A general stable approach to modeling and coupling multilayered sys- tems with various types of layers Guochenhao Song 1 Zhuang Mo 2 J. Stuart Bolton 3 Ray W. Herrick Lab, Purdue University 177 South Russell Street, West Lafayette, IN 47907, USA

ABSTRACT In this article, a general method is proposed to model layered systems with two-by-two transfer ma- trices, and further, to solve for the acoustic absorption, reflection, and transmission coefficients. Since the proposed method uses the matrix representation of various layers and interfaces from the Transfer Matrix Method (TMM), the equation system can be established efficiently. However, the traditional TMM can lose stability when there is a large disparity between the magnitudes of the waves traveling in opposite directions within the layers (i.e., at higher frequencies, for a thick layer, or for extreme parameter values). In such cases, the contribution of the most attenuated wave can be masked by numerical errors and can induce instability when solving the system. Therefore, in the proposed method, to stabilize the calculated acoustic properties of the system, the principle is to ensure the accuracy of the wave attenuation terms by decomposing each layer’s transfer matrix and reformulating the equation system. This method can couple different layer types in a general way and is easy to assemble and implement with numerical code. The predicted acoustic properties of layered systems calculated using the proposed method have been validated by comparison with those pre- dicted by other existing methods.

1. INTRODUCTION

The modeling of layered structures is an essential topic in a wide range of areas: i.e., noise control, vibration control, geodynamics, etc. Numerical methods like finite element and boundary element methods have become popular for systems with complex geometry, although their computational cost is relatively expensive. On the contrary, analytical models are usually developed under certain as- sumptions: e.g., each sub-element layer has a relatively large lateral dimension and is homogeneous. As a result, they usually can produce much higher efficiency and make it possible to perform various computationally intensive tasks: e.g., parameter studies and design optimizations.

Previously, several methods have been proposed to couple different types of layers. The “arbitrary coefficient method” (ACM) [1,2] proposed by Bolton et al. first expresses the field variables with the

1 song520@purdue.edu

2 mo26@purdue.edu

3 bolton@purdue.edu

2022

amplitude of the propagating waves based on the governing equations of each layer. Then, the equa- tion system to be solved is constructed according to the boundary conditions at interfaces. The ACM is a general method to model multilayered systems, even with poro-elastic layers. However, the method can be time-consuming when redeveloping the equation system for a different layer layout.

On the contrary, the “global transfer matrix” (GTM) [3,4] method proposed by Brouard et al. models the multilayered system using a developed transfer matrix representation. Therefore, the equa- tion system can be easily constructed for different layouts. In the GTM procedure, a set of homoge- neous governing equations is first formed with respect to the layer and boundary condition infor- mation: i.e., transfer matrices, boundary condition matrices, and the impedance at the back of the system. Based on the equation system, a global transfer matrix is defined. The acoustic properties of the layered system (i.e., surface impedance, reflection coefficient, absorption coefficient, and trans- mission coefficient) can be predicted from this global transfer matrix by calculating the determinant of some of its submatrices. GTM is known to be a general method to model and couple any layers with the knowledge of their transfer matrices and boundary condition matrices.

More recently, Xue et al. have proposed a modified transfer matrix method (modified TMM) [5] to couple elastic-solid or poro-elastic layers with fluid layers on both sides (or bonded to a rigid wall) and further to reduce the dimension of the layered system to a two-by-two matrix. Matrix decompo- sition methods have been implemented to extract two important modes so as to reduce the system dimension. As for this modified TMM, the idea of simplifying complex systems into two-by-two transfer matrices makes it extremely easy to couple a series of subsystems together. Based on this standard two-by-two transfer matrix, the acoustic properties of the system can then be predicted [6,7].

However, all of ACM, GTM, and modified TMM will lose stability when there is a significant disparity between the magnitudes of the waves traveling in opposite directions within the layers (i.e., at higher frequencies, for a thick layer, or extreme parameter values). In such cases, the contribution of the most attenuated wave can be masked by numerical errors and can induce instability when solving the system. Therefore, Dazel et al. proposed a method to decompose the transfer matrix to extract the exponential terms and then to use a recursive approach to model layered systems stably [8]. The principle of their approach is to work with the information vectors rather than the state vector, which involves exponential growing/decaying terms. As a result, Dazel et al. ’s method can predict the acoustic properties stably and does not lead to an enlargement of the system size. Since the fun- damental theory for the governing equations and boundary conditions is the same in all approaches, the predicted acoustic properties also agree with each other when the results are stable.

In this paper, inspired by Xue et al. ’s work [5], the proposed stabilized transfer matrix method (stabilized TMM) also solves the two-by-two transfer matrix that relates the state vectors on the top and bottom of the layered system. In addition, in order to couple and model various types of layers in a general way, the proposed stabilized TMM constructs the equation system in a similar way as GTM. Since the stabilized TMM only solves the relationship between the first and last state vectors, the additional impedance information at the back of the system is unnecessary. More importantly, the proposed method introduces an additional step to decompose the global matrix to avoid the inaccurate numerical evaluation of the exponential terms (i.e., wave attenuation terms). Thus, it can stably pre- dict the acoustic properties of the system. Compared with another stable approach, Dazel et al. ’s method [8], the stabilized TMM would inevitably lead to an enlargement of the system dimension when the number of layers increases. However, it’s arguable that, in Dazel et al. ’s method, the whole theory of the recursive approach is abstract, and the information vector does not directly represent the field variables in the layered systems. Instead, the methodology used in the stabilized TMM is more straightforward and directly works with the field variables.

In this paper, Section 2 introduces the transfer matrix notation for various layers. Then in Section 3, the approach to stably predict the acoustic properties of multi-layered systems is illustrated. In Section 4, The stabilized TMM is further verified by comparing the predicted acoustic properties from it with those from the other previously proposed methods. Finally, Section 5 concludes on the benefits and potential applications of the stabilized TMM.

2. THE MATRIX REPRESENTATIONS

2.1. Transfer matrix of different layers A multilayered system with fluid, elastic-solid, or poro-elastic layers is often modeled using the trans- fer matrix method. This method reformulates the governing equations of a layer into a matrix repre- sentation. In other words, it can relate the state vectors at two surfaces, 𝑽 𝟏 ! , and 𝑽 𝟐 " , with a transfer matrix, [𝑻] . Figure 1 shows the general transfer matrix representation of a one-layer system.

Figure 1: The general transfer matrix representation of one layer.

To be more specific, the acoustic field in a fluid layer only consists of two waves (a single dilata- tional wave propagating forward and backward). Therefore, as listed in Table 1, the field at two sur- faces can be described with two-by-one state vectors, 𝑉 # ! $ and 𝑉 % "

$ . Similarly, the elastic-solid layer, in which one dilatational and one rotational waves propagate forward and backward, has a four-by- one state vector [3], 𝑉 # ! & and 𝑉 % " & ; and the poro-elastic layer, in which two dilatational and one rota- tional waves propagate forward and backward, has a six-by-one state vector [3,9], 𝑉 # ! ' and 𝑉 % "

' . In Table 1, 𝑝 refers to pressure, 𝜐 to velocities, and 𝜎 to stresses; superscripts and subscripts denote the medium and the directions, respectively.

Table 1: Transfer matrix notation and corresponding state vectors associated with different layers.

Layer type Transfer matrix State vector

+

Fluid +𝑻 𝒇 , 𝟐×𝟐 𝑽 𝒇 = +𝑝 𝜐 *

$ ,

& ] +

& 𝜎 **

& 𝜏 .*

& 𝜐 *

Elastic-solid [𝑻 𝒔 ] 𝟒×𝟒 𝑽 𝒔 = [𝜐 .

$ , +

Poro-elastic [𝑻 𝒑 ] 𝟔×𝟔 𝑽 𝒑 = +𝜐 .

$ 𝜎 **

& 𝜐 *

& 𝜐 *

& 𝜏 .*

& 𝜎 **

2.2 Boundary conditions at the interface of different layers Since the dimension of the transfer matrix and the definition of the state vector are different for dif- ferent types of layers, a set of boundary conditions is required to connect two adjacent layers. There- fore, as shown in Figure 2, the boundary condition matrices have been introduced. The matrix repre- sentation for the boundary conditions at the i th interface is:

[𝑩 𝒊 " ]𝑽 𝒊 " = [𝑩 𝒊 ! ]𝑽 𝒊 ! , (1)

where [𝑩 𝒊 " ] and [𝑩 𝒊 ! ] denote the boundary condition matrices for the upper and lower interface, and 𝑽 𝒊 " and 𝑽 𝒊 ! denote the state vectors at the upper and lower interfaces. Note that the definition of boundary condition matrices here differs from that in previous literature [5]. However, with this rep- resentation, the implementation of the proposed stabilized TMM is easier.

Var = (Tax + Ver PATE TEI], aie sate vestry = [7] State vector V;- a Layer depth

Figure 2: Boundary condition matrices and the state vectors at the interface.

Since the boundary conditions that govern different interfaces are different, the boundary condition matrices, [𝑩 𝒊 " ] and [𝑩 𝒊 ! ] , are different as well. The boundary condition matrices for the interfaces connecting the fluid, elastic-solid, or poro-elastic layers have already been derived in the literature. Note that the definition of the state vectors in the present work is slightly different from the one used in deriving boundary condition matrices [5]. Hence, some minor differences would occur in the cor- responding matrices.

3. METHOD TO STABILIZE THE TMM

The traditional transfer matrix method will lose stability when one of the waves is more attenuated than the other waves (i.e., in the higher frequencies or for a thick layer). In such cases, the minor truncation errors in the transfer matrix are comparable to the contribution of the most attenuated wave. Although these errors seem not to play a role in the transfer matrix, they will cause instability in the process of inverting the matrix to solve for the dimension-reduced transfer matrix. The instability can further affect the calculation of reflection, absorption, and transmission coefficients. Therefore, in- stead of directly using the transfer matrix, which has already introduced some errors, a process to analytically decompose the matrix is required to control the wave attenuation terms: i.e.,

[𝐓] = [𝚽][𝚲][𝚽] 2𝟏 , (2)

where the diagonal matrix, [𝚲] , consists of the attenuation terms, 𝑒 34 # 5 , of all propagating waves. When one of the waves is highly attenuated, this diagonal matrix can be nearly singular. Some minor truncation errors will be introduced when multiplying matrices on both sides of the diagonal matrix [𝚲] , and further induce instability. Thus, the transfer matrix decomposition is essential to stabilize the transfer matrix approach. This approach is also used in Dazel et al. ’s approach to stably model multilayered systems [8]. Built upon the decomposed matrices, the method to stabilize the transfer matrix approach is proposed here to model and couple multilayered structures.

As for a multilayered structure with different types of layers, since the dimensions of the transfer matrices are different, they cannot be modeled directly with simple multiplications. Hence, the issue is to operate with different types of layers while still keeping the result stable. The conventional methods like arbitrary coefficient method (ACM) and global transfer matrix method (GTM) can lose stability in a one-layer system, let alone in multilayered systems. Thus, it is valuable to propose a method that can stably model and couple multilayered structures. Here, a stable method is proposed to reduce the dimension of the system and relate the two-by-one state vectors at the top and the bottom of the system, 𝑽 𝟏 ! , and 𝑽 𝒏7𝟏 " , with a two-by-two transfer matrix, [𝑻] 𝟐×𝟐 . Based on [𝑻] 𝟐×𝟐 , the ma- terial acoustic properties (i.e., transmission, reflection, and absorption coefficients) can be further solved with a standardized transfer matrix procedure [6].

For a general 𝑛 -layer multilayered system shown in Figure 3, boundary conditions at each inter- face can be expressed as:

[𝑩 𝟏 " ]𝑽 𝟏 " = [𝑩 𝟏 ! ]𝑽 𝟏 ! , [𝑩 𝟐 " ][𝑻 𝟏 ] 2𝟏 𝑽 𝟐 ! = [𝑩 𝟐 ! ]𝑽 𝟐 ! ,

⋯ [𝑩 𝒏 " ][𝑻 𝒏2𝟏 ] 2𝟏 𝑽 𝒏2𝟏 ! = [𝑩 𝒏 ! ]𝑽 𝒏 ! , [𝑩 𝒏7𝟏 " ][𝑻 𝒏 ] 2𝟏 𝑽 𝒏 ! = [𝑩 𝒏7𝟏 ! ]𝑽 𝒏7𝟏 ! .

(3)

DWO“SS -\\ 7 interface |

Figure 3: The general transfer matrix representation for a multilayered structure. Further, the governing equations can be collected into a matrix form: i.e.,

⎣ ⎢ ⎢ ⎢ ⎢ ⎡ [𝑩 𝟏 " ] −[𝑩 𝟏 ! ] ⋯ [𝟎] [𝟎] [𝟎] [𝑩 𝟐 " ][𝑻 𝟏 ] 2𝟏 ⋯ [𝟎] [𝟎] ⋯ ⋯ ⋯ ⋯ ⋯ [𝟎] [𝟎] ⋯ [𝑩 𝒏 " ][𝑻 𝒏2𝟏 ] 2𝟏 −[𝑩 𝒏 ! ] [𝟎] [𝟎] ⋯ [𝟎] [𝑩 𝒏7𝟏 " ][𝑻 𝒏 ] 2𝟏 ⎦

⎣ ⎢ ⎢ ⎢ ⎡ 𝑽 𝟏 "

⎣ ⎢ ⎢ ⎢ ⎡ 0 0 ⋯

⎥ ⎥ ⎥ ⎥ ⎤

⎥ ⎥ ⎥ ⎤

⎥ ⎥ ⎥ ⎤

𝑽 𝟏 !

⋯ 𝑽 𝒏2𝟏 !

=

𝑽 𝒏7𝟏 ! . (4)

0 [𝑩 𝒏7𝟏 ! ]⎦

𝑽 𝒏 ! ⎦

The higher-dimension square matrix in Equation (4) is defined as the global matrix [𝑨] . The de- composed transfer matrix [𝑻 𝒊 ] = [𝚽 𝒊 ][𝚲 𝒊 ][𝚽 𝒊 ] 2𝟏 is implemented analytically and the global matrix [𝑨] is further decomposed to increase the stability: i.e.,

⎣ ⎢ ⎢ ⎢ ⎡ [𝑰] [𝟎] ⋯ [𝟎] [𝟎] [𝟎] [𝚽 𝟏 ] 2𝟏 ⋯ [𝟎] [𝟎] ⋯ ⋯ ⋯ ⋯ ⋯ [𝟎] [𝟎] ⋯ [𝚽 𝒏2𝟏 ] 2𝟏 [𝟎] [𝟎] [𝟎] ⋯ [𝟎] [𝚽 𝒏 ] 2𝟏 ⎦

⎥ ⎥ ⎥ ⎤

[𝑨] = [𝑨 𝟏 ]

, (5)

where

⎣ ⎢ ⎢ ⎢ ⎢ ⎡ [𝑩 𝟏 " ] −[𝑩 𝟏 ! ][𝚽 𝟏 ] ⋯ [𝟎] [𝟎] [𝟎] [𝑩 𝟐 " ][𝚽 𝟏 ][𝚲 𝟏 ] 2𝟏 ⋯ [𝟎] [𝟎] ⋯ ⋯ ⋯ ⋯ ⋯ [𝟎] [𝟎] ⋯ [𝑩 𝒏 " ][𝚽 𝒏2𝟏 ][𝚲 𝒏2𝟏 ] 2𝟏 −[𝑩 𝒏 ! ][𝚽 𝒏 ] [𝟎] [𝟎] ⋯ [𝟎] [𝑩 𝒏7𝟏 " ][𝚽 𝒏 ][𝚲 𝒏 ] 2𝟏 ⎦

⎥ ⎥ ⎥ ⎥ ⎤

[𝑨 𝟏 ] =

. (6)

The decomposed global matrix [𝑨] can be considered to be, for 𝑖 ranges from 1 to 𝑛 , mapping the boundary conditions [𝑩 𝒊 ! ] and [𝑩 𝒊7𝟏 " ] to [𝑩 𝒊 ! ][𝚽 𝒊 ] and [𝑩 𝒊7𝟏 " ][𝚽 𝒊 ] and the state vectors [𝑽 𝒊 ! ] to [𝚽 𝒊 ] 2𝟏 [𝑽 𝒊 ! ] . It can be proved that [𝚽 𝒊 ] 2𝟏 can translate the state vector to a vector consisting of the amplitudes of all the waves at 𝑖 2 interface. As a result, the relationship between two surfaces of the layer can be simply characterized by a diagonal matrix [𝚲 𝒊 ] , which describes the attenuation of each

Vy+ = [Tlax2 + Vaei- (>, = r r| ch ae State vector V+ = [>], [B,+] — _ State veet 1] Vi- = [Ti] - Ver | | a Lean S State v vector 1 Vy: y+ i Vo- = [Tz] -V3+ = 2 vector r V5 + ss — : = NS vector \\ 3 : n n =([T]- Viasat 7 = = State v vector K 1+ NS 1*| 1+ 2+ 2- 3+ 3- n+1* [P] Bar] er nas Air/rigid backing State vector V,44- =

wave in the layer. On the other hand, this process also avoids introducing additional numerical trun- cation errors when inverting the global matrix [𝑨] , hence, increasing stability. The proposed stabi- lized TMM is, to some degree, similar to the ACM method [1], but the reformulated matrix ensures a stable result. In addition, as the proposed method incorporates the transfer matrix method, the equa- tion system can be constructed easily. The inversion of the global matrix [𝑨] can be written as:

⎣ ⎢ ⎢ ⎢ ⎡ [𝑰] [𝟎] ⋯ [𝟎] [𝟎] [𝟎] [𝚽 𝟏 ] ⋯ [𝟎] [𝟎] ⋯ ⋯ ⋯ ⋯ ⋯ [𝟎] [𝟎] ⋯ [𝚽 𝒏2𝟏 ] [𝟎] [𝟎] [𝟎] ⋯ [𝟎] [𝚽 𝒏 ]⎦

⎥ ⎥ ⎥ ⎤

[𝑨] 2# =

[𝑨 𝟏 ] 2𝟏 ,

(7)

∗ , +𝑨 𝟏,𝟐

∗ , ⋯ +𝑨 𝟏,𝒏2𝟏

∗ , +𝑨 𝟏,𝒏

∗ ,

⎣ ⎢ ⎢ ⎢ ⎢ ⎡ +𝑨 𝟏,𝟏

⎥ ⎥ ⎥ ⎥ ⎤

∗ , +𝑨 𝟐,𝟐

∗ , ⋯ +𝑨 𝟐,𝒏2𝟏

∗ , +𝑨 𝟐,𝒏

∗ , ⋯ ⋯ ⋯ ⋯ ⋯ +𝑨 𝒏2𝟏,𝟏

+𝑨 𝟐,𝟏

=

.

∗ , +𝑨 𝒏2𝟏,𝟐

∗ , ⋯ +𝑨 𝒏2𝟏,𝒏2𝟏

∗ , +𝑨 𝒏2𝟏,𝒏

∗ ,

∗ , +𝑨 𝒏,𝟐

∗ , ⋯ +𝑨 𝒏,𝒏2𝟏

∗ , +𝑨 𝒏,𝒏 ∗ , ⎦

+𝑨 𝒏,𝟏

According to Equation (4), the dimension-reduced transfer matrix that relates the state vectors [𝑽 𝟏 " ] and [𝑽 𝒏7𝟏 ! ] can be solved with a sub-element of the inversion of the constructed matrix, +𝑨 𝟏,𝒏

∗ , , and the last unused boundary condition matrix, [𝑩 𝒏7𝟏 ! ] ,: i.e.,

[𝑻] %×% = D 𝑇 ## 𝑇 #% 𝑇 %# 𝑇 %% F = +𝑨 𝟏,𝒏

∗ ,[𝑩 𝒏7𝟏 ! ], (8)

so that the state vectors at two surfaces have a relation of [𝑽 𝟏 " ] = [𝑻] %×% [𝑽 𝒏7𝟏 ! ] . With this dimen- sion-reduced transfer matrix, the transmission coefficient, 𝑇 , reflection coefficient, 𝑅 , absorption co- efficient, 𝛼 , and transmission loss, 𝑇𝐿 , can be easily calculated from a traditional transfer matrix method [6,7]. For example, a layer with fluid on both sides:

𝑇= 2𝑒 34 $ 5

, (9a)

𝑇 ## + 𝑇 #% cos𝜃/𝜌 : 𝑐+ 𝑇 %# 𝜌 : 𝑐/ cos 𝜃+ 𝑇 %%

𝑅= 𝑇 ## + 𝑇 #% cos 𝜃/𝜌 : 𝑐−𝑇 %# 𝜌 : 𝑐/cos 𝜃−𝑇 %%

, (9b)

𝑇 ## + 𝑇 #% cos 𝜃/𝜌 : 𝑐+ 𝑇 %# 𝜌 : 𝑐/cos 𝜃+ 𝑇 %%

𝛼= 1 −|𝑅| % , (9c)

1 |𝑇| , (9d)

𝑇𝐿= 20 log #:

where 𝜃 denotes incidence angle (being 0 ∘ for normal incidence), 𝑘 * = 𝑘cos 𝜃 denotes the wave- number component in the z-direction, 𝑑 denotes the layer thickness, and 𝜌 : 𝑐 : denotes the character- istic impedance of the fluid. In addition, for a layer fixed on a rigid wall, the reflection coefficient is given as:

𝑅= 𝑇 ## cos 𝜃/(𝑇 %# 𝜌 : 𝑐) −1

𝑇 ## cos 𝜃/(𝑇 %# 𝜌 : 𝑐) + 1 . (10)

4. EXAMPLES

The acoustic properties of two different configurations of layered systems (as shown in Figure 4) were predicted by the proposed stabilized TMM and verified by comparison with other methods (i.e., Xue et al. ’s method, GTM, and Dazel et al. ’s method). The first configuration is a 10 cm poro-elastic

layer bonded to a rigid wall, and the other configuration is a 10 cm poro-elastic layers bonded to a 10 cm solid-elastic layer. The properties of the 10 cm poro-elastic layer (i.e., flow resistivity, 𝜎 , porosity, 𝜙 , tortuosity, 𝛼 < , viscous characteristic length, Λ , thermal characteristic length, Λ′ , frame density, 𝜌 # , Young's modulus, 𝐸 , mechanical lossing factor, 𝜂 , and Poisson's ratio, 𝜈 ) are given in Table 2, and the properties of the 10 cm solid-elastic layer (i.e., bulk density, 𝜌 = , Young's modulus, 𝐸 , me- chanical lossing factor, 𝜂 , and Poisson's ratio, 𝜈 ) are given in Table 3. Table 2: Parameters for poro-elastic layer

𝝈 – Rayls/m 𝝓 𝜶 < 𝚲 – m 𝚲′ – m 𝝆 𝟏 – kg/m 3 𝑬 - Pa 𝜼 𝝂

4 × 10 > 0.4 1.75 9.3 × 10 2> 2.0 × 10 2? 120 4 × 10 @ 0.2 0.3

Table 3: Para meters for solid-elastic layer

𝝆 𝒃 – kg/m 3 𝑬 - Pa 𝜼 𝝂

1000 1.0 × 10 ? 0.5 0.4

As shown in the one-layered example in Figure 4, the absorption coefficient of a 10 cm poro- elastic layer bonded to the wall was predicted by the stabilized TMM, and three methods from liter- ature (i.e., Xue et al. ’s method, GTM, and Dazel et al. ’s method). The predicted absorption coeffi- cients from all the methods agree with each other in the low-frequency region. However, both Xue et al. ’s method and GTM give a noisy prediction of acoustic properties at higher frequencies (i.e., > 500 Hz). In contrast, by ensuring the accuracy of the wave attenuation terms, the proposed stabilized TMM produces the same stable results as Dazel et al. ’s approach.

In the two-layered example in Figure 4, the predicted absorption coefficient of a 10 cm poro-elastic layer bonded with a 10 cm solid-elastic layer is presented. Since the Xue et al. procedure was not designed for this type of configuration and Dazel et al. also did not provide the interface operators between the poro-elastic and elastic-solid layers, only the predictions from stabilized TMM and GTM are compared. Similar conclusions can be drawn from the multi-layered case as for the one-layer case. That is, both methods produce identical stable absorption coefficient predictions in the low-frequency region, while the stabilized TMM shows its benefits as it ensures a stable prediction of acoustic prop- erties at higher frequencies without increasing the computational complexity. 5. CONCLUSIONS

A stabilized transfer matrix method has been developed to model and couple a multi-layered system consisting of various layers with a simple two-by-two transfer matrix and further to predict the ab- sorption coefficient of the layered system. The method decomposes the global matrix analytically so that the exponential wave attenuation terms are controlled. Therefore, the contribution of the most attenuated wave will not be masked by truncation errors, thus, increasing the stability when inverting the global matrix [𝑨] .

In addition, similar to Xue et al. ’s method, the stabilized TMM models the layered system with a two-by-two transfer matrix. Hence, it can be conveniently connected to other systems with same- dimension transfer matrices by simple multiplications. This makes redesigning a complicated system much easier. Since it also formulates the equation system in a way similar to the GTM, compared with Xue et al. ’s method, the method can model and couple multi-layered systems in a more general way. Overall, this method is stable, general, robust, and straightforward. As a modeling tool, it can be useful when some of the layers in the system are thick or when the high-frequency absorption coefficients are of the most interest. Further work will be needed to develop the transfer matrix rep- resentation for some special types of layers (i.e., stiff panels) to extend the applications of the method.

Figure 4: Predicted absorption coefficient of multi-layered systems. Left: one-layer system (a 10 cm

poro-elastic layer) bonded to a rigid wall. Right: two-layer system (10 cm poro-elastic layer + 10

cm elastic-solid layer) with air on both sides. 6. ACKNOWLEDGMENTS

We are grateful to our colleagues at the 3M Company for their financial support of this work.

7. REFERENCES

1. Bolton, J. S., Shiau, N. M., and Kang, Y. J., 1996, “Sound Transmission through Multi-Panel

Structures Lined with Elastic Porous Materials,” Journal of Sound and Vibration , 191 (3), pp. 317–347. 2. Bolton, J. S. and Shiau, N.-M., 1987, “Oblique Incidence Sound Transmission through Multi-

Panel Structures Lined with Elastic Porous Materials,” 11th Aeroacoustics Conference , p. 2660. 3. Allard, J. F., and Atalla, N., 2009, Propagation of Sound in Porous Media: Modelling Sound

Absorbing Materials . 4. Brouard, B., Lafarge, D., and Allard, J. F., 1995, “A General Method of Modelling Sound

Propagation in Layered Media,” Journal of Sound and Vibration , 183 (1), pp. 129–142. 5. Xue, Y., Bolton, J. S., and Liu, Y., 2019, “Modeling and Coupling of Acoustical Layered Systems

That Consist of Elements Having Different Transfer Matrix Dimensions,” Journal of Applied Physics , 126 (16). 6. ASTM, 2019, “Standard Test Method for Normal Incidence Determination of Porous Material

Acoustical Properties Based on the Transfer Matrix Method E2611,” American Society for Testing of Materials , pp. 1–14. 7. Song, B. H., and Bolton, J. S., 2000, “A Transfer-Matrix Approach for Estimating the

Characteristic Impedance and Wave Numbers of Limp and Rigid Porous Materials,” The Journal of the Acoustical Society of America , 107 (3), pp. 1131–1152. 8. Dazel, O., Groby, J. P., Brouard, B., and Potel, C., 2013, “A Stable Method to Model the Acoustic

Response of Multilayered Structures,” Journal of Applied Physics , 113 (8). 9. Biot, M. A., 1956, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid II.

Higher Frequency Range,” Journal of the Acoustical Society of America , 28 (2), pp. 179–191.