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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 PII: DOI: Reference: S0165-2125(14)00173-5 http://dx.doi.org/10.1016/j.wavemoti.2014.12.002 WAMOT 1901 To appear in: Wave Motion Received date: 12 October 2014 Revised date: December 2014 Accepted date: December 2014 Please cite this article as: P.C Vinh, T.T Tuan, M.A Capistran, Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers, Wave Motion (2014), http://dx.doi.org/10.1016/j.wavemoti.2014.12.002 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Research highlights The reflection and transmission of one-component waves through a stack of arbitrary number of layers is considered The explicit formulas for the reflection and transmission coefficients of one-component waves are obtained From these formulas, the explicit expressions of the reflection and transmission coefficients for an FGM layer are derived Approximate formulas are established for a stack of thin layers and for a thin FGM layer It is shown that they are good approximations Click here to view linked References Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers Pham Chi Vinha ∗, Tran Thanh Tuana , Marcos A Capistranb a Faculty of Mathematics, Mechanics and Informatics Hanoi University of Science 334, Nguyen Trai Str., Thanh Xuan, Hanoi,Vietnam b CIMAT, Guanajuato Gto, Mexico 36000, Mexico December 1, 2014 Abstract The transmission and reflection of one-component elastic, acoustic, optical waves on a stack of arbitrary number of different homogeneous layers have been intensively studied in the literature However, all obtained formulas 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 transmission coefficients of one-component waves through an FGM layer Based on the obtained exact formulas, approximate formulas for the reflection 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- Introduction Studies of the wave propagation in layered media have long been of interest to researchers in the fields of geophysics, acoustics, and electromagnetics Applications of these 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 subject has 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] Among the wave propagation problems in layered media, the problem of wave propagation in a stack of arbitrary number n of different layers is very significant due 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 parameters continuously vary in the thickness variable), researchers often approximate the FGM layer 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 provide higher accuracies The study of reflection and transmission of the one-component elastic, acoustic and optical waves through a stack of arbitrary number of different homogeneous layers 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 to obtain 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 on the 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], where the reflection and transmission of SH waves through a stack of arbitrary number of different elastic layers were considered One can find that the transfer matrix method (or its reformulated forms) was again utilized and no other approaches providing explicit expressions were mentioned in these papers Because no explicit expression of the entries of the global transfer has been provided so far, to the best of knowledge of the authors, all obtained expressions for the reflection and transmission coefficients of one-component (elastic, acoustic, optical) waves on a stack of arbitrary number n of different layers are therefore not explicit The main aim of this paper is to derive the explicit formulas for the reflection and transmission 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 immediately arrive at the explicit formulas for the reflection and transmission coefficients of one-component waves through an FGM layer Using the derived exact formulas, approximate formulas for the reflection 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 Formulas (24)-(27) are the explicit version of the implicit formulas mentioned above: 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) in Brekhovskikh [2]; The results obtained in this paper are therefore new 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 2.1 Explicit formulas for the reflection and transmission coefficients of SH waves Equations and continuity conditions Consider a stack of arbitrary number n of different homogeneous isotropic elastic layers By hk , ρk , µk and βk = 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 two homogeneous isotropic elastic half-spaces numbered by ”0” and ”(n + 1)” as shown in Fig The half-space ”0” (the upper half-space), with the mass density ρ0 , the shear modulus µ0 and the transverse wave velocity β0 = µ0 /ρ0 , occupies the domain z ≤ 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 = µn+1 /ρn+1 , occupies the domain z ≥ h u(0) in ρ0, µ0 φ0 φ u(0) r z0=0 ρ1, µ1, h1 z ρ ,µ ,h 2 z2 ρ ,µ ,h ρ 3 ,µ n−1 x z3 z ,h n−1 n−1 n−2 z ρn, µn, hn n−1 z =h n ρ ,µ n+1 n+1 z φn+1 u(n+1) t Figure 1: A stack of n homogeneous elastic layers sandwiched between two halfspaces Here we use the orthogonal coordinate system 0xyz in which the z-axis is perpendicular to the layers and x-axis is parallel to the layers By ρ, µ and β = µ/ρ we denote the mass density, the shear modulus and the transverse wave velocity, respectively, of the composite layer They depend on z and are expressed as:   ρ1 , µ1 , β1 for z0 ≤ z ≤ z1 ρ(z), µ(z), β(z) =   ρn , µn , βn for zn−1 ≤ z ≤ zn (1) 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 of the form 0, u(0) (x, z, t), T T and 0, u(n+1) (x, z, t), for the half-space ”0” and the T half-space ”n + 1”, respectively, and 0, u(x, z, t), for the composite layer, where:  (1)  u (x, z, t) for z0 ≤ z ≤ z1 (2) u(x, z, t) =   (n) u (x, z, t) for zn−1 ≤ z ≤ zn = h Then, the equation governing on the SH motion is: ∂ u(k) ∂ u(k) ∂ u(k) + = , k = 0, n + ∂x2 ∂z βk2 ∂t2 (3) for the half-spaces and: µ(z) ∂ 2u ∂ ∂u ∂ 2u + µ(z) = ρ(z) ∂x2 ∂z ∂z ∂t2 (4) for the composite layer The traction vector on the plane z = const has the form T 0, τ (0) (x, z, t), , 0, τ (n+1) (x, z, t), T for the half-spaces and 0, τ (x, z, t), T for the composite layer, where: τ (k) = µ(k) ∂u(k) ∂u , k = 0, n + and τ = µ ∂z ∂z (5) Suppose that the layers and the half-spaces are perfectly bonded to each other Then, the displacement and stress are required to be continuous through the interfaces z = zk , in particular:   u(0) = u(1) , τ (0) = τ (1) at z0 =    (1) (2) (1) (2)   at z1 u = u , τ = τ    u(n−1) = u(n) , τ (n−1) = τ (n) at zn−1    u(n) = u(n+1) , τ (n) = τ (n+1) at z = h n 2.2 (6) Reflection and transmission coefficients Assume that an incident plane SH wave with the unit amplitude propagates in the half-space ”0” and it is of the form: (0) uin = ei(k0 x sin φ0 +k0 z cos φ0 ) where φ0 is the incident angle, see Fig 1, k0 = (7) ω is the wave number of the halfβ0 plane ”0” and and ω is the circular frequency Then, the reflected and transmitted waves have the form: (n+1) i(k0 x sin φ0 −k0 z cos φ0 ) u(0) , ut r = Re where k(n+1) = ω βn+1 = T ei(k(n+1) x sin φ(n+1) +k(n+1) z cos φ(n+1) ) (8) is the wave number of the half-plane ”(n + 1)”, φ(n+1) is the transmission angle, see Fig 1, that is defined by the Snell law (see Brekhovskikh [2]): k0 sin φ0 = k(n+1) sin φ(n+1) := ξ, R and T are the reflection coefficient and the transmission coefficient, respectively, and they are needed to be determined In Eqs (7), (8) and in everywhere, we omit the factor e−iωt for the sake of brevity Note (0) (0) (n+1) that uin and ur satisfy Eq (3) for k = 0, while ut k = n + is a solution of Eq (3) for 2.3 Explicit formulas for the reflection and transmission coefficients 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) The solution of Eq (4) is sought in the form: u = U (z)eiξx (9) where U (z) is a unknown function needed to be determined Introducing the representation of solution (9) into Eq (4) and the first and the last of Eq (6) and taking (0) (0) (n+1) into account (5), (7), (8) and the fact that u(0) = uin + ur , u(n+1) = ut d dU µ(z) + [ρ(z)ω − µ(z)ξ ]U = 0, ≤ z ≤ h dz dz , yield: (10) and: U (0) = + R, U (h) = T eiηn+1 h , dU (0) iη0 µ0 = (1 − R), dz µ1 dU (h) iηn+1 µn+1 iηn+1 h = e dz µn (11) where ηj = kj cos φj (j = 0, n + 1) Note that, in addition to (11), according to (6)2 -(6)n , U (z) and µ(z)dU/dz are required to be continuous at zk , k = 1, n − Introducing new unknown functions given by: y1 (z) = µ(z) dU , y2 (z) = U (z) dz (12) we can write Eq (10) as an equation for the unknown vector Y (z) = [y1 (z), y2 (z)]T , namely: dY (z) = A(z)Y (z), ≤ z ≤ h dz (13) Then, from (18) and H (n+1) = ehn An H (n) , according to (22), we have: (n+1) H11 (n) = cos θn+1 H11 + sin θn+1 (n) H21 an+1 (n) (28) (n) Introducing the expressions (24) and (26) of H11 and H21 , respectively, into (28) yields: n+1 (n+1) H11 = cos θi i=1   i∈{1, ,n} a a a i2j−1  i1 i3 + (−1)j sin θi1 sin θi2 sin θi2j cos θi cos θn+1  a a a i i i 2j j=1 i1