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.
Effective sound insulation in underwater environments and high-speed transport vehicles must balance broadband noise attenuation with adaptability to complex flow and boundary conditions [
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 [
Prior research on convective flow effects in airborne applications has provided valuable insights. Koval [
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 [
We adopt a Cartesian ( x , y , z ) system with the z ‑axis normal to the plates. The plates themselves extend in the x - y plane over a rectangular area of size a × 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 φ i with respect to the z ‑axis (i.e. the angle between its propagation direction and the x - y plane normal) and an azimuth angle θ in the x - y plane (measured from the x ‑axis). By analogy with Snell’s law, the same pair of angles ( φ i , θ ) describes the direction of each plane‑wave component in all layers if there is no mean flow to break the in‑plane wavevector continuity.
| Layer | Material | Properties/Thickness |
|---|---|---|
| 1 | Water (incident field) | ( ρ i , c i ) |
| 2 | Aluminum plate 1 | Thickness t 1 |
| 3 | Air gap 1 | Height H g 1 |
| 4 | Poroelastic core | Thickness H p |
| 5 | Air gap 2 | Height H g 2 |
| 6 | Aluminum plate 2 | Thickness t 2 |
| 7 | Air (transmission field) | ( ρ t , c t ) |
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 ( φ i , θ ), plate and layer geometry ( t 1 , t 2 , H g 1 , H p , H g 2 , a , b ), and material properties.
As depicted in
Φ i = e j ω t − j ( k i x x + k i y y + k i z z ) + R e j ω t − j ( k i x x + k i y y − k i z z ) , (1)
where the incident wave has a unit amplitude, R represents the complex reflection coefficient, ω denotes the angular frequency, and j = − 1 . The Cartesian components of the incident wavenumber are defined as:
k i x = k i sin φ i cos θ , k i y = k i sin φ i sin θ , k i z = k i 2 − ( k i x 2 + k i y 2 ) .
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:
∂ 2 Φ g ^ ∂ t 2 = c g ^ 2 ∇ 2 Φ g ^ , (2)
where g ^ ∈ { g 1 , g 2 } indicates either air gap, and c g ^ is the speed of sound in the respective gap. The harmonic form of the acoustic velocity potential in each air gap is given by:
Φ g ^ = Φ gi ^ + Φ gr ^ = I g ^ e j ω t − j ( k g ^ x x + k g ^ y y + k g ^ z z ) + R g ^ e j ω t − j ( k g ^ x x + k g ^ y y − k g ^ z z ) , (3)
where I g ^ and R g ^ represent the amplitudes of the incident and reflected wave components in air gap g ^ .
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:
Φ t = T e j ω t − j ( k t x x + k t y y + k t z z ) , (4)
where T is the complex amplitude of the transmitted wave, and the wavenumber k t in the stationary transmission medium is given by k t = ω / c t .
As illustrated in
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:
( i ) v i z = D w t 1 D t , ( ii ) v g 1 z = ∂ w t 1 ∂ t , ( iii ) p i − p g 1 = w t 1 [ B 1 ( k x 2 + k y 2 ) 2 − ω 2 m s 1 ] . (5)
Here, v i z and p i represent the normal acoustic particle velocity and acoustic pressure at the surface exposed to the incident field, while v g 1 z and p g 1 denote their equivalents in the adjacent quiescent air gap. The symbol w t 1 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 s 1 .
At each interface between the poroelastic material and its surrounding air gaps (both assumed to be quiescent), four boundary conditions must be enforced:
( i ) − β p g ^ = s , ( ii ) − ( 1 − β ) p g ^ = σ z , ( iii ) v g ^ z = ( 1 − β ) ∂ u z ∂ t + β ∂ U z ∂ t , ( iv ) τ x z = τ y z = 0 , (6)
where p g ^ and v g ^ z are the acoustic pressure and normal particle velocity at the air gap-poroelastic interface, which are expressed using the velocity potential Φ g ^ for the stationary gap medium:
p g ^ = ρ g ^ ∂ Φ g ^ ∂ t = j ω ρ g ^ Φ g ^ , v g ^ z = j k g ^ z Φ g ^ . (7)
The coefficients β and 1 − β are used to distribute the pressure contribution between the fluid and solid phases of the poroelastic medium, while s and σ 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 τ x z and τ y z are set to zero based on the assumption of inviscid flow in the stationary air gaps.
Similarly, the following three boundary conditions must be fulfilled at the mid-surface of the transmitting plate:
( i ) v g 2 z = ∂ w t 2 ∂ t , ( ii ) v t z = ∂ w t 2 ∂ t , ( iii ) p g 2 − p t = w t 2 [ B 2 ( k x 2 + k y 2 ) 2 − ω 2 m s 2 ] . (8)
Here, p t and v t z , associated with the transmitted field, are defined via the potential function Φ t as:
p t = ρ t ∂ Φ t ∂ t = j ω ρ t Φ t , v t z = j k t z Φ t . (9)
In these expressions, w t 2 is the transverse displacement of the transmitting plate, with bending stiffness B 2 and mass per unit area m s 2 . 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.
The equations governing the flexural motions of two coupled plates with two air gaps are described as follows [
B 1 ∇ 4 w t 1 + m s 1 ∂ 2 w t 1 ∂ t 2 = p i − p g 1 , (10a)
B 2 ∇ 4 w t 2 + m s 2 ∂ 2 w t 2 ∂ t 2 = p g 2 − p t , (10b)
where ∇ 4 = ( ∂ 2 / ∂ x 2 + ∂ 2 / ∂ y 2 ) 2 = ( k x 2 + k y 2 ) 2 and ∂ 2 / ∂ t 2 = − ω 2 . The terms are defined as follows: p g 1 = ρ g 1 ∂ Φ g 1 / ∂ t , and p g 2 = ρ g 2 ∂ Φ g 2 z / ∂ t . The term B i ∇ 4 w t i represents the classical biharmonic operator for bending, while m s i ∂ 2 w t i / ∂ t 2 accounts for the inertial resistance of the plate. The flexural stiffness of the plates can be articulated as
B i = E i h i 3 ( 1 + j η i ) 12 ( 1 − ν i 2 ) ( i = 1 , 2 ) , (11)
where E i , h i , η i , and ν i denote the Young’s modulus, thickness, loss factor, and Poisson’s ratio of the i th plate, respectively. This expression for complex flexural rigidity is commonly utilized to incorporate damping effects through the loss factor η i . The inclusion of the complex term ( 1 + j η i ) is a standard approach in harmonic analysis to account for losses in a phenomenological manner. Following previous studies [
w t 1 ( x , y , t ) = ∑ m , n ϕ m n ( x , y ) α 1 , m n e j ω t , w t 2 ( x , y , t ) = ∑ m , n ϕ m n ( x , y ) α 2 , m n e j ω t , (12)
where α 1 , m n and α 2 , m n are the modal displacement coefficients of the upper and bottom plates, respectively.
Given the expressions for the incident and transmitted waves, Φ i and Φ t , in Equations (1) and (4), we can now utilize the continuity of normal velocity at
| Boundary Condition | Description | Modal Function ϕ m n ( x , y ) |
|---|---|---|
| CCCC | All edges clamped (fully clamped) | ( 1 − cos ( 2 m π x a ) ) ( 1 − cos ( 2 n π y b ) ) |
| SSSS | All edges simply supported | sin ( m π x a ) sin ( n π y b ) |
| FFFF | All edges free (free-free plate) | cos ( ( m − 1 ) π x a ) cos ( ( n − 1 ) π y b ) |
| CCSS | Mixed clamped and simply supported | ( 1 − cos ( 2 m π x a ) ) sin ( n π y b ) |
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 ( x , y ) = 1 − e j ( k x x + k y y ) k i z ∑ m , n ( ω ϕ m n − j V ⋅ ∇ ϕ m n ) α 1 , m n , (13)
T ( x , y ) = ω k t z e j ( k x x + k y y + k t z ( L g 1 + H p + L g 2 ) ) ∑ m , n ϕ m n α 2 , m n . (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 Φ i and Φ 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 g 1 , R g 1 , I g 2 and R g 2 ) can be determined by a 8 × 8 matrix equation:
M C = S (15)
where
C = [ C 1 C 2 C 3 C 4 I g 1 R g 1 I g 2 R g 2 ] t , (16)
S = [ ∑ m , n α 1 , m n ϕ m n e j ( k x x + k y y ) 0 0 0 0 0 0 ∑ m , n α 2 , m n ϕ m n e j ( k x x + k y y ) ] t . (17)
To determine the unknown modal coefficients α 1 , m n , α 2 , m n , and α 3 , m n , the principle of virtual work is employed. The virtual work δ 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:
δ i = δ α i , m n ϕ m n ( x , y ) ( i = 1 , 2 ) . (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:
∫ 0 b ∫ 0 a ( B 1 ∇ 4 w t 1 + m s 1 ∂ 2 w t 1 ∂ 2 t − p i + p g 1 ) δ 1 d x d y = 0 , (19)
∫ 0 b ∫ 0 a ( B 2 ∇ 4 w t 2 + m s 2 ∂ 2 w t 2 ∂ 2 t − p g 2 + p t ) δ 2 d x d y = 0 . (20)
The variables Φ i , Φ g 1 , Φ g 2 , Φ t in p i , p g 1 , p g 2 and p t , and w t 1 , w t 2 have been expressed in terms of α 1 , m n and α 2 , m n using the solution of the unknown vector C .
Sound intensity in the incident or transmitted field is given by:
I = 1 2 Re ( p v * ) , (21)
where v * is the complex conjugate of acoustic velocity v , and Re ( ⋅ ) denotes the real part. The incident and transmitted acoustic power over plate surface area S are:
Π i = 1 2 Re [ ∫ S p i v i z * d S ] , Π t = 1 2 Re [ ∫ S p t v t z * d S ] , (22)
where p i , p t , v i z , and v t z represent pressure and velocity fields in the incident and transmitted domains. The power transmission coefficient for a single incident wave is:
τ ( φ i , θ ) = Π t Π i . (23)
Sound transmission loss (STL), expressed in decibels, is:
STL = 10 log ( 1 τ ) . (24)
This metric evaluates acoustic system performance, with details in prior studies [
To study external flow and poroelastic core effects on STL in a double-plate sandwich structure, material parameters from [
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
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.
The impact of air gap height ( H g , defined as the sum of H g 1 + H 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 ( φ i = 0 ∘ ) and oblique incidence ( φ i = 45 ∘ ).
By comparing
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 mm and t 2 = 3.9 mm outperforms the t eq = 4 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 mm , t 2 = 3.9 mm ) matches the STL of a t eq = 20 mm monolithic plate, driven by mass-stiffness asymmetry and optimal impedance.
- The case with t 1 = 3 mm and t 2 = 1 mm resembles the uniform t eq = 4 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.
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 mm ) and thick lower plate ( t 2 = 3.9 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.
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.
The authors declare no conflicts of interest regarding the publication of this paper.