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Accepted ManuscriptExplicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers Pham Chi Vinh, Tran Thanh Tu

Accepted Manuscript

Explicit formulas for the reflection and transmission coefficients of

one-component waves through a stack of an arbitrary number of layers

Pham Chi Vinh, Tran Thanh Tuan, Marcos A Capistran

DOI: http://dx.doi.org/10.1016/j.wavemoti.2014.12.002

To appear in: Wave Motion

Received date: 12 October 2014

Revised date: 2 December 2014

Accepted date: 4 December 2014

Please cite this article as: P.C Vinh, T.T Tuan, M.A Capistran, Explicit formulas for thereflection and transmission coefficients of one-component waves through a stack of an arbitrarynumber of layers, Wave Motion (2014), http://dx.doi.org/10.1016/j.wavemoti.2014.12.002

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Research highlights

1 The reflection and transmission of one-component waves through a stack of arbitrary number of layers is considered

2 The explicit formulas for the reflection and transmission coefficients

of one-component waves are obtained

3 From these formulas, the explicit expressions of the reflection and transmission coefficients for an FGM layer are derived

4 Approximate formulas are established for a stack of thin layers and for a thin FGM layer

5 It is shown that they are good approximations

*Research Highlights

Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers

Hanoi University of Science

334, Nguyen Trai Str., Thanh Xuan, Hanoi,Vietnam

December 1, 2014

Abstract The transmission and reflection of one-component elastic, acoustic, op- tical waves on a stack of arbitrary number of different homogeneous layers have been intensively studied in the literature However, all obtained formu- las for the reflection and transmission coefficients are in implicit form In this paper, we provide the explicit formulas for them From these formulas

we immediately arrive at the explicit formulas for the reflection and mission coefficients of one-component waves through an FGM layer Based

trans-on the obtained exact formulas, approximate formulas for the reflectitrans-on and transmission coefficients are established for a stack of thin layers and for a thin FGM layer It is numerically shown that they are good approximations Since the obtained formulas are totally explicit, they are useful in evaluating, not only numerically but also analytically, the transmission and reflection coefficients of one-component waves.

Key words: One-component waves, Reflection coefficient, Transmission coefficient,

A stack of arbitrary number of layers A composite layer, An FGM layer

∗ Corresponding author: Tel:+84-4-35532164; Fax:+84-4-38588817; E-mail address: inh@vnu.edu.vn (P C Vinh)

pcv-*Manuscript (Clear)

Click here to view linked References

1 Introduction

Studies of the wave propagation in layered media have long been of interest to searchers in the fields of geophysics, acoustics, and electromagnetics Applications ofthese studies include such technologically important areas as earthquake prediction,underground fault mapping, oil and gas exploration, architectural noise reduction,and the design of ultrasonic transducer Extensive review of works on this subjecthas been reported in the literature as is evidenced from the books by Ewing et al.[1], Brekhovskikh [2], Yeh [3], Brekhovskikh & Gordin [4], Nayfeh [5], Born & Wolf[6], Borcherdt [7]

re-Among the wave propagation problems in layered media, the problem of wavepropagation in a stack of arbitrary number n of different layers is very significantdue to two reasons Firstly, this structure (i.e the stack of arbitrary number

n of different layers) is a realistic model for many practical problems Secondly,when studying the wave propagation in an FGM layer (whose material parameterscontinuously vary in the thickness variable), researchers often approximate the FGMlayer by a set of arbitrary number n of homogeneous layers welded to each other.The accuracy of the approximate solution depends on n: lager values of n providehigher accuracies

The study of reflection and transmission of the one-component elastic, acousticand optical waves through a stack of arbitrary number of different homogeneouslayers plays important role in practical applications The main aim of this study is

to derive the expressions of the reflection and transmission coefficients In order toobtain these expressions, the matrix transfer method is employed (see Thomson [8],Haskell [9]) The reflection and transmission coefficients are then expressed in terms

of the entries of the global matrix which is a product of n local transfer matrices.See, for examples, Brekhovskikh [2], Eqs (3.38) and (3.47); Yeh [3], Eq (5.1-27);Born & Wolf [6], Eq (41), page 62; Borcherdt [7], Eq (9.1.25); Levine [10], Eq.(27.5); Ben-Menahem & Singh [11], Eq (3.162), the well-known monographs onthe topic of waves in layered media See also the recently published papers: Goto

et al [12], Chen et al [13], Wang & Gross [14], Golub et al [15, 16], wherethe reflection and transmission of SH waves through a stack of arbitrary number

of different elastic layers were considered One can find that the transfer matrixmethod (or its reformulated forms) was again utilized and no other approachesproviding explicit expressions were mentioned in these papers

Because no explicit expression of the entries of the global transfer has beenprovided so far, to the best of knowledge of the authors, all obtained expressionsfor the reflection and transmission coefficients of one-component (elastic, acoustic,optical) waves on a stack of arbitrary number n of different layers are therefore notexplicit

The main aim of this paper is to derive the explicit formulas for the reflection andtransmission coefficients of one-component (elastic, acoustic, optical) waves through

a stack of arbitrary number n of different layers In order to obtain these

formu-las, the mathematical induction is employed From these formulas we immediatelyarrive at the explicit formulas for the reflection and transmission coefficients ofone-component waves through an FGM layer Using the derived exact formulas,approximate formulas for the reflection and transmission coefficients are establishedfor a stack of thin layers and for a thin FGM layer It is numerically shown thatthey are good approximations.

Formulas (24)-(27) are the explicit version of the implicit formulas mentionedabove: Eq (5.1-27) in Yeh [3]; Eq (41), page 62 in Born & Wolf [6]; Eq (9.1.25)

in Borcherdt [7]; Eq (27.5) in Levine [10]; Eq (3.162) in Ben-Menahem & Singh[11] Formulas (21) are explicit form of the implicit formulas (3.38) and (3.47) inBrekhovskikh [2]; The results obtained in this paper are therefore new

Since the obtained formulas are totally explicit, they are useful in evaluating, notonly numerically but also analytically, the transmission and reflection coefficients ofone-component waves

trans-mission coefficients of SH waves

Consider a stack of arbitrary number n of different homogeneous isotropic elasticlayers By hk, ρk, µk and βk = p

µk/ρk we denote the thickness, the mass density,the shear modulus and the transverse wave velocity, respectively, of the kth layer(k = 1, n) The stack is called shortly the composite layer and its thickness is

h = h1+ h2+ + hn Suppose that the composite layer is sandwiched between twohomogeneous isotropic elastic half-spaces numbered by ”0” and ”(n + 1)” as shown

in Fig 1 The half-space ”0” (the upper half-space), with the mass density ρ0,the shear modulus µ0 and the transverse wave velocity β0 = p

µ0/ρ0, occupies thedomain z ≤ 0 The half-space ”n+1” (the lower half-space), with the mass density

ρn+1, the shear modulus µn+1 and the transverse wave velocity βn+1 =p

µn+1/ρn+1,occupies the domain z ≥ h

ρ0, µ0

z1z

u in (0)

u r

(0)

φ0

φn+1

zn−2z

respectively, of the composite layer They depend on z and are expressed as:

where z0 = 0, zk = h1+ h2+ + hk, k = 1, n (zn= h) and ρk, µk, βk are constants

We are interested in the propagation of SH waves whose displacement vector is ofthe form£

u(n −1) = u(n), τ(n −1) = τ(n) at zn−1

u(n) = u(n+1), τ(n) = τ(n+1) at zn= h

(6)

Assume that an incident plane SH wave with the unit amplitude propagates in thehalf-space ”0” and it is of the form:

u(0)in = ei(k0 x sin φ 0 +k 0 z cos φ 0 ) (7)

where φ0 is the incident angle, see Fig 1, k0 = ω

β0 is the wave number of the plane ”0” and and ω is the circular frequency Then, the reflected and transmittedwaves have the form:

k = n + 1

2.3 Explicit formulas for the reflection and transmission

co-efficients

In order to determine the reflection and transmission coefficients R and T , we have

to find the solution of Eq (4) that satisfies the continuity conditions (6) Thesolution of Eq (4) is sought in the form:

where U (z) is a unknown function needed to be determined Introducing the sentation of solution (9) into Eq (4) and the first and the last of Eq (6) and takinginto account (5), (7), (8) and the fact that u(0)= u(0)in + u(0)r , u(n+1) = u(n+1)t , yield:

repre-ddz

·µ(z)dUdz

¸+ [ρ(z)ω2− µ(z)ξ2]U = 0, 0≤ z ≤ h (10)and:

2×2 be the fundamental solution matrix of Eq (13) satisfying

F (0) = I, I is the 2×2 identity matrix (see Chicone [17]) Then, the general solution

Y (z) of Eq (13) is given by:

Y (z) = F (z) Y0, Y0 =£

A, B¤T

(16)where A and B are constants such that |A|2+|B|2 6= 0 It is not difficult to verifythat (see also Aki and Rechards [18], Eq (7.44)):

The matrix e(z−z m−1 )A m is also given by Eq (18) in which θm is replaced by θ∗

m =(z− zm −1)ηm Substituting (16) into (15)1 and taking into account F (0) = I we getimmediately:

defined as:

H(n) := ehn A nehn−1 A n−1 eh1 A 1 (22)Note that that H(n) = F (h) according to (17) and by (18): detH(n)= 1 It is clearfrom (21) that the expressions of R and T will become explicit if we obtain explicitformulas for Hij(n) We will prove that Hij(n) are expressed by the following explicitformulas using the mathematical induction:

+) As H(1) = eh 1 A 1 given by (18), (23) is clearly true

+) It is easy to check that Eqs (24)-(27) are valid for n = 2 using (18) and (22).+) Assume that Eqs (24)-(27) hold for n = 2m where m is integer and m ≥ 1

Then, from (18) and H(n+1) = eh n A nH(n), according to (22), we have:

By using the equality:

Cn+12j = Cn2j+ Cn2j−1one can see that:

+) Analogously, one can prove that Eqs (25)- (27) are true for(n + 1) and then

Eqs (24)- (27) hold for (n+2) By the mathematical induction Eqs (24)-(27) aretrue for any value of n.

The formulas (21), in which Hij(n) given by (23)-(27), are the desired explicitformulas for the reflection and transmission coefficients

θm= hmηm, ηm =

vu

t ω2

β2

m −c

(m) 66

of mth-layer and c(m)ij (m = 0, n + 1) are those of the half-spaces ”0” and ”(n + 1)”,respectively

coeffi-cients for longitudinal waves

Again suppose that the half-spaces and the layers be made of orthotropic materialswhose principal axes are identical Consider a coming plane longitudinal wave trav-eling in the half-space ”0” in the z-direction (φ0 = 0) with the unit amplitude andits displacements are of the form:

and µj (j = 0, n + 1) in Eqs (21), (47) is replaced by c(j)33 (j = 0, n + 1), ηj = kj (j =

0, n + 1)

trans-mission coefficients of SH waves for an FGM layer

In stead of a composite layer as defined above, we now consider an isotropic FGM(functionally grade material) layer with the thickness h and its material parame-ters are continuous functions of the thickness variable z: µ∗ = µ∗(z), ρ∗ = ρ∗(z),

z ∈ [0, h] The quantities associated to the FGM layer are distinguished by a star.Corresponding to the FGM layer we have Eqs (13)-(15) in which Y , A(z), µ(z),ρ(z), R and T are replaced by Y∗, A∗(z), µ∗(z), ρ∗(z), R∗ and T∗, respectively Weare interested in deriving the explicit formulas for the reflection and transmissioncoefficients R∗ and T∗ of SH waves through this layer by using the formulas obtained

in the previous Section for a composite layer To this end, the FGM layer is proximately replaced by a composite layer consisting n homogeneous layers with thesame thickness ∆h = h/n They are perfectly bonded to each other and the m-thlayer is characterized by the constant material matrix Am= A∗(zm), m = 1, n Let

ap-F∗(z) be the fundamental solution matrix corresponding to the FGM layer satisfying

F∗(0) = I and H∗ = F∗(h) It is not difficult to prove that:

F (z)⇒ F∗(z) in the interval [0, h] as n→ ∞

Hij∗ = lim

where Hij∗ are the entries of matrix H∗ and Hij(n) are calculated by (23)-(27) Since

H11(n) can be expressed by:

according to (24), it is clear that H∗

˜i = ρi/ρ0, ˜βi = βi/β0, Ci2 = 1− β2

i sin2φ0/β02Taking the limit two sides of (40) as n → ∞ (equivalently ∆h → 0) and using the

dimensionless variable ˜z = z/h we obtain:

Ij11 =

Z 1 0

d˜z1

˜ρ(˜z1) ˜β2(˜z1)

Z 1

˜ 1

˜ρ(˜z2)C2(˜z2)d˜z2· · ·

Z 1

˜2j−2

d˜z2j−1

˜ρ(˜z2j−1) ˜β2(˜z2j−1)

Z 1

˜2j−1

˜ρ(˜z2j)C2(˜z2j)d˜z2j

Z 1

˜ 1

d˜z2

˜ρ(˜z2) ˜β2(˜z2)· · ·

Z 1

˜ 2j−3

d˜z2j−2

˜ρ(˜z2j−2) ˜β2(˜z2j−2)

Z 1

˜ 2j−2

˜ρ(˜z2j −1)C2(˜z2j −1)d˜z2j −1,

(44)

and:

Ij21 =

Z 1 0

d˜z1

˜ρ(˜z1) ˜β2(˜z1)

Z 1

˜ 2

˜ρ(˜z2)C2(˜z2)d˜z1· · ·

Z 1

˜2j−3

˜ρ(˜z2j−2)C2(˜z2j−2)d˜z2j−2

Z 1

˜2j−2

d˜z2j−1

˜ρ(˜z2j−1) ˜β2(˜z2j−1)

Z 1

˜ 1

d˜z2

˜ρ(˜z2) ˜β2(˜z2)· · ·

Z 1

˜ 2j−2

˜ρ(˜z2j−1)C2(˜z2j−1)d˜z2j−1

Z 1

˜ 2j−1

d˜z2j

˜ρ(˜z2j) ˜β2(˜z2j),

pre-ij

Remark 2: i) Following the same procedure and using the results obtained inSubsections 2.4 and 2.5 we derive the explicit exact formulas for the reflection andtransmission coefficients of SH elastic waves and longitudinal elastic waves for anorthotropic FGM layer The corresponding coefficients R and T are given by (21)

in which H∗

ij are defined by similar expressions to (41) and (43)

ii) Analogously, one can obtain the explicit exact formulas for the reflection andtransmission coefficients of one-component acoustic and optical waves, for both acomposite layer and an FGM layer

For a composite layer consisting of thin layers:

Suppose that the layers are all thin, i.e 0 < εi << 1∀ i = 1, n, where εi = k0hi.From (19)1,2 and ξ = k0sin θ0 it follows 0 <|θi| << 1 ∀ i = 1, n Using (39) and theapproximations: cos θi = 1− θ

2 i

2, tan θi= θi−θ

3 i

β2 i

˜iβ˜2 i

˜jβ˜2 j

εiεj,ˆ

H12(n) =−ρ0β2

0k0³Pn i=1

˜iC2

iεi+13

n

P

i=1

˜iC4 i

˜

β2 i

ε3

i −nP−2

i=1

nP−1 j=i+1

n

P

k=j+1

˜iC2 i

˜jβ˜2 j

˜kC2

kεiεjεk´

, (48)ˆ

H21(n) = 1

ρ0β2

0k0

³Pn i=1

εi

˜iβ˜2 i

+ 13

n

P

i=1

C2 i

˜iβ˜4 i

ε3

j −nP−2

i=1

nP−1 j=i+1

n

P

k=j+1

˜jC2 j

˜iβ˜2

i˜kβ˜2 k

Figure 2: Dependence of the modulus|R| of the reflection coefficient on the sionless frequency ε = k0h in the interval [0, 1] for the composite layer whose shearmodulus is given by (49)

dimen-As an example, we consider a composite layer consisting of 9 homogeneousisotropic elastic layers with the same thicknesses that is sandwiched between twoidentical homogeneous isotropic half-spaces Further, we suppose that the mass den-