External mean flow influence on sound transmission through finite clamped double-wall sandwich panels

Journal of Sound and Vibration 405 (2017) 269–286

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External mean flow influence on sound transmission through finite clamped double-wall sandwich panels Yu Liu a,b,n, Jean-Cédric Catalan b,1 a b

Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, PR China Department of Mechanical Engineering Sciences, University of Surrey, Guildford GU2 7XH, UK

a r t i c l e i n f o

abstract

Article history: Received 22 December 2016 Accepted 28 May 2017 Handling Editor: L.G. Tham

This paper studies the influence of an external mean flow on the sound transmission through finite clamped double-wall sandwich panels lined with poroelastic materials. Biot's theory is employed to describe wave propagation in poroelastic materials and vari20ous configurations of coupling the poroelastic layer to the facing plates are considered. The clamped boundary of finite panels are dealt with by the modal superposition theory and the weighted residual (Garlekin) method, leading to a matrix equation solution for the sound transmission loss (STL) through the structure. The theoretical model is validated against existing theories of infinite sandwich panels with and without an external flow. The numerical results of a single incident wave show that the external mean flow has significant effects on the STL which are coupled with the clamped boundary effect dominating in the low-frequency range. The external mean flow also influences considerably the limiting incidence angle of the panel system and the effect of the incidence angle on the STL. However, the influences of the azimuthal angle and the external flow orientation are negligible. & 2017 Elsevier Ltd. All rights reserved.

Keywords: Double-wall sandwich panels External mean flow Sound insulation Poroelastic material Clamped boundary condition

1. Introduction Double-wall sandwich panels have been widely used in many applications (e.g. transportation vehicles, modern buildings, and aerospace structures) due to their superior sound insulation properties over a wide frequency range and excellent mechanical properties. Poroelastic materials have been applied as the sandwich core within a double-wall (or multilayered) panel in order to improve the sound insulation performance. Based on Biot's theory [1], Bolton et al. [2] developed an analytical model for sound transmission through laterally infinite double-wall panels lined with poroelastic materials and have validated the model experimentally; this model was then extended by Liu [3] to triple-wall sandwich panels. Lee et al. [4] simplified the method of Bolton et al. by considering only the energetically dominant wave with negligible shear wave contributions and the poroelastic material was treated as a layer of equivalent fluid. Tanneau et al. [5] proposed an optimisation method for maximising sound transmission loss (STL) of infinite multilayered panels including solid, fluid and porous components, and Lee et al. [6] studied the optimal poroelastic layer sequencing using a topology optimisation method.

n

Corresponding author at. E-mail address: [email protected] (Y. Liu). 1 Visiting student from ENSTA ParisTech, École Nationale Supérieure de Techniques Avancées, Université Paris-Saclay, Palaiseau, France.

http://dx.doi.org/10.1016/j.jsv.2017.05.049 0022-460X/& 2017 Elsevier Ltd. All rights reserved.

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The modelling of sound transmission through realistic sandwich panels, however, needs to take into account the finite nature of the structure. Brunskog [7] examined the influence of finite cavities on the sound insulation of periodically framed double-wall structures. Villot et al. [8] proposed an approximate technique based on spatial windowing of plane waves to predict sound radiation and transmission of finite multilayered structures. Leppington et al. [9] employed a modal superposition theory to model the sound transmission through a pair of rectangular elastic plates with simply supported boundary condition. Using the same method, Xin et al. [10] studied theoretically the vibroacoustic response of a clamped double-panel partition enclosing an air cavity. Xin and Lu [11] then considered both fully clamped and simply supported boundary conditions through an analytical and experimental investigation. Liu and Daudin [12,13] extended these studies to consider clamp mounted double-wall panels lined with poroelastic materials. Panneton and Atalla [14] and Sgard et al. [15] carried out numerical predictions of the STL through finite double-wall panels with poroelastic linings using finite element method and boundary element method, respectively. In many practical applications (for example, aircraft and high-speed trains) of sandwich panels for sound insulation purposes, the presence of an external flow is common and hence its influence on sound transmission must be considered. Koval [16] found that the external flow, panel curvature and internal pressurisation affect significantly the STL of an infinite single-wall panel. Xin and Lu modelled analytically the sound transmission through a finite single panel with convective flows on both sides [17] and through an infinite double-wall panel with an external mean flow [18]. Meng et al. [19] studied this problem for infinite double-wall panels lined with acoustic absorptive materials. Zhou et al. [20] extended the work of Bolton et al. [2] to account for the external mean flow effect on the STL of double-wall panels with poroelastic linings. More recently, Liu and Sebastian [21] considered the effects of both an external and an internal mean flow on the sound transmission through double-wall sandwich panels. In addition, the similar problem for the convective effect of such an external mean flow has also been studied extensively for double-wall sandwich shells [22–28]. All these studies have shown that the presence of an external flow improves the sound insulation performance of the structures. However, as far as sound transmission through double-wall panels is concerned, the problem considering finite dimensions, poroelastic materials, and an external mean flow has yet to be addressed theoretically, to the best knowledge of the authors. In spirit of the previous works (e.g. [11,13,20]), the present study aims to develop such a theoretical model for the sound transmission across clamp mounted finite double-wall panels lined with poroelastic materials, with a focus on the influence of an external mean flow on the STL as well as the coupled effects of the external mean flow and the clamped boundary. The remaining paper presents in Section 2 a theoretical formulation of the vibroacoustic problem. Section 3 determines the sound transmission loss and the limiting angle of incidence. The validation of the theoretical model and the numerical results on the STL, the finite extent effect and the external flow influence are discussed in detail in Section 4. The conclusions with a summary of the findings are made in Section 5.

2. Theoretical formulation 2.1. Description of the system As illustrated in Fig. 1, the double-wall sandwich panel system consists of two parallel homogeneous thin elastic plates lined with a poroelastic layer. An air gap exists in this configuration between the poroelastic material and the facing plate. The system is situated in ambient air and a plane acoustic wave of unit amplitude is incident on the first (upper) plate and transmits through the system. The incidence field above the first plate and the transmission field underneath the second (bottom) plate are supposed to be semi-infinite. The air properties, i.e. air density and speed of sound, in the incident field, gap field and transmission field are assumed to be identical and are denoted as ρ 0 and c. The two rectangular plates are supposed to be fully clamped on each side to an infinite rigid baffle and the dimensions are a and b along the x-axis and y-axis, respectively. In the present study, an external mean flow is considered in the incident field (see Fig. 1(b)) while the media in the air gap and the transmission side are stationary. The mean flow with the constant velocity V is assumed to be uniform in the external incident field and the boundary layer effect is ignored which is common in many simplified modelling of sound

Fig. 1. Schematic diagram of sound wave transmission through the double-wall panel of the BU configuration: (a) side view, (b) perspective view.

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transmission problems [17–21]. Therefore, the vibroacoustic problem is formulated in a three-dimensional (3-D) form using an (x, y, z ) Cartesian system, the z-axis being aligned in the normal of the panel as shown in Fig. 1(b). The direction of the incident wave can be described by two angles: i.e. the incidence angle φ (0° ≤ φ ≤ 90°) between the incident wave and the normal of the x-y plane, and the azimuthal angle θ (0° ≤ θ ≤ 360°) between the projection of the incident wave on the x-y plane and the x-axis. The constant external flow is oriented at an angle of θ e from the x-axis. The subsequent derivations in Section 2.2 are based on the panel configuration shown in Fig. 1 as it is a representative case for the general theoretical formulation. 2.2. Velocity potential and wavenumber On the incident side, the sound field can be decomposed as an incident wave and a reflected wave; thus the acoustic velocity potential can be expressed in a harmonic form:

Φ1 = Φi + Φr = e jωt − j(k ixx + k iyy + k izz ) + Re jωt − j(k ixx + k iyy − k izz ) ,

(1)

where Φ i is the incident velocity potential with unit amplitude, Φ r the velocity potential of the reflected wave with the amplitude R, ω the angular frequency and the symbol j = −1 . Note that the influence of the external mean flow is included in the wavenumber ki of the acoustic waves in the incident field. The acoustic velocity potential in the external incident field satisfies the wave equation that considers a convective term [17,21]:

D2Φ1 Dt 2

= c 2∇2Φ1,

(2)

in which the material derivate is D/Dt = ∂/∂t + V·∇, and V = (V cos θe, V sin θe, 0) is the velocity vector of the external flow. Thus Eq. (2) can be expressed as 2 ⎛∂ ∂ ∂ ⎞ + V sin θe ⎟ Φ1 = c 2∇2Φ1. ⎜ + V cos θe ⎝ ∂t ∂x ∂y ⎠

(3)

Substituting the wave harmonic form (1) of Φ 1 into the above equation yields the wavenumber ki as

ki =

kis , 1 + Msin φ cos(θ − θe )

(4)

where the corresponding wavenumber in the stationary medium k is = ω/c and the Mach number of the external mean flow M ¼ V/c. The components of the wavenumber ki can be expressed as

kix = ki sin φ cos θ ,

kiy = ki sin φ sin θ ,

kiz =

(

)

ki2− kix2 + kiy2 .

(5)

In the air gap field, an incident wave and a reflected wave exist and the harmonic velocity potential can be expressed as

Φg = Φgi + Φgr = Ig e jωt − j(kgxx + kgyy + kgzz ) + Rg e jωt − j(kgxx + kgyy − kgzz ) ,

(6)

where Ig and Rg are the respective amplitudes of the incident and reflected waves in the air gap. Due to the stationary medium in this region, the wavenumber of the acoustic waves can be expressed as

kg = kis = ω/c .

(7)

As illustrated in Fig. 1(a), the acoustic waves inside the air gap make an angle φ g with the normal of the x-y plane. Thanks to the conservation of the tangential wavenumber components imposed by the Snell's law, the azimuthal angle θ remains unchanged across the panel system [3,20,21,29]. Therefore the components of the wavenumber kg are obtained as

kgx = kg sin φg cos θ ,

kgy = kg sin φg sin θ ,

kgz =

(

)

2 2 kg2− kgx + kgy .

(8)

An anechoic termination is assumed for the semi-infinite transmission field. Hence only a single transmitted wave exists in this region and the harmonic expression of the velocity potential is

Φt = Te jωt − j(ktxx + ktyy + ktzz ) ,

(9)

where T is the amplitude of the transmitted wave, and similarly the components of the wavenumber kt = ω/c are

ktx = kt sin φt cos θ ,

kty = kt sin φt sin θ ,

ktz =

(

)

2 kt2− ktx2 + kty .

(10)

Note that the air properties in both the air gap and the transmission field are identical; hence the wavenumber and the incidence angle remain the same in these two fields, i.e. kt ¼ kg, φt = φg .

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In a stationary medium, the acoustic pressure and the normal acoustic particle velocity can be related to the corresponding velocity potential by

p = ρ0

∂Φ = jωρ0 Φ, ∂t

vz = −

∂Φ = jk z Φ; ∂z

(11)

whereas in the presence of an external mean flow, the acoustic pressure in the incident field is subject to the convective effect and hence

p = ρ0

DΦ = j(ω − Vk x cos θe − Vk y sin θe )ρ0 Φ. Dt

(12)

2.3. Wave propagation in porous materials Biot's theory [1] is employed in this study to describe wave propagation in poroelastic materials. In this theory, porous materials are modelled as a solid and a fluid phase and two dilatational (frame and airborne) waves and one rotational (shear) wave are propagating in both phases. The positive and negative propagation of these waves along the z-axis introduce six wave components within the porous layer characterised by the amplitudes C1–C6. The displacement components, ux , uy , uz and Ux, Uy, Uz of the solid and fluid phases respectively, and the stress components σz , s, τxz , τyz can be expressed in terms of these constants. Using the boundary conditions of a specific configuration, a matrix equation, for example Eq. (24), is obtained that leads to the analytical solutions of the wave amplitudes C1–C6 and thus the displacement and stress components ux , uy , uz , Ux, Uy, Uz , and σz , s, τxz , τyz . The expressions of all these variables can be found in detailed descriptions in Refs. [2,3,20]. Among the six wave components, the constants C1–C4 are related to the dilatational waves whereas C5 and C6 define the rotational strains. Lee et al. [4] compared the energy associated with the three wave types and concluded that the contribution of the rotational wave is always negligible compared to those of the frame and airborne waves. Moreover, the clamped boundary conditions as shown in Fig. 1 restrict the mid-surface, in-plane motion of the panel and the shear wave propagation in the system. Therefore in this study, the shear components C5, C6 are excluded leaving only four components C1–C4 of the two dilatational waves, and the in-plane displacements of the plates and the porous layer (i.e. ux, uy and Ux, Uy) and the shear stresses τxz , τyz are also ignored. 2.4. Boundary conditions and configurations 2.4.1. Boundary conditions and plate motion Following previous studies [2,9,20,21,25–28], two basic methods of coupling a porous layer to the facing plate are considered, i.e. (1) B (Bonded): the porous layer and the plate are bonded together, and (2) U (Unbonded): an air gap separates the porous layer and the plate. The fluid-structure interaction introduces the boundary conditions (BC) of continuous normal velocity, displacement and stress at the interface between the plate, porous layer or the air medium. These boundary conditions can be arranged into three BC types depending on the coupling method. For the bonded coupling (B), a unique type of boundary conditions BC (I) applies to the first plate (for example) bonded to the porous layer and includes three equations:

(i) v1z =

Dw1 , Dt

(ii) uz = w1,

(iii) Uz = w1,

(13)

where w1 is the normal (transverse) displacement of the first plate, and v1z is the normal acoustic particle velocity in the incident field. For the unbonded coupling (U), two types of boundary conditions apply to the porous layer and the facing plate, respectively. The first type BC (II) includes three conditions which must be satisfied at the interface of the porous layer and the air gap:

(i) − βpg = s ,

(ii) − (1 − β )pg = σz ,

(iii) vgz = (1 − β )

∂uz ∂U + β z, ∂t ∂t

(14)

where pg and vgz are the acoustic pressure and the normal acoustic particle velocity at the open surface of the porous layer, respectively. The second type of boundary conditions for the U coupling, BC (III), are satisfied at the mid-surface of the second plate (for example), i.e.

(i) vgz =

∂w2 , ∂t

(ii) vtz =

∂w2 , ∂t

(15)

where w2 is the normal displacement of the second plate, and vtz is the normal acoustic particle velocity in the transmission field. Note that the BC types (II) and (III) need to be modified accordingly if they are applied to a different plate; for example when applying BC (III) to the first plate, the condition (15)(ii) now becomes Eq. (13)(i) that takes into account the external flow effect.

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273

Fig. 2. Side view of sound wave transmission through the double-wall sandwich panel: (a) the UU configuration, (b) the BB configuration.

For the two plates with the B and U coupling methods, the governing equations of their flexural motions are given by [10–13]:

D1∇4 w1 + m1

∂ 2w1

D2∇4 w2 + m2 4

2

∂t 2

= ρ0

∂ 2w2 ∂t 2

2

2

DΦ1 + σz + s , Dt

= ρ0

∂Φg ∂t

− ρ0

(16a)

∂Φt , ∂t

(16b)

2 2

where ∇ = (∂ /∂x + ∂ /∂y ) , Dj and mj are the flexural stiffness per unit width and mass per unit area of the jth plate, respectively. The plate flexural stiffness can be written as

Dj =

(

)

(

)

Ejh3j 1 + jηj 12 1 − νj2

(j = 1, 2), (17)

where Ej, hj, ηj and νj are the Young's modulus, thickness, loss factor and Poisson's ratio of the jth plate. Again corresponding modifications are required for these governing equations if a different coupling method is applied to the plate.

2.4.2. Panel configurations Three different configurations were considered in previous studies [2,9,20,21,25–28] for the double-wall sandwich panel by combining the B and U coupling methods, i.e. BU, UU and BB. The schematic diagram of the BU configuration has been shown in Fig. 1, and the schematics of the UU and BB cases are shown in Figs. 2(a) and (b), respectively. This study will focus on the UU configuration because it presents the best overall sound insulation performance (see Section 4.3). The BU and BB cases, however, will be considered in the validation of the numerical calculations in Section 4.2.1 and their theoretical formulations are given in the appendices. As shown in Fig. 2(a), the UU configuration has two air gaps with the gap depths of L1 and L2 for the upper and bottom gaps, respectively. The two air gaps are assumed to have identical physical properties, and thus the velocity potential expressions of the sound waves within the two air gaps are

ErrorconvertingfromMathMLtoLaTeX

(18)

Note that the incidence angles and wave numbers as well as their components in the two air gaps are identical to those in the transmission field, i.e. φg = φt , kg = kt , as aforementioned. The expressions of the wave number components in the air gaps can be found in Eq. (8). 2.5. Clamped boundary 2.5.1. Modal decomposition and matrix equation Since the bottom and upper plates are fully clamped to the rigid baffle, both the rotation moment and transverse displacement are equal to zero along the edges:

x = 0, a,

∀

0 < y < b,

w1 = w2 = 0,

∂w1 ∂w2 = = 0, ∂x ∂x

(19a)

y = 0, b,

∀

0 < x < a,

w1 = w2 = 0,

∂w1 ∂w2 = = 0. ∂y ∂y

(19b)

Following previous studies [10–13], the transverse displacements w1 and w2 can be written as a modal decomposition:

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w1(x, y , t ) =

∑

ϕmn(x, y)α1, mne jωt ,

m, n

w2(x, y , t ) =

∑

ϕmn(x, y)α2, mne jωt ,

(20)

m, n

where α1, mn and α2, mn are the modal displacement coefficients of the upper and bottom plates, respectively. The modal (or basic) function

⎛ 2mπx ⎞⎟⎜⎛ 2nπy ⎞⎟ ϕmn(x, y) = ⎜ 1 − cos 1 − cos ⎝ a ⎠⎝ b ⎠

(21)

ensures that the clamped boundary conditions specified in Eqs. (19a) and (19b) are satisfied, provided that a sufficiently large number of modes are taken in the summation of Eq. (20). For the UU configuration shown in Fig. 2, the type of boundary conditions BC (II) is applied on each side of the porous layer, and BC (III) on both plates. First, the solutions of the amplitudes of the reflected and transmitted waves, R and T, can be obtained by using the continuity of normal velocity on the surfaces of the two thin plates. Substituting the expressions of Φ 1 and Φ t in Eqs. (1) and (9) into Eqs. (13)(i) and (15)(ii) yields the expressions of R and T as

R(x, y) = 1 −

T (x, y) =

e j(k xx + k yy) kiz

∑ ⎡⎣ ωϕmn − jV·∇ϕmn⎤⎦α1, mn, m, n

ω j(k xx + k yy + ktzHt ) e ∑ ϕmnα2, mn, ktz m, n

(22)

(23)

where Ht is the total thickness of the porous foam and air gap(s), and Ht = L1 + H + L2 for the UU configuration. Note that the above expressions of R and T are universal for all configurations. Substituting the expressions of uz, Uz, σz , s in Refs. [2,3,20], Φg1, Φg 2, Φt and w1, w2 into Eqs. (14)(i)–(iii) and (15)(i) on the surfaces of the porous layer and both plates, one can obtain a system of 8 equations for 8 unknowns which can be rearranged into a matrix form as

(24)

AC = B,

where A is an 8  8 transfer matrix, C is the 8  1 vector of the unknown amplitudes, and B is the 8  1 forcing vector. For the UU configuration, the vectors C and B are obtained as T C = ⎡⎣ C1 C2 C3 C4 Ig1 Rg1 Ig2 Rg2⎤⎦ ,

⎡ B = ⎢ ∑ ϕmnα1, mn 0 0 0 0 0 0 ⎢⎣ m, n

(25) ⎤T

∑ ϕmnα2, mn⎥⎥ m, n

⎦

e j(k xx + k yy) . (26)

The elements of the transfer matrix A can be found in Appendix A for all configurations. Therefore, the unknown vector C can be solved simultaneously in terms of αj, mn (j ¼ 1, 2) by inverting the matrix Eq. (24).

2.5.2. The weighted residual method It remains to solve the modal coefficients α1, mn and α2, mn for the closure of the entire problem. The weighted residual (Galerkin) method is used that sets the integral of a weighted residual of the modal function to zero and yields an arbitrarily accurate double-series solution [11]. The integral equations are obtained by applying Eq. (16b) to the two plates and considering the convective flow on the external side, i.e. b

∫0 ∫0 b

∫0 ∫0

a

a

⎛ ∂Φg1 ⎞ ∂ 2w DΦ1 ⎜⎜ D1∇4 w1 + m1 2 21 − ρ0 ⎟⎟·ϕ (x, y) dxdy = 0, + ρ0 ∂t ⎠ mn Dt ∂ t ⎝

(27a)

⎛ ∂Φ ∂ 2w ∂Φ ⎞ ⎜⎜ D2∇4 w2 + m2 22 − ρ0 g2 + ρ0 t ⎟⎟·ϕmn(x, y) dxdy = 0, ∂t ∂t ⎠ ∂t ⎝

(27b)

The variables Φ1, Φg1, Φg 2, Φt and w1, w2 can be expressed in terms of α1, mn and α2, mn using the solution of the unknown vector C . Substituting these expressions into Eqs. (27a) and (27b), one can obtain a set of infinite algebraic equations for the unknown modal coefficients α1, mn and α2, mn . To solve these equations numerically, a truncation of 1 ≤ m , n ≤ N is needed resulting in a matrix of 2N2 equations:

Y. Liu, J.-C. Catalan / Journal of Sound and Vibration 405 (2017) 269–286

⎡ T1, mn T2, mn ⎤ ⎧F ⎫ ⎧ α1, mn ⎫ ⎥ ⎨ ⎬ Tα = ⎢ . = ⎨ mn ⎬ ⎢⎣ T3, mn T4, mn ⎥⎦ 2 2⎩ α2, mn ⎭2N2× 1 ⎩ 0 ⎭ 2 2N × 1 2N × 2N

275

(28)

Once the modal coefficients α1, mn and α2, mn are solved through the matrix Eq. (28), the unknown vector C and the amplitudes R and T can be determined which are required to assess the STL of the double-wall sandwich panel. The expressions of the matrix T and the vector F are presented in Appendix B and Appendix C for all configurations.

3. Transmission loss and limiting incidence angle 3.1. Transmission loss For a single incident wave, the power transmission coefficient is given by:

τ (φ, θ ) =

Πt , Πi

(29)

where Π i and Π t are the sound power associated with the incident wave and the transmitted wave respectively, and are defined as [10,11,14]:

Π i, t =

1 ⎡ Re 2 ⎢⎣

⎤

∫S pi,t vi*,t dS⎥⎦,

(30)

where Re[·] and the asterisk represent the real part and the complex conjugate of the argument. In the case of a harmonic wave, the sound pressure and the acoustic particle velocity are related as v = p/(ρ0 c ). Hence the sound power of the incident or transmitted wave can be written as

Π i, t =

1 2ρ0 c

∫S │pi,t │2

dS .

(31)

Note that the presence of an integral over the plate surface area S ¼ ab is due to the finite nature of the panel. Therefore, applying Eqs. (11) and (12) to pt and pi respectively, the power transmission coefficient can be obtained as

τ (φ, θ ) =

1 ab

│p │2

b

∫S │pt│2 dS = [1 + M sin φ cos(θ − θe)]2 ab1 ∫0 ∫0

a

│T│2 dxdy ,

i

(32)

where the transmitted wave amplitude T is dependent on the local position on the bottom plate as can be seen in Eq. (23). It is explicit in Eq. (32) the influence of the external mean flow on the transmission coefficient τ through the Mach number M and also implicitly through the solution of T. Finally, the sound transmission loss for a single incident wave is calculated in decibels as

1 STL = 10 log . τ

(33)

The decibel equivalent of τ is widely used in practice to measure the performance of acoustic barriers in isolating sound waves from one space to another, and will be adopted in the following to evaluate the STL of double-wall sandwich panels in the presence of an external mean flow. 3.2. Limiting angle of incidence The incidence angles in the air gap(s) and the transmission field remain the same, i.e. φg = φt , due to the identical air properties in the two regions. This identical incidence angle can be solved through the conservation of the tangential wavenumber components at the interfaces of the panel system [3,21], namely

kix = kgx = k x,

kiy = kgy = k y,

(34)

which can be simplified as

ki sin φ = kg sin φg .

(35)

Substituting Eqs. (4) and (7) into the relation (35), the incidence angle φ g can be obtained as

⎡ ⎤ sin φ φg = sin−1⎢ ⎥. ⎣ 1 + M sin φ cos (θ−θe ) ⎦

(36)

It is obvious from Eq. (36) that the incidence angle remains unchanged through the double-wall sandwich panel, i.e.

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φg = φt = φ , as long as the external flow Mach number M ¼ 0. In the presence of an external flow, however, the discontinuous acoustic impedance of the air medium throughout the system may lead to a total internal reflection. As a result, evanescent waves with a pure imaginary wavenumber component of kgz decay exponentially in the z-direction and hence there is no sound transmission through the sandwich panel. The limiting angle of incidence, φlim , is hence defined as the critical angle above which no incident waves are transmitted. This critical incidence angle occurs when φg,max = 90°. Substituting this relation into Eq. (36), the limiting incidence angle is obtained as

⎡ ⎤ 1 φlim = sin−1 ⎢ ⎥ if │1 − M cos (θ − θe )│ > 1. ⎣ 1 − M cos (θ − θe ) ⎦

(37)

The condition in Eq. (37) ensures that the incident wave is not evanescent and can be reduced to

M cos (θ − θe ) < 0

(38)

because

1− M cos( θ − θe ) < −1

⇒

M cos ( θ − θe ) > 2

(39)

and such high Mach numbers are not achievable under normal flight conditions. When the condition (38) is not satisfied, the limiting angle is taken as φlim = 90° since there is no total reflection [21]. Therefore, the limiting incidence angle takes the value as

⎧ ⎡ ⎤ 1 ⎪ sin−1⎢ ⎥ if M cos (θ − θe ) < 0, ⎣ 1 − M cos (θ − θe ) ⎦ φlim = ⎨ ⎪ if M cos (θ − θe ) ≥ 0. ⎩ 90°

(40)

The variation of φlim with the azimuthal angle θ − θe relative to the flow direction for a series of external flow Mach numbers is shown in Fig. 3. It can be observed that in the case of downstream incident sound waves (i.e. θ − θe ∈ [0°, 90°] ∪ [270°, 360°]), the limiting angle is always 90° for all Mach numbers indicating that all incident waves will transmit through the panel system despite the existence of the external mean flow. However for upstream incident waves (i.e. θ − θe ∈ [90°, 270°]), the increasing external flow Mach number progressively reduces the limiting angle and eventually leads to the total reflection of those sound waves with large incidence angles φ > φlim . Therefore in a diffuse sound field with random incidence, it is predicted that fewer incident sound waves can transmit through the double-wall panel system (i.e. improved sound insulation performance) due to the influence of the external mean flow.

4. Results and discussion 4.1. Parameters and convergence check Parameters used in previous works on double-wall sandwich panels [2,3,20,21] are taken in the present study to show the influence of the external mean flow on sound transmission loss. Table 1 summarises the property parameters of the plates, porous layer and ambient air. The two thin plates are made of aluminium and have identical properties for the three configurations with the thicknesses of h1 ¼ 1.27 mm and h2 ¼ 0.762 mm for the first and second plates, respectively. The thickness of the polyurethane foam is H ¼ 41 mm for the BB configuration and H ¼ 27 mm for BU and UU; the depth of the air gap(s) varies from L ¼ 14 mm for BU to L1 = L2 = 7 mm for UU, resulting in the same total thickness Ht ¼ 41 mm and hence a fair comparison among all configurations. The ambient air properties of standard atmosphere at sea level are taken

Fig. 3. Variation of the limiting incidence angle with the relative azimuthal angle for a number of external mean flow Mach numbers.

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Table 1 Property parameters used for the double-wall sandwich system. Symbol

Description

Value

Aluminium plate properties ρ

material density

2700 kg m−3

E

Young's modulus

ν η

Poisson's ratio loss factor

7 × 1010 Pa 0.33 0.01

Porous layer properties ρ1

bulk density of solid phase

30 kg m−3

Em

bulk Young's modulus

νs ηs s

bulk Poisson's ratio loss factor flow resistivity

8 × 105 Pa 0.4 0.265

ϵ′ β H

geometrical structure factor porosity foam thickness

25,000 MKS Rayls m−1 7.8 0.9 27 mm

Ambient air properties ρ0

density

1.21 kg m−3

c

speed of sound

Pr γ

Prandtl number ratio of specific heats

343 m s−1 0.71 1.4

in the incident field, air gap and transmission field as given in Table 1. Without loss of generality, the azimuthal angle of the incident wave is fixed at θ = 45°, and the external mean flow is aligned along the x-axis (i.e. θe = 0°) unless otherwise stated. Due to the double-series solution of the modal coefficients α1, mn, α2, mn as a result of modal decomposition, a truncation of the infinite series with a sufficiently large number of modes is required to ensure a converged solution of the matrix Eq. (28) with sufficient accuracy. An iterative process is performed for the convergence check by including progressively more terms in the double-series summation. The convergence criterion is prescribed such that the contribution from the successive mode in the summation to the STL calculation falls within a preset error bound of 0.1 dB. When this criterion is met, the maximum mode number Nm is determined and the STL solution is regarded as converged. The maximum mode number Nm increases with the frequency and panel dimensions [13,27,30]; namely, once a value of Nm is found for a given frequency and a given panel size, it also ensures a convergent solution for all frequencies and panel dimensions below. The highest frequency used in the numerical study is 10 kHz, and Fig. 4 shows the convergence check for this frequency and three panels of different dimensions excited by an obliquely incident wave in the presence of an external mean flow M ¼ 0.5. Within the range of external flow Mach number from M ¼ 0 to 1, the maximum mode number is identified as Nm ¼ 39, 27 and 14 for the three square panels of a = b = 1.0 m , 0.5 m and 0.2 m, respectively, to ensure the convergence of the double-series 2 ) of α1, mn, α2, mn in the matrix equation (28). solution, i.e. a total of 3042, 1458 and 392 unknown terms (2Nm 4.2. Validation of the model The theoretical model and numerical code used in this study are validated against the works of Bolton et al. [2] and Zhou et al. [20] who studied the transmission loss for infinite sandwich panels without and with an external mean flow,

Fig. 4. Convergence of the STL solution for a series of finite panels, UU configuration, M ¼ 0.5, φ = 30° , θ = 45° , f = 10 kHz .

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Fig. 5. Comparison of STL spectra between the present study and Bolton et al. [2] for the BB, BU and UU configurations, a = b = 1.0 m, M ¼ 0, θ = 45°, (a) φ = 0°, (b) φ = 30° .

respectively. The original data published in Bolton et al. [2] and Zhou et al. [20] were for random transmission loss in a diffuse field, and the results of a single incident wave are calculated using their theories for the following comparisons in Figs. 5 and 6. For the validation cases discussed in this section, a large panel of a = b = 1.0 m is used; to be consistent with the parameters used in Refs. [2,20], the foam thickness H ¼ 27 mm for the BB configuration and the air gap depths L1 = 2 mm, L2 = 6 mm for the UU configuration. 4.2.1. Without the external flow Figs. 5(a) and (b) compare the calculated STL spectra of the current study with those from Bolton et al.'s model [2] for a normal incident wave (φ = 0°) and an oblique incident wave (φ = 30°), respectively. In the case of normal incidence, the agreement between the current model and Bolton et al.'s theory is satisfactory for all configurations, especially at low frequencies below 100 Hz. The current model for finite panels slightly underestimates the STL levels compared with the theory for infinite panels [2]. Moreover, in the mid-high frequency range, dense peaks and dips over the STL spectra are visible for the current model. This phenomenon is clearly associated with the inherent modal behaviour of the finite doublewall panel system and is dominated by the bottom plate on the transmission side [11]. The relatively large dips at about 250 Hz for the BU and UU configurations and at about 1100 Hz for the BB configuration, however, are due to the mass-airmass resonance which are insensitive to the lateral dimension of the panel and the clamped boundary condition [11,21]. The case of oblique incidence in Fig. 5(b) shows similar comparison between the results of the current study and Bolton et al. [2]. The current model produces slightly higher STL levels than Bolton et al.'s theory, and a large discrepancy can be observed around 600 Hz in particular for the BB configuration. Nevertheless, the overall agreement between the two models is reasonably good. Among all the configurations, the UU case appears to be the overall best configuration in sound insulation, particularly at very high frequencies. Therefore for simplicity this configuration is selected for the numerical calculations thereafter. Further discussion on this configuration choice can be found in Section 4.3. 4.2.2. With the external mean flow The comparisons between the current model and Zhou et al.'s [20] in the presence of an external mean flow for normal incidence (φ = 0°) and oblique incidence (φ = 30°) are shown in Figs. 6(a) and (b), respectively. It can be seen that for frequencies above the mass-air-mass resonance frequency ( f > 250 Hz) the agreement between the two models is similar to

Fig. 6. Comparison of STL spectra between the present study and Zhou et al. [20] for the UU configuration and three flow Mach numbers, a = b = 1.0 m, θ = 45° , (a) φ = 0° , (b) φ = 30° .

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that in Figs. 5(a) and (b), i.e. satisfactory match of STL, numerous resonances due to the finite panel, and slightly underestimated and overestimated STL by the current model for normal incidence and oblique incidence, respectively. The striking feature of the finite panel, however, is the remarkably high STL below the mass-air-mass resonance frequency. This phenomenon is inherently due to the clamped boundary imposed on the finite panel as has been observed previously (e.g. [10–12]), and the presence of the external mean flow magnifies the effect of finite dimensions and shifts up the critical frequency where this effect occurs, which will be discussed further in Section 4.4. The two test cases of Bolton et al. [2] and Zhou et al. [20] have verified that the accuracy of the theoretical model developed in this study is acceptable. 4.3. Influence of panel configurations As has been shown in Figs. 5(a) and (b), the three configurations of the double-wall sandwich panel exhibit different STL results. Apart from these configurations, another configuration with only an air gap in between the two facing plates, referred to as the AA configuration, is considered as a reference to assess the effects of the poroelastic material and the air gap. The STL results of all four configurations are calculated based on a finite square panel of 0.5 m × 0.5 m and the same total thickness, as shown in Fig. 7. In the low-frequency range ( f < 150 Hz), the STL levels of all configurations appear very close because the clamped boundary imposes a rigorous constraint on panel vibration and governs the STL characteristics at low frequencies. In the mid-high frequency range above 500 Hz, the BB configuration exhibits the lowest STL levels since there is no wave reflection at the interface between the porous layer and the air gap. Comparing the AA configuration with the others, it can be seen in Fig. 7 that the dips in the STL spectra are attenuated by the damping effect of the porous material, in particular the strong dips due to the standing wave resonance [3,10,11] at about 4500 Hz and 9000 Hz, which is consistent with previous observations [19,25]. These results suggest that both the air gap and the poroelastic material are critical to the double-wall sandwich panel for superior sound insulation performance. Moreover, the UU configuration exhibits the best overall sound insulation properties among the four configurations because it provides the highest STL levels at high frequencies yet with an acceptable sacrifice of STL in the low-mid frequency range [12]. The excellent high-frequency performance can be attributed to the extra wave reflection induced by the two air gaps and hence enhanced sound absorption within the poroelastic layer.

Fig. 7. Comparison of STL spectra among the four panel configurations, a = b = 0.5 m, M ¼ 0.5, φ = 30° , θ = 45° .

Fig. 8. Effects of finite dimensions on the STL spectra for the UU configuration, φ = 30° , θ = 45°, (a) M ¼ 0, (b) M ¼ 0.5.

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4.4. Influence of finite dimensions In order to quantify the finite dimension effect of the clamp mounted panel, Figs. 8(a) and (b) compare the STL spectra among three finite panels with different lateral extensions as well as an infinite panel with and without the external mean flow, respectively. The width of the three square panels varies from 1.0 m, 0.5 m to 0.2 m, and the incident wave is at oblique angles of φ = 30° and θ = 45°. The STL results of the infinite panel are calculated from the models of Bolton et al. [2] and Zhou et al. [20]. Consider first the case of no external flow in Fig. 8(a). The STL spectrum of the large panel of a = b = 1.0 m can be well approximated by that of the infinite panel over almost the entire frequency range, as has been shown in Fig. 5 (b). Apart from the dense dips at high frequencies as a result of complex modal behaviours, the finite extension of the panel comes into effect distinctly only at sufficiently low frequencies below the fundamental natural frequency f (1, 1) where the wave length of the sound wave becomes relatively long compared with the size of the panel [12]. For the same reason, as the panel dimension decreases gradually this effect of finite dimensions becomes more obvious in terms of the significantly enhanced STL levels and the enlarged frequency range of this effect. Xin and Lu [11] compared the clamped and simply supported boundary conditions and have shown that the increased constraint on panel vibration of the clamped condition raises the panel stiffness and hence increases significantly the (1, 1) modal frequency and the STL levels below this frequency. The reduction of the panel dimensions, therefore, behaves similarly to the increased constraint by increasing the panel stiffness and hence the STL below f (1, 1). Beyond the (1, 1) modal frequency, however, the STL spectra of the finite panels exhibit similar trends as the infinite counterpart which can be regarded as an asymptotic form of the finite panels.

4.5. Influence of external flow Mach numbers Fig. 8(b) shows this comparison among the finite and infinite panels in the presence of an external mean flow with the Mach number M ¼ 0.5. It has been known that a convective mean flow can enhance the transmission loss across an elastic panel due to the aerodynamic (or acoustic radiation) damping effect (e.g. [17,21,31,32]). The fluid-structure interaction adds a convective fluid loading on the structure and hence raises the panel stiffness, which in turn increases the sound power radiated to the convective fluid and reduces the sound power transmitted through the panel. The effect of the external mean flow is evident in Fig. 8(b) through the further increased panel stiffness which elevates significantly the STL levels of the three finite panels below the (1, 1) modal frequency. Comparing the STL spectra in Figs. 8(a) and (b), it is shown that the convective flow effect of added stiffness is huge for the large square panel of 1.0 m width with a STL gain about 35 dB at 10 Hz, for example. However, this effect of the external mean flow becomes progressively less distinct as the panel dimensions decrease; the STL increment declines to only 15 dB at 10 Hz for the small panel of 0.2 m width which is already very stiff compared with the 1.0 m × 1.0 m counterpart. The effect of the external flow Mach number on the transmission loss can be seen more obviously in Fig. 9, when the sandwich panel of a = b = 0.5 m is excited by an oblique wave (φ = 30°, θ = 45°) at a series of Mach numbers M = 0, 0.25, 0.5, 1. As the external flow Mach number increases, three phenomena can be observed: (1) the fundamental natural frequency f (1, 1) is shifted up to higher frequencies until it reaches about 250 Hz for M ¼ 1; (2) the noticeable resonances at low frequencies below 250 Hz in the case of M ¼ 0 are attenuated gradually due to the aerodynamic damping effect exerted by the external flow; and most importantly, (3) the STL levels are enhanced significantly below the (1, 1) modal frequency. The STL spectra above the frequency of 250 Hz, however, exhibit the dips and peaks characteristic of the finite panel yet with similar general trends and slightly enhanced STL levels with the increasing Mach number.

Fig. 9. Effect of external flow Mach numbers on the STL spectra for the UU configuration, a = b = 0.5 m, φ = 30° , θ = 45°.

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Fig. 10. Effect of incidence angles φ on the STL spectra for the UU configuration, a = b = 0.5 m, θ = 45°, (a) M ¼ 0, (b) M ¼ 0.5.

4.6. Influence of incident wave direction The sound wave incidence angle has been found to influence significantly the STL of a double-wall panel [10,17–19,30]. This effect is analysed for the problem in the present study with the clamped boundary, poroelastic core and external mean flow. Fig. 10(a) compares the STL spectra for four different incidence angles (φ = 0°, 30°, 60°, 75°) in the absence of the external mean flow. Note that φ < φlim is satisfied for all φ values since the limiting incidence angle is always 0° at the fixed azimuthal angle of θ − θe = 45° from the data shown in Fig. 3. The results of Fig. 10(a) demonstrate that the sound insulation properties of the double-wall sandwich panel are remarkably sensitive to the incidence angle of the sound wave. As φ increases from normal incidence (φ = 0°) to highly oblique incidence (φ = 75°), the STL levels decrease remarkably in particular at very high frequencies above 4000 Hz. This indicates that incident sound waves with smaller incidence angles, which may interfere with structural bending waves destructively [30], transmit through the structure more difficultly than those with larger φ values. Moreover, the oblique incident waves also show a damping effect on the resonance dips at low frequencies below 250 Hz. The influence of the incidence angle on the STL in the presence of the external mean flow (M ¼ 0.5) is presented in Fig. 10(b). The external flow effect on the STL spectrum of normal incidence (i.e. φ = 0°) can be first seen through the increased STL level particularly in the low-frequency range with a huge gain of about 28 dB at 10 Hz, for example. The STL spectra of the oblique incidence angles are also elevated significantly and the averaged levels appear very similar among all φ cases. At low frequencies dominated by the clamped boundary condition, the order of the STL spectra is reversed compared with that in Fig. 10(a); namely the STL level increases with more oblique incidence angles, which is in accordance with the previous finding by Xin and Lu [17]. In this frequency range, oblique incident waves are easier to transmit through the double-wall sandwich panel than the normal incident wave due to the stronger effect of the external mean flow for the former. With regard to the azimuthal angle of the incident wave, it has been found that this angle has insignificant influence on the transmission loss of a panel system immersed in stationary fluid, while its effect becomes noticeable in the case of a moving fluid [10,18,19]. Figs. 11(a) and (b) present the STL spectra for a fixed incidence angle (φ = 30°) and a number of azimuthal angles (θ = 0°, 60°, 120°, 180°) without and with the external mean flow, respectively, and the results tend to support the previous finding. Note that the range of azimuthal angles [0°, 180°] is considered because the panel system is symmetrical to the x-z plane when the external flow is aligned along the x-axis (i.e. θe = 0°). When the external fluid is stationary (i.e. M ¼ 0), due to the symmetry of the square panel to the y-z plane, the STL spectra of θ = 0° and 60° coincide

Fig. 11. Effect of azimuthal angles θ on the STL spectra for the UU configuration, a = b = 0.5 m, φ = 30° , (a) M ¼ 0, (b) M ¼ 0.5.

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with those of θ = 180° and 120°, respectively. As shown in Fig. 11(a), the azimuthal angle plays an insignificant role on the STL spectra with negligible influence in the low-mid frequency range below 1000 Hz. Above this frequency, the STL levels of θ = 60° and 120° exceed those of θ = 0° and 180° because the structural modal behaviours become more complex due to the extra y-component of the incident sound wave. The presence of the external mean flow (M ¼ 0.5), however, causes the panel system asymmetrical to the y-z plane and hence the variation of STL results with the azimuthal angle, as can be seen in Fig. 11(b). When the azimuthal angle changes from θ < 90° (downstream incident) to θ > 90° (upstream incident), the STL levels decrease consistently over the entire frequency range with a considerable and constant STL drop by up to 6 dB at low frequencies, which implies that the downstream incident waves (i.e. −90° < θ < 90° in the case of θe = 0°), in particular those closer to the flow direction, are more difficult to transmit through the panel system and therefore are preferable from the perspective of sound insulation. Consistent results of the azimuthal angle effect on the STL for an infinite sandwich panel were presented by Meng et al. [19] who ascribed this influence mainly to the refraction effect of the mean flow as can be seen in Eq. (4). Again, the relation φ < φlim is ensured for all cases. As can be seen from Fig. 3, the smallest limiting incidence angle for M ¼ 0.5 is φlim ≈ 42° at θ − θe = 180°, greater than the chosen incidence angle of φ = 30°. 4.7. Influence of flow orientation Finally, the influence of the external flow orientation on the STL spectra is shown in Fig. 12 for the Mach number M ¼ 0.5. A normal incident wave (i.e. φ = 0°) is considered in order to eliminate the azimuthal angle effect, and the range of flow orientation θe ∈ [0°, 45°] is taken since the system of the square panel is now symmetrical to both the x-z and y-z planes. The results in Fig. 12 demonstrate that over the entire frequency range the orientation of the external mean flow has a negligible effect on the sound transmission through clamped finite panel systems. In the low-mid frequency range below about 2000 Hz, the STL spectra of the four cases studied (θe = 0°, 15°, 30°, 45°) almost fall onto one master curve. At higher frequencies, however, small variations of STL with the flow direction angle are caused by more complicated structural modal behaviours of the system, as seen in other figures. Therefore, the clamped double-wall panels of finite extent are insensitive to different flow orientations in terms of sound insulation properties.

Fig. 12. Effect of the external flow orientation on the STL spectra for the UU configuration, a = b = 0.5 m, M ¼ 0.5, φ = 0° .

5. Conclusions In this work, a theoretical model has been developed to study the influence of an external mean flow on the sound transmission through finite double-wall sandwich panels lined with poroelastic materials and clamped to an infinite rigid baffle. Biot's [1] theory was used to describe the wave propagation in the poroelastic material, and three configurations based on the coupling method of the poroelastic layer to the facing plates are considered. A modal decomposition and the weighted residual (Galerkin) method [11,13] were employed to account for the clamped boundary of finite panels that involve solving the matrix equations of the modal coefficients and hence wave amplitudes. The theoretical model has been validated against previous results [2,20] of infinite double-wall sandwich panels with and without an external mean flow, and the discrepancies particularly at low frequencies can be ascribed to the finite nature of the large panel considered. The numerical results of various panel configurations have shown that the UU case owns the optimal overall sound insulation performance and hence this configuration is selected for further analysis. The influence of the external mean flow on the STL has been found to be significant in the low-frequency range coupled with the effects of the finite dimensions and clamped boundary. The added panel stiffness by the aerodynamic damping of the external flow shifts up the fundamental

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natural frequency of the finite panel system and raises the STL levels significantly below this modal frequency along with the attenuation of low-frequency resonances. The increase of the incidence angle lowers the STL spectra but this trend is reversed by the existence of the external mean flow which enhances the STL of oblique incident waves more than that of normal incident waves. The azimuthal angle of the incident waves and the orientation of the external flow have shown negligible influences on the sound transmission across the clamped sandwich panels. Moreover, the external mean flow also reduces the limiting incidence angle of the panel system for the upstream incident waves, which implies a further improvement of the sound insulation performance in a diffuse sound field.

Acknowledgements The authors would like to acknowledge the Erasmus þ Programme for the financial support to this project.

Appendix A. Expressions of the matrix Eq. (24) A.1. UU case

(A.1)

A.2. BU case

⎤ ⎡ −b1κ1 −b2κ2 b1κ1 b2κ2 0 0 ⎥ ⎢ κ1 −κ1 κ2 −κ2 0 0 ⎥ ⎢ ⎥ ⎢ f −jk H −jkgz H jkgz H f jkIz H f −jkIIz H f jkIIz H I z ε1 e ε2 e ε2 e jβρ0 ωe jβρ0 ωe ⎥ ⎢ ε1 e A = ⎢ s −jkIzH −jkgz H s jkIz H s −jkIIz H s jkIIz H jkgz H ⎥ ε1 e ε2 e ε2 e j(1 − β )ρ0 ωe j(1 − β )ρ0 ωe ⎥ ⎢ ε1 e ⎥ ⎢ −jkIz H −jkIIz H −jkgz H jkIz H jkIIz H jkgz H −ωκ1β1e ωκ2β2e −ωκ2β2e −kgz e kgz e ⎥ ⎢ ωκ1β1e ⎢ −jkgz (H + L) jkgz (H + L) ⎥ −kgz /ωe 0 0 0 0 kgz /ωe ⎦ ⎣

(A.2)

T

C = ⎡⎣ C1 C2 C3 C4 Ig Rg ⎤⎦ ⎡ B = ⎢ ∑ ϕmnα1, mn ⎢⎣ m, n

∑ ϕmnα1, mn m, n

(A.3) ⎤T 0 0 0 ∑ ϕmnα2, mn⎥ e j(k xx + k yy) ⎥⎦ m, n

(A.4)

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A.3. BB case

⎡ bκ −b1κ1 −b2κ2 ⎤ b2κ2 1 1 ⎥ ⎢ κ − κ κ −κ2 1 1 2 ⎥ ⎢ A=⎢ b1κ1e−jkIzH −b1κ1e jkIzH b2κ2e−jkIIzH −b2κ2e jkIIzH ⎥ ⎥ ⎢ ⎢⎣ κ e−jkIzH −κ1e jkIzH κ2e−jkIIzH −κ2e jkIIzH ⎥⎦ 1

(A.5)

T C = ⎡⎣ C1 C2 C3 C4⎤⎦

(A.6)

⎡ B = ⎢ ∑ ϕmnα1, mn ⎢⎣ m, n

⎤T ∑ ϕmnα2, mn⎥ e j(kxx + kyy) ⎥⎦ m, n

∑ ϕmnα1, mn ∑ ϕmnα2, mn m, n

m, n

(A.7)

The expressions of the coefficients in this appendix, e.g. ε1s , ε1f , ε2s , ε2f , κ1, κ2, kIz , kIIz , β1, β2, b1, b2, can be found in Ref. [13].

Appendix B. Expressions of the matrix T 4 ⎡⎣ T ⎤ *1 *2 + Δ1*3 − Q 11 Δ2*1 + Δ2*2 + Δ2*3 + Δ2*4 + γ Δ3*1 + Δ3*2 + γ Δ3*3 + Δ3*4 11, mn⎦N 2× N 2 = 4π D1ab Δ1 + Δ1 1 2

(

)

(

⎡⎣ T ⎤ *1 *2 + Δ2*3 + Δ2*4 12, mn⎦N 2× N 2 = −Q 12 Δ2 + Δ2

(

⎡⎣ T ⎤ *1 *2 + Δ2*3 + Δ2*4 21, mn⎦N 2× N 2 = Q 21 Δ2 + Δ2

(

) (

γ1 = j cos2 θe

ρ0 V 2π 2b kiza

,

)

γ2 = j sin2 θe

⎤ ⎡ λ *1 ⎥ ⎢ 3 1 ⎥ ⎢ λ3* Δ3*1 = 3⎢ ⎥ ⋱ ⎥ ⎢ 1 ⎢⎣ λ3* ⎥⎦ 2 2 N ×N

)

) (

)

(B.1)

ρ0 V 2π 2a

λ3*1

⎡ 0 λ *2 ⋯ λ *2 ⎤ 3 3 ⎥ ⎢ 2 ⎢ * *2 ⎥ ⋯ λ λ 0 3 Δ3*2 = 2⎢ 3 ⎥ ⋮ ⎥ 0 ⎢ ⋮ ⎥ ⎢ *2 ⋯ λ3*2 0 ⎦ 2 2 ⎣ λ3 N ×N

(

)

4 ⎡⎣ T ⎤ *1 *2 + Δ1*3 + Q 22 Δ2*1 + Δ2*2 + Δ2*3 + Δ2*4 22, mn⎦N 2× N 2 = 4π D2ab Δ1 + Δ1

(

)

kizb

(B.2)

⎤ ⎡ 12 ⎥ ⎢ 2 ⎥ ⎢ 2 = ⎥ ⎢ ⋱ ⎢⎣ 2⎥ N ⎦N × N

(B.3)

⎤ ⎡ 12 ⎥ ⎢ 2 ⎥ 2 λ3*2 = ⎢ ⎥ ⎢ ⋱ ⎥ ⎢⎣ N2 ⎦N × N

⎤ ⎡ λ *3 ⎥ ⎢ 3,1 ⎥ ⎢ *3 λ 3 3,2 * Δ3 = 3⎢ ⎥ ⋱ ⎥ ⎢ ⎢ λ3,*N3 ⎥⎦ 2 2 ⎣ N ×N

⎤ ⎡1 ⎥ ⎢ 3 2 1 * ⎥ λ3, n = n ⎢ ⋱ ⎥ ⎢ ⎣ 1⎦N × N

⎤ ⎡ λ *4 ⎥ ⎢ 3,1 ⎥ ⎢ *4 λ3,2 Δ3*4 = 2⎢ ⎥ ⋱ ⎥ ⎢ ⎢ λ3,*N4 ⎥⎦ 2 2 ⎣ N ×N

⎡ 0 1 ... 1⎤ ⎥ ⎢ ⋮⎥ 1 0 λ3,*n4 = n2⎢ ⎢⋮ 0 ⋮⎥ ⎥ ⎢⎣ 1 ... 1 0⎦N × N

The expressions of the submatrices Δ1*i (i = 1, 2, 3) and Δ2*i (i = 1, 2, 3, 4) can be found in Ref. [11].

(B.4)

(B.5)

(B.6)

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B.1. UU case

(

)

−1 −1 Q 11 = m1ω2 − jρ0 ω2/kiz − jρ0 ω A51 + A61

(

Q 12 = −jρ0 ω

−1 A58

)

−1 A68

+

⎡ −1 −jkgz( L1+ H + L2) −1 jkgz ( L1+ H + L2)⎤ Q 21 = −jρ0 ω⎣⎢ A71 e + A81 e ⎦⎥ ⎡ −1 −jkgz( L1+ H + L2) −1 jkgz ( L1+ H + L2)⎤ Q 22 = −m2ω2 + jρ0 ω2/kgz − jρ0 ω⎢⎣ A78 e + A88 e ⎥⎦

(B.7)

B.2. BU case

( = Λ (A

)

( )

)

−1 −1 −1 −1 −1 −1 −1 −1 Q 11 = Λ1 A11 + A12 + A21 + A22 − Λ2 A31 + A32 + A 41 + A 42 + m1ω2 − jρ0 ω2/kiz

Q 12

1

−1 16

+

)+ (

−1 A26

−1 Λ2 A36

+

−1 A 46

⎡ ⎤ −1 −1 −jkgz ( H + L) −1 −1 Q 21 = − jρ0 ω⎢ A51 + A52 e + A61 + A62 e jkgz( H + L)⎥ ⎣ ⎦

(

)

(

)

⎡ −1 −jkgz( H + L) −1 jkgz ( H + L)⎤ Q 22 = − m2ω2 + jρ0 ω2/kgz − jρ0 ω⎢⎣ A56 e + A66 e ⎥⎦

(B.8)

B.3. BB case

( = Λ (A ⎡ = Λ ⎣(A ⎡ = Λ ⎣(A

) ( ) + Λ (A

) )

−1 −1 −1 −1 −1 −1 −1 −1 Q 11 = Λ1 A11 + A12 + A21 + A22 + Λ2 A31 + A32 + A 41 + A 42 + m1ω2 − jρ0 ω2/kiz

Q 12 Q 21 Q 22

1

−1 13

+

−1 A14

+

−1 A23

)e )e

+

1

−1 11

+

−1 A12

−jkIzH

1

−1 13

−1 + A14

−jkIzH

−1 A24

2

( + (A

+

−1 33

−1 A21

+

−1 23

+

)e )e

−1 A22

−1 + A24

−1 A34

+

−1 A 43

+

−1 A 44

jkIzH ⎤

⎡ −1 −1 −jkIzH −1 −1 jkIIzH ⎤ e + A 41 + A 42 ⎦+ Λ2 ⎣ A31 + A32 e ⎦

jkIzH

( ⎤ ⎡ ⎦+ Λ ⎣ ( A 2

−1 33

) )e

−1 + A34

Again, the expressions of the coefficients Λ1, Λ2 can be found in Ref. [13];

−jkIzH

Aij−1

( + (A

−1 43

) )e

−1 + A 44

jkIIzH ⎤

2 2 ⎦ − m2ω + jρ0 ω /kgz

(B.9)

is the element of the inverse of the transfer matrix A .

Appendix C. Expression of the vector F

{ Fmn}N2×1 = ⎡⎣ F11

( fmn ( k x, k y)

F21 … FN1 F12 F22 … FN2 F1N F2N ... FNN ⎤⎦

(C.1)

) (

(C.2)

Fmn = 2jρ0 ω−V cos θe k x−V sin θe k y fmn k x, k y b

( )

a

(

−j k xx + k yy

= ∫ ∫ ϕmn x, y e 0 0

)

) dxdy

⎧ ab ⎪ ⎪ 4jn2π 2a 1 − e−jk yb ⎪ ⎪ k k 2b2 − 4n2π 2 ⎪ y y ⎪ = ⎨ 4jm2π 2b 1 − e−jk xa ⎪ ⎪ k x k x2a2 − 4 m2π 2 ⎪ ⎪ 16m2n2π 4 1 − e−jk xa 1 − e−jk yb ⎪− ⎪ k k k 2a2 − 4m2π 2 k 2b2 − 4n2π 2 x y x y ⎩

(

)

(

)

(

)

(

)

(

(

)(

)(

) )

for k x = 0, k y = 0 for k x = 0, k y ≠ 0

for k x ≠ 0, k y = 0

for k x ≠ 0, k y ≠ 0

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