An expression for the sound absorption of the coating was derived, aiming to achieve broadband sound absorption without increasing the overall structural size. The experimental results show that the variation trend of the experimental values is in good agreement with that of the theoretical values, and the amplitude consistency of the acoustic absorption coefficient between the two is also high. This verifies, to a certain extent, the correctness of the analytical theory and simulation method. Within the frequency range of 10–8000 Hz, the effective bandwidth with an acoustic absorption coefficient greater than 0.8 covers 786–7152 Hz and 7483–8000 Hz. In particular, the sound absorption coefficient in the low-frequency band (1800–3000 Hz) remains stable above 0.8, which solves the traditional bottleneck of small-sized structures being difficult to cover the low-frequency broadband.

(2)

This study systematically investigated the influences of two categories of key factors: first, the structural and material parameters, including unit cell length, thickness of each layer, cavity radii, elastic modulus, Poisson’s ratio, damping loss factor, and density; second, the phase transformation behavior of Ni₅₀Ti₅₀ alloy. Through parametric analysis of the aforementioned variables, the following regularity conclusions were drawn:

(i)

Regarding the effects of structural parameters: With the increase in unit cell length, the first sound absorption peak shifts gradually toward the high-frequency range; when the thickness of each layer increases, the variation trend is consistent, all leading to the shift in the first sound absorption peak toward the low-frequency range; among the cavity radii, the increase in r₂ and r₃ causes a slight low-frequency shift in the first sound absorption peak, whereas the increase in r₁ results in a decrease in the peak sound absorption coefficient.

(ii)

Regarding the effects of material parameters: As the elastic modulus increases gradually, the first sound absorption peak of the acoustic coating tends to shift toward the high-frequency range; with the increase in Poisson’s ratio of each structural layer, the first sound absorption peak in the sound absorption coefficient curve also shifts toward the high-frequency range; the material damping loss factor has a minor effect on the peak frequency of the sound absorption curve, and its main role is to modify the amplitude of the sound absorption peak; density exerts a weak overall influence on the sound absorption coefficient curve, only causing a slight low-frequency shift in the first sound absorption valley.

(iii)

Regarding the effects of phase transformation behavior: With the increase in temperature, Ni₅₀Ti₅₀ alloy undergoes phase transformation, and the sound absorption performance of the multi-level coating is accordingly enhanced.

It should be noted that the current research has certain limitations. First, the factor of hydrostatic pressure in the actual service scenarios of underwater structures has not been considered. Second, due to the limitations of experimental equipment, the actually verified frequency band is 10–8000 Hz, while the sound absorption performance in the 8000–20,000 Hz high-frequency range is based on theoretical and simulation predictions. Third, the (SMA) model in this research has been simplified: the Ni₅₀Ti₅₀ alloy layer is treated as a homogeneous layer in the four-port network theoretical modeling, and the influence of the actual martensite–austenite phase transformation kinetics and microstructure of the Ni₅₀Ti₅₀ alloy on the sound absorption performance in the low-frequency range remains to be discussed.

In future work, we plan to conduct further in-depth research on the following two aspects. First, we will comprehensively consider the effects of water pressure, temperature, and seawater corrosion to study the sound absorption and sound insulation properties of claddings for deep-sea environments. Second, we will investigate temperature control to adjust material properties, thereby realizing the active control of shape memory alloy (SMA) composite acoustic coatings.