Chao Shen^1,2,3, Feng Zeng⁴, Qianfeng Wang⁵, Ying Hu^3*
1School of Automation Science and Engineering, South China University of Technology, Guangzhou, China.
²School of Chemical Engineering and Energy Technology, Dongguan University of Technology, Dongguan, China.
³Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China.
⁴School of Automation and Electrical Engineering, Chengdu Technological University, Chengdu, China.
⁵Aerospace Nanhu Electronic Information Technology Co., Ltd., Jingzhou, China.
DOI: 10.4236/jamp.2025.137143   PDF    HTML   XML   112 Downloads   479 Views   Citations

Abstract

This research investigates the sound transmission loss (STL) characteristics of finite double-plate sandwich structures featuring poroelastic cores and air gaps under external water flow conditions. A theoretical framework, based on Biot’s theory and modal decomposition, is developed to analyze fluid-structure interactions, convective flow effects, and various boundary conditions. The study highlights the influence of air gap resonances, the added-mass effect from hydrodynamic loading, and asymmetric plate thickness distributions on STL performance. Results indicate that optimized plate asymmetry enhances STL by leveraging mass-stiffness imbalances, while the directional dependence of incident acoustic waves significantly affects transmission outcomes. These insights offer practical design strategies for developing lightweight, broadband underwater acoustic insulation systems tailored for dynamic marine environments.

Keywords

Sound Transmission Loss, Air Gap Resonances, Fluid-Structure Interaction, Underwater Acoustics, Poroelastic Material

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1. Introduction

Effective sound insulation in underwater environments and high-speed transport vehicles must balance broadband noise attenuation with adaptability to complex flow and boundary conditions [1]-[4]. For example, submarine hulls require high sound transmission loss (STL) to ensure stealth, while underwater pipelines demand noise reduction to safeguard marine ecosystems [5]. Similarly, lightweight acoustic structures in marine vessels enhance passenger comfort by minimizing interior noise.

In air, the significant impedance mismatch between solids (e.g., metals) and air facilitates effective sound reflection, even with thin structures. However, underwater, this impedance contrast is substantially reduced, typically to a ratio of around 10 [6]. For instance, a 4 mm-thick aluminum plate ( $Z_{\text{Al}}=1.73\times {10}^7\text{N}\cdot \text{s}/{\text{m}}^3$ ) reflects approximately 71% of acoustic energy at 100~Hz in water, compared to over 99.99% in air, due to the smaller impedance difference between air ( $Z_{\text{air}}=412\text{N}\cdot \text{s}/{\text{m}}^3$ ) and water ( $Z_{\text{water}}=1.48\times {10}^6\text{N}\cdot \text{s}/{\text{m}}^3$ ). This reduced contrast poses significant challenges for acoustic performance in marine applications [7] [8].

Prior research on convective flow effects in airborne applications has provided valuable insights. Koval [9] identified STL degradation in curved plates due to flow, while Tang et al. [10] [11] and Zhou et al. [12] extended these findings to sandwich and poroelastic systems. Liu et al. [13] [14] developed flow-acoustic coupling models for infinite structures, demonstrating STL improvements through shear stress and resonance suppression. However, the impact of external water flows on finite double-plate systems, particularly under varied boundary conditions, remains largely unexplored. Currently, no comprehensive framework addresses the interaction between finite sandwich structures and external water flows, accounting for factors such as plate thickness distribution with constant total mass, varying poroelastic core heights, and diverse boundary conditions. This gap limits the development of effective underwater sound insulation systems and high-performance panels for submerged or water-exposed environments [15].

This study introduces a validated theoretical model for waterborne sound insulation in finite double-plate sandwich structures with poroelastic cores and air gaps. Grounded in Biot’s theory and modal decomposition [16], the model integrates convective flow dynamics and fluid-structure interactions. It systematically evaluates the effects of boundary constraints, resonance phenomena, and hydrostatic pressure on STL. The research offers three key contributions: 1) it demonstrates how varying plate thickness distributions, while maintaining constant total mass, induces an added-mass effect that significantly alters acoustic behavior; 2) it incorporates external water flow dynamics into STL modeling for poroelastic sandwich systems; and 3) it analyzes STL performance across four distinct boundary conditions, providing insights into optimizing acoustic performance in submerged settings. These findings pave the way for designing advanced underwater sound insulation systems and high-performance acoustic barriers for flow-exposed structures.

2. Vibro-Acoustic Theoretical Modeling

Figure 1 shows the geometry and notation for the sound transmission loss problem of a double-plate sandwich with a poroelastic core, immersed in water on the upper side in Figure 1(a) and immersed in water on the bottom side in Figure 1(b). As shown in Figure 1(a), a plane acoustic wave of unit amplitude is incident from the upper water domain (semi-infinite, density ${\rho }_{\text{i}}$ , sound speed $c_{\text{i}}$ ) onto the first aluminium plate (thickness $t_1$ ), then propagates through an air gap of height $H_{\text{g}1}$ , the poroelastic layer of thickness $H_{\text{p}}$ , a second air gap of height $H_{\text{g}2}$ , and finally through the second aluminium plate (thickness $t_2$ ) into the lower air domain (semi‑infinite, density ${\rho }_{\text{t}}$ , sound speed $c_{\text{t}}$ ). The description about the layering of materials is shown in Table 1.

We adopt a Cartesian $\left(x,y,z\right)$ system with the $z$ ‑axis normal to the plates. The plates themselves extend in the $x\text{-}y$ plane over a rectangular area of size $a\times b$ ; each of their four edges may be subjected to arbitrary boundary conditions (simply supported, clamped, free, etc.). The incident wave makes an elevation angle ${\phi }_{\text{i}}$ with respect to the $z$ ‑axis (i.e. the angle between its propagation direction and the $x\text{-}y$ plane normal) and an azimuth angle $\theta$ in the $x\text{-}y$ plane (measured from the $x$ ‑axis). By analogy with Snell’s law, the same pair of angles $\left({\phi }_{\text{i}},\theta \right)$ describes the direction of each plane‑wave component in all layers if there is no mean flow to break the in‑plane wavevector continuity.

All fluid layers (water domains and air gaps) are assumed quiescent for default unless otherwise noted, so only acoustic pressure continuity and normal-velocity continuity are enforced at each interface. The poroelastic layer is modeled in full three-dimensional Biot theory, and the two plates are treated as thin, linearly elastic shells. The goal of the theoretical formulation that follows is to compute the complex transmission coefficient through this six‑layer stack and hence the STL as a function of frequency, incidence angles ( ${\phi }_{\text{i}},\theta$ ), plate and layer geometry ( $t_1,t_2,H_{\text{g}1},H_{\text{p}},H_{\text{g}2},a,b$ ), and material properties.

2.1. Velocity Potential and Wavenumber

As depicted in Figure 1(a), the acoustic field in the upper water region (incident field) comprises an incoming wave and its reflection resulting from interaction with the first aluminum plate. Due to the presence of a uniform external flow in this region, the acoustic velocity potential is expressed in harmonic form as:

${\Phi }_{\text{i}}={\text{e}}^{\text{j}\omega t-\text{j}\left(k_{\text{i}x}x+k_{\text{i}y}y+k_{\text{i}z}z\right)}+R{\text{e}}^{\text{j}\omega t-\text{j}\left(k_{\text{i}x}x+k_{\text{i}y}y-k_{\text{i}z}z\right)}$ , (1)

where the incident wave has a unit amplitude, $R$ represents the complex reflection coefficient, $\omega$ denotes the angular frequency, and $\text{j}=\sqrt{-1}$ . The Cartesian components of the incident wavenumber are defined as:

$k_{\text{i}x}=k_{\text{i}}\sin {\phi }_{\text{i}}\cos \theta ,k_{\text{i}y}=k_{\text{i}}\sin {\phi }_{\text{i}}\sin \theta ,k_{\text{i}z}=\sqrt{k_{\text{i}}^2-\left(k_{\text{i}x}^2+k_{\text{i}y}^2\right)}$ .

Within the two stationary air gaps (Air gap 1 and Air gap 2) located between the plates and the poroelastic core, the acoustic field consists of forward- and backward-propagating waves. In the absence of flow, the velocity potential in these air gaps adheres to the standard Helmholtz wave equation:

$\frac{{\partial }^2{\Phi }_{\text{<mn> g </mn> <mo> ^ </mo>}}}{\partial t^2}=c_{\text{<mn> g </mn> <mo> ^ </mo>}}^2{\nabla }^2{\Phi }_{\text{<mn> g </mn> <mo> ^ </mo>}},$ (2)

where $\text{<mn> g </mn> <mo> ^ </mo>}\in \left\{{\text{g}}_1,{\text{g}}_2\right\}$ indicates either air gap, and $c_{\text{<mn> g </mn> <mo> ^ </mo>}}$ is the speed of sound in the respective gap. The harmonic form of the acoustic velocity potential in each air gap is given by:

${\Phi }_{\text{<mn> g </mn> <mo> ^ </mo>}}={\Phi }_{\widehat{\text{gi}}}+{\Phi }_{\widehat{\text{gr}}}=I_{\text{<mn> g </mn> <mo> ^ </mo>}}{\text{e}}^{\text{j}\omega t-\text{j}\left(k_{\text{<mn> g </mn> <mo> ^ </mo>}x}x+k_{\text{<mn> g </mn> <mo> ^ </mo>}y}y+k_{\text{<mn> g </mn> <mo> ^ </mo>}z}z\right)}+R_{\text{<mn> g </mn> <mo> ^ </mo>}}{\text{e}}^{\text{j}\omega t-\text{j}\left(k_{\text{<mn> g </mn> <mo> ^ </mo>}x}x+k_{\text{<mn> g </mn> <mo> ^ </mo>}y}y-k_{\text{<mn> g </mn> <mo> ^ </mo>}z}z\right)},$ (3)

where $I_{\text{<mn> g </mn> <mo> ^ </mo>}}$ and $R_{\text{<mn> g </mn> <mo> ^ </mo>}}$ represent the amplitudes of the incident and reflected wave components in air gap $\text{<mn> g </mn> <mo> ^ </mo>}$ .

On the transmission side, the semi-infinite lower fluid domain is anechoically terminated, resulting in a single transmitted acoustic wave. Without external flow in this region, the velocity potential of the transmitted wave is expressed as:

${\Phi }_{\text{t}}=T{\text{e}}^{\text{j}\omega t-\text{j}\left(k_{\text{t}x}x+k_{\text{t}y}y+k_{\text{t}z}z\right)}$ , (4)

where $T$ is the complex amplitude of the transmitted wave, and the wavenumber $k_{\text{t}}$ in the stationary transmission medium is given by $k_{\text{t}}=\omega /c_{\text{t}}$ .

2.2. Boundary Conditions

As illustrated in Figure 1(a), the poroelastic core is positioned between two air gaps, which in turn are bounded by the incident and transmitting plates. This structure introduces two distinct types of interfaces: 1) the plate-air gap interface and 2) the air gap-poroelastic interface. The governing boundary conditions at these interfaces are derived based on fluid-structure and fluid-poroelastic coupling principles [12], and they can be broadly categorized into two groups: (a) continuity of acoustic velocity and structural displacement, and (b) dynamic equilibrium between acoustic pressure forces and structural response.

2.2.1. Incident Plate Boundary Conditions

At the mid-surface of the incident plate, three boundary conditions must be satisfied to account for the interaction with the incident acoustic field (which includes mean flow) and the adjacent stationary air gap. These are:

$\left(\text{i}\right)v_{\text{i}z}=\frac{\text{D}{w}_{t1}}{\text{D}t},\left(\text{ii}\right)v_{\text{g}1z}=\frac{\partial w_{t1}}{\partial t},\left(\text{iii}\right)p_{\text{i}}-p_{\text{g}1}=w_{t1}\left[B_1{(k_x^2+k_y^2)}^2-{\omega }^2{m}_{s1}\right].$ (5)

Here, $v_{\text{i}z}$ and $p_{\text{i}}$ represent the normal acoustic particle velocity and acoustic pressure at the surface exposed to the incident field, while $v_{\text{g}1z}$ and $p_{\text{g}1}$ denote their equivalents in the adjacent quiescent air gap. The symbol $w_{t1}$ indicates the transverse displacement of the plate. The first two conditions ensure the continuity of normal velocity and displacement across the plate interface, while the third condition describes the transverse dynamics of the plate as dictated by the Euler-Bernoulli beam equation, characterized by bending stiffness $B_1$ and surface mass density $m_{s1}$ .

2.2.2. Poroelastic Layer Interface Boundary Conditions

At each interface between the poroelastic material and its surrounding air gaps (both assumed to be quiescent), four boundary conditions must be enforced:

$\begin{array}{l}\left(\text{i}\right)-\beta p_{\text{<mn> g </mn> <mo> ^ </mo>}}=s,\left(\text{ii}\right)-\left(1-\beta \right)p_{\text{<mn> g </mn> <mo> ^ </mo>}}={\sigma }_z,\\ \left(\text{iii}\right)v_{\text{<mn> g </mn> <mo> ^ </mo>}z}=\left(1-\beta \right)\frac{\partial u_z}{\partial t}+\beta \frac{\partial U_z}{\partial t},\left(\text{iv}\right){\tau }_{xz}={\tau }_{yz}=0,\end{array}$ (6)

where $p_{\text{<mn> g </mn> <mo> ^ </mo>}}$ and $v_{\text{<mn> g </mn> <mo> ^ </mo>}z}$ are the acoustic pressure and normal particle velocity at the air gap-poroelastic interface, which are expressed using the velocity potential ${\Phi }_{\text{<mn> g </mn> <mo> ^ </mo>}}$ for the stationary gap medium:

$p_{\text{<mn> g </mn> <mo> ^ </mo>}}={\rho }_{\text{<mn> g </mn> <mo> ^ </mo>}}\frac{\partial {\Phi }_{\text{<mn> g </mn> <mo> ^ </mo>}}}{\partial t}=\text{j}\omega {\rho }_{\text{<mn> g </mn> <mo> ^ </mo>}}{\Phi }_{\text{<mn> g </mn> <mo> ^ </mo>}},v_{\text{<mn> g </mn> <mo> ^ </mo>}z}=\text{j}{k}_{\text{<mn> g </mn> <mo> ^ </mo>}z}{\Phi }_{\text{<mn> g </mn> <mo> ^ </mo>}}.$ (7)

The coefficients $\beta$ and $1-\beta$ are used to distribute the pressure contribution between the fluid and solid phases of the poroelastic medium, while $s$ and ${\sigma }_z$ represent the fluid pressure and normal stress in the solid skeleton. The third condition in Equation (6). ensures conservation of the normal velocity at the interface. Finally, the shear stresses ${\tau }_{xz}$ and ${\tau }_{yz}$ are set to zero based on the assumption of inviscid flow in the stationary air gaps.

2.2.3. Transmitting Plate Boundary Conditions

Similarly, the following three boundary conditions must be fulfilled at the mid-surface of the transmitting plate:

$\left(\text{i}\right)v_{\text{g}2z}=\frac{\partial w_{t2}}{\partial t},\left(\text{ii}\right)v_{\text{t}z}=\frac{\partial w_{t2}}{\partial t},\left(\text{iii}\right)p_{\text{g}2}-p_{\text{t}}=w_{t2}\left[B_2{(k_x^2+k_y^2)}^2-{\omega }^2{m}_{s2}\right].$ (8)

Here, $p_{\text{t}}$ and $v_{\text{t}z}$ , associated with the transmitted field, are defined via the potential function ${\Phi }_{\text{t}}$ as:

$p_{\text{t}}={\rho }_{\text{t}}\frac{\partial {\Phi }_{\text{t}}}{\partial t}=\text{j}\omega {\rho }_{\text{t}}{\Phi }_{\text{t}},v_{\text{t}z}=\text{j}{k}_{\text{t}z}{\Phi }_{\text{t}}$ . (9)

In these expressions, $w_{t2}$ is the transverse displacement of the transmitting plate, with bending stiffness $B_2$ and mass per unit area $m_{s2}$ . As before, the first two conditions maintain velocity and displacement continuity across the plate-air interface, and the third describes the dynamic response of the plate under differential acoustic loading.

2.3. Modal Decomposition

The equations governing the flexural motions of two coupled plates with two air gaps are described as follows [17]-[19]:

$B_1{\nabla }^4{w}_{t1}+m_{s1}\frac{{\partial }^2{w}_{t1}}{\partial t^2}=p_{\text{i}}-p_{\text{g}1},$ (10a)

$B_2{\nabla }^4{w}_{t2}+m_{s2}\frac{{\partial }^2{w}_{t2}}{\partial t^2}=p_{\text{g}2}-p_{\text{t}}$ , (10b)

where ${\nabla }^4={\left({\partial }^2/\partial x^2+{\partial }^2/\partial y^2\right)}^2={\left(k_x^2+k_y^2\right)}^2$ and ${\partial }^2/\partial t^2=-{\omega }^2$ . The terms are defined as follows: $p_{\text{g}1}={\rho }_{\text{g}1}\partial {\Phi }_{\text{g}1}/\partial t$ , and $p_{\text{g}2}={\rho }_{\text{g}2}\partial {\Phi }_{\text{g}2z}/\partial t$ . The term $B_i{\nabla }^4{w}_{ti}$ represents the classical biharmonic operator for bending, while $m_{si}{\partial }^2{w}_{ti}/\partial t^2$ accounts for the inertial resistance of the plate. The flexural stiffness of the plates can be articulated as

$B_i=\frac{E_i{h}_i^3\left(1+\text{j}{\eta }_i\right)}{12\left(1-{\nu }_i^2\right)}\left(i=1,2\right)$ , (11)

where $E_i$ , $h_i$ , ${\eta }_i$ , and ${\nu }_i$ denote the Young’s modulus, thickness, loss factor, and Poisson’s ratio of the $i\text{th}$ plate, respectively. This expression for complex flexural rigidity is commonly utilized to incorporate damping effects through the loss factor ${\eta }_i$ . The inclusion of the complex term $\left(1+\text{j}{\eta }_i\right)$ is a standard approach in harmonic analysis to account for losses in a phenomenological manner. Following previous studies [17]-[20], the transverse displacements $w_{t1}$ and $w_{t2}$ can be written as a modal decomposition:

$w_{t1}\left(x,y,t\right)={\displaystyle {\sum }_{m,n}{\varphi }_{mn}}\left(x,y\right){\alpha }_{1,mn}{\text{e}}^{\text{j}\omega t},w_{t2}\left(x,y,t\right)={\displaystyle {\sum }_{m,n}{\varphi }_{mn}}\left(x,y\right){\alpha }_{2,mn}{\text{e}}^{\text{j}\omega t}$ , (12)

where ${\alpha }_{1,mn}$ and ${\alpha }_{2,mn}$ are the modal displacement coefficients of the upper and bottom plates, respectively. Table 2 provides the details for other types of edge supports.

Given the expressions for the incident and transmitted waves, ${\Phi }_{\text{i}}$ and ${\Phi }_t$ , in Equations (1) and (4), we can now utilize the continuity of normal velocity at

the interfaces of the two plates. Substituting these expressions into the boundary conditions given by Equations 5(i) and 8(ii) allows us to derive the amplitudes of the reflected and transmitted waves, $R$ and $T$ , as:

$R\left(x,y\right)=1-\frac{{\text{e}}^{\text{j}\left(k_{x}x+k_{y}y\right)}}{k_{\text{i}z}}{\displaystyle {\sum }_{m,n}\left(\omega {\varphi }_{mn}-\text{j}V\cdot \nabla {\varphi }_{mn}\right){\alpha }_{1,mn}}$ , (13)

$T\left(x,y\right)=\frac{\omega }{k_{\text{t}z}}{\text{e}}^{\text{j}\left(k_{x}x+k_{y}y+k_{\text{t}z}\left(L_{\text{g}1}+H_{\text{p}}+L_{\text{g}2}\right)\right)}{\displaystyle {\sum }_{m,n}{\varphi }_{mn}{\alpha }_{2,mn}}$ . (14)

The detailed expressions of the boundary condition equations in Equations (5)-(8) can be seen in Appendix A. Using the Equations (A.1) and (A.10) one can obtain the expressions of ${\Phi }_{\text{i}}$ and ${\Phi }_t$ : The other Equations (A.2)-(A.9) together lead to a system of 8 equations. Therefore, the complex constants ( $C_1,C_2,C_3,C_4,I_{\text{g}1},R_{\text{g}1},I_{\text{g}2}$ and $R_{\text{g}2}$ ) can be determined by a $8\times 8$ matrix equation:

$MC=S$ (15)

where

$C={\left[C_1{C}_2{C}_3{C}_4{I}_{\text{g}1}{R}_{\text{g}1}{I}_{\text{g}2}{R}_{\text{g}2}\right]}^t$ , (16)

$S={\left[{\displaystyle {\sum }_{m,n}{\alpha }_{1,mn}{\varphi }_{mn}{\text{e}}^{\text{j}\left(k_{x}x+k_{y}y\right)}}000000{\displaystyle {\sum }_{m,n}{\alpha }_{2,mn}{\varphi }_{mn}{\text{e}}^{\text{j}\left(k_{x}x+k_{y}y\right)}}\right]}^t$ . (17)

2.4. The Galerkin Method

To determine the unknown modal coefficients ${\alpha }_{1,mn}$ , ${\alpha }_{2,mn}$ , and ${\alpha }_{3,mn}$ , the principle of virtual work is employed. The virtual work ${\delta }_i$ resulting from a force acting on a particle through a virtual displacement (an infinitesimal change in the particle’s position consistent with the system’s constraints) is given by:

${\delta }_i=\delta {\alpha }_{i,mn}{\varphi }_{mn}\left(x,y\right)\left(i=1,2\right).$ (18)

The sum of these virtual work contributions over the entire system must equal zero. Applying the Galerkin method to the equations of motion for the plate, Equations (10a) and (10b), involves integrating weighted functions over the domain of the double-plate configuration. The weak form of the principle of virtual work can be expressed as:

${\displaystyle {\int }_0^b{\displaystyle {\int }_0^a\left(B_1{\nabla }^4{w}_{t1}+m_{s1}\frac{{\partial }^2{w}_{t1}}{{\partial }^{2}t}-p_{\text{i}}+p_{\text{g}1}\right){\delta }_1\text{d}x\text{d}y}}=0$ , (19)

${\displaystyle {\int }_0^b{\displaystyle {\int }_0^a\left(B_2{\nabla }^4{w}_{t2}+m_{s2}\frac{{\partial }^2{w}_{t2}}{{\partial }^{2}t}-p_{\text{g}2}+p_{\text{t}}\right){\delta }_2\text{d}x\text{d}y}}=0$ . (20)

The variables ${\Phi }_{\text{i}},{\Phi }_{\text{g}1},{\Phi }_{\text{g}2},{\Phi }_t$ in $p_{\text{i}},p_{\text{g}1},p_{\text{g}2}$ and $p_{\text{t}}$ , and $w_{t1},w_{t2}$ have been expressed in terms of ${\alpha }_{1,mn}$ and ${\alpha }_{2,mn}$ using the solution of the unknown vector $C$ .

2.5. Sound Transmission Loss

Sound intensity in the incident or transmitted field is given by:

$I=\frac{1}{2}\text{Re}\left(p{v}^{*}\right),$ (21)

where $v^{*}$ is the complex conjugate of acoustic velocity $v$ , and $\text{Re}(\cdot )$ denotes the real part. The incident and transmitted acoustic power over plate surface area $S$ are:

${\Pi }_{\text{i}}=\frac{1}{2}\text{Re}\left[{\displaystyle {\int }_S{p}_{\text{i}}{v}_{\text{i}z}^{*}\text{d}S}\right]$ , ${\Pi }_{\text{t}}=\frac{1}{2}\text{Re}\left[{\displaystyle {\int }_S{p}_{\text{t}}{v}_{\text{t}z}^{*}\text{d}S}\right],$ (22)

where $p_{\text{i}},p_{\text{t}},v_{\text{i}z}$ , and $v_{\text{t}z}$ represent pressure and velocity fields in the incident and transmitted domains. The power transmission coefficient for a single incident wave is:

$\tau \left({\phi }_{\text{i}},\theta \right)=\frac{{\Pi }_{\text{t}}}{{\Pi }_{\text{i}}}$ . (23)

Sound transmission loss (STL), expressed in decibels, is:

$\text{STL}=10\log \left(\frac{1}{\tau }\right).$ (24)

This metric evaluates acoustic system performance, with details in prior studies [12] [21].

3. Results and Discussion

3.1. Parameters of the Systems

To study external flow and poroelastic core effects on STL in a double-plate sandwich structure, material parameters from [12] [21]-[23] are used. Both aluminum face plates have identical properties, with thicknesses $t_1=t_2=2$ mm. The poroelastic layer, made of polyurethane foam, follows parameters in [21] [23]. The incident and transmission media are water at standard room temperature properties. Air gaps have heights $H_{\text{g}1}=H_{\text{g}2}=0.01$ m, with no internal flow, yielding a total system height $H_{\text{t}}=0.041$ m. Unless specified, the incident wave’s azimuthal angle is $\theta =0^{\circ }$ , and external flow aligns with the $x$ -axis ( ${\theta }_{\text{e}}=0^{\circ }$ ).

3.2. Impact of Boundary Conditions and Plate Dimensions on STL

The influence of boundary conditions and plate dimensions on the STL of rectangular plates subjected to normal sound incidence in the absence of external flow ( $M=0$ ) is depicted in Figure 2. Four distinct boundary conditions are examined: (a) fully clamped (CCCC), (b) simply supported (SSSS), (c) free-free (FFFF), and (d) mixed clamped and simply supported (CCSS). The STL is plotted against frequency for four different square plate dimensions: $a=b=0.125$ m, $a=b=0.25$ m, $a=b=0.5$ m, and $a=b=1$ m.

The effect of boundary conditions on STL varies with frequency. The CCCC condition shows a significant STL dip at the fundamental resonance due to reduced damping from rigid edge clamping, while the SSSS condition yields the highest STL below this resonance. Smaller plates ( $a=b=0.125$ m) exhibit higher STL at frequencies above 1000 Hz, outperforming larger plates ( $a=b=1$ m), which show lower STL in this range. This is attributed to smaller plates having higher natural frequencies, leading to more pronounced resonant STL dips at higher frequencies.

3.3. Effect of Air Gap Height and External Water Flow

The impact of air gap height ( $H_{\text{g}}$ , defined as the sum of $H_{\text{g}1}+H_{\text{g}2}$ in the absence of a poroelastic layer) and external grazing flow ( $V$ ) on sound transmission loss (STL) is analyzed for two angles of sound incidence: normal incidence ( ${\phi }_{\text{i}}=0^{\circ }$ ) and oblique incidence ( ${\phi }_{\text{i}}={45}^{\circ }$ ). Figure 3(a) presents the STL results for normal incidence. When there is no flow ( $V=0$ m/s), increasing the air gap height from 0.01 m to 0.04 m causes the characteristic peaks and dips in the STL spectrum to shift to lower frequencies. For example, the first significant dip in STL moves from around 280 Hz for $H_{\text{g}}=0.01$ m to below 150 Hz for $H_{\text{g}}=0.04$ m. This behavior is typical of cavity-backed acoustic systems, where the air gap functions as a quarter-wavelength resonator at its harmonic frequencies. When comparing the no-flow condition ( $V=0$ m/s) to the case with grazing flow ( $V=17$ m/s) for each gap height, the presence of flow leads to improved STL at very low frequencies (e.g., below 70 Hz). However, the dominant factor influencing the shift in resonance frequencies continues to be the height of the air gap. At frequencies higher than the extra low frequency, the impact of the grazing water flow at this velocity is quite limited.

By comparing Figure 3(a) and Figure 3(b), it becomes clear that raising the air gap height effectively adjusts the STL performance to lower frequencies by shifting the plate-air-plate resonances. The angle of incidence only slightly affects the overall shape and magnitude of the STL spectrum. Ultimately, the air gap height is the main factor influencing the position of the resonance frequencies.

3.4. Influence of Plate Thickness Distribution and Added-Mass Effect

Figure 4(a) shows the STL behavior of a double-plate system with fully clamped (CCCC) boundary conditions, without a porous core. The structure consists of two aluminum plates with a fixed total thickness of 4 mm, varying between the upper ( $t_1$ ) and lower ( $t_2$ ) plates. The upper medium is water, the inter-plate cavity and lower medium are air, with acoustic excitation at normal incidence ( ${\phi }_{\text{i}}=0^{\circ }$ ), no mean flow ( $M=0$ ), and an air gap height of $H_{\text{g}}=0.02\text{m}$ .

Added-Mass Effect

STL generally follows the mass-law, increasing by 6 dB per doubling of surface mass density. However, fluid-structure interaction with water induces an added-mass effect, increasing the plates’ apparent mass and inertial impedance. This enhances STL in the mid-frequency range, exceeding mass-law predictions. For example, the configuration with $t_1=0.1\text{mm}$ and $t_2=3.9\text{mm}$ outperforms the $t_{\text{eq}}=4\text{mm}$ mass-law prediction below 500 Hz, lowering resonance frequencies by 50 Hz and boosting STL by 5 - 10 dB in the 200 - 500 Hz range, compared to an air-only scenario with reduced inertial resistance.

Air Gap Resonances

In the 200 Hz - 10 kHz band, STL surpasses mass-law predictions due to the air gap’s acoustic behavior, which creates impedance mismatch and supports internal reflections and standing waves, enhancing modal suppression. STL is sensitive to plate thickness distribution:

- The asymmetric case ( $t_1=0.1\text{mm}$ , $t_2=3.9\text{mm}$ ) matches the STL of a $t_{\text{eq}}=20\text{mm}$ monolithic plate, driven by mass-stiffness asymmetry and optimal impedance.

- The case with $t_1=3\text{mm}$ and $t_2=1\text{mm}$ resembles the uniform $t_{\text{eq}}=4\text{mm}$ case, with less STL enhancement.

These results highlight that STL depends not only on total mass but also on structural impedance distribution and cavity dynamics.

4. Conclusions

This study provides a detailed examination of waterborne sound transmission through finite double-plate sandwich structures with poroelastic cores and air gaps, incorporating the effects of external water flow. Utilizing a robust theoretical model grounded in Biot’s theory and modal decomposition, the research uncovers critical mechanisms governing STL performance. Key findings include:

1) Asymmetric plate thickness distributions, maintaining constant total mass, significantly enhance STL through mass-stiffness asymmetry, with configurations like a thin upper plate ( $t_1=0.1\text{mm}$ ) and thick lower plate ( $t_2=3.9\text{mm}$ ) achieving performance comparable to much thicker monolithic plates.

2) The added mass effect from hydrodynamic loading increases effective plate mass, reducing resonance frequencies and boosting STL by 5 - 10 dB in the 200 - 500 Hz range.

3) Air gap heights critically influence resonance frequencies, shifting STL peaks and dips to lower frequencies as gap height increases.

These findings emphasize the importance of integrating fluid-structure interactions and geometric optimization in designing efficient underwater sound insulation systems. Future work could explore graded poroelastic materials and advanced optimization techniques to further improve STL while minimizing structural weight, advancing applications in marine engineering and submerged environments.

Acknowledgements

The authors acknowledge the support from National Natural Science Foundation of China (Grant No. 12202102), Guangdong Provincial University Innovation Team Project, No. 2023KCXTD038, Guangdong Provincial Key Laboratory of Distributed Energy Systems and the Center for Computational Science and Engineering at the Southern University of Science and Technology.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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