By Don Dissanayake

ISBN-10: 9533071117

ISBN-13: 9789533071114

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20) and (21) into Eq. (39), and in view of Eq. (40) and Tv = Tv−1 , one obtains the local phase relation of a typical layer JK (KJ) JK JK JK ⎤ ⎪⎧d KJ ⎪⎫ ⎡ P JK 0 ⎪⎧a ⎪⎫ ⎡⎢ exp ( − Λ − h ) ⎥⎨ = =⎢ ⎨ KJ ⎬ JK ⎬ JK JK 0 exp ( Λ + h ) ⎥⎦ ⎩⎪d ⎭⎪ ⎣ 0 ⎩⎪ a ⎭⎪ ⎢⎣ 0 ⎤⎡ 0 ⎥⎢ P KJ ⎦ ⎣ I dJK JK I aJK ⎤ ⎪⎧d ⎪⎫ ⎥ ⎨ KJ ⎬ 0 ⎦ ⎩⎪d ⎭⎪ (41) where the naJK × naJK and ndJK × ndJK diagonal matrices P JK = exp( −Λ −JK h JK ) and P KJ = exp( Λ +JK h JK ) are referred to as local phase matrices, and I aJK and I dJK are identity matrices of order naJK and ndJK , respectively.

The proposed formulation of MRRM includes all wave modes in the structure and possesses good numerical stability by properly excluding exponentially growing function and matrix inversion operation. It is therefore suitable for the accurate analysis of acoustic waves in complex multilayered anisotropic structures by a uniform computer program. In comparison with the well-known traditional transfer matrix method, the present MRRM is unconditionally numerically stable, irrespective of the total number of layers, the thickness of individual layers and the frequency.

1 Governing equations and state vectors Consider a homogeneous, arbitrarily anisotropic elastic medium. From the threedimensional linear elasticity (Synge, 1956; Stroh, 1962; Nayfeh, 1995) we have the constitutive relations σ ij = cijklε kl (1) ε kl = (uk , l + ul , k ) / 2 (2) the strain-displacement relations 28 Acoustic Waves and the equations of motion (in absence of body forces) i σ ij , j = ρ u (3) where the comma in the subscripts and superposed dot imply spatial and time derivatives, σ ij , ε kl and ui are respectively the stress, strain and displacement tensors, c ijkl are the elastic constants, and ρ is the material density.

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