Don Dissanayake's Acoustic Waves PDF
By Don Dissanayake
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Additional info for Acoustic Waves
It should be noted that the exponentially growing functions, which usually cause numerical instability (such as in the TMM) for large values of the frequency-thickness product, have been completely excluded from the phase matrices P JK and P KJ , since we always have Re(λ−JK h JK ) > 0 or Re(λ−JK h JK ) = 0, Im(λ−JK h JK ) > 0 ( Re(λ+JK h JK ) < 0 or Re( λ+JK h JK ) = 0, Im( λ+JK h JK ) < 0 ). As indicated by Eq. (41), there are nv equations in the local phase relation of each layer. 2 Global phase relation of the structure Grouping together the local phase relations for all layers from up to down yields the global phase relation with nv × N equations a = Pd = PUd (42) where the (nv × N ) × (nv × N ) block diagonal matrices P , named the global phase matrix, is composed of Reverberation-Ray Matrix Analysis of Acoustic Waves in Multilayered Anisotropic Structures P =< P 12 , P 21 , P 23 ," , P JI , P JK ," , P N ( N + 1) , P( N + 1) N > 37 (43) the variant of the global departing wave vector d is related to the wave vector d by the (nv × N ) × (nv × N ) block diagonal matrix U , which is referred to as the global permutation matrix, to account for the different sequence of components arrangement between d and d .
6. 0 (c) Phase velocity-frequency spectra Fig. 7. Comparison of dispersion curves of the composite with different boundary conditions 44 Acoustic Waves 5. Conclusion We present a unified formulation of the method of reverberation-ray matrix (MRRM) for the analysis of acoustic wave propagation in multilayered anisotropic elastic/piezoelectric structures based on the state space formalism and Fourier transforms in the framework of three-dimensional elasticity or piezoelectricity. 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.
Therefore, the matrices Λ − and Φ − and the vector a correspond to the eigenvalues λi , which satisfy Re( −λi ) < 0 or Re( −λi ) = 0,Im( −λi ) < 0 , while the matrices Λ + and Φ + and the vector d are associated with the remaining eigenvalues. It is easily seen that we always have na + nd = nv with nv = 6 for elastic materials and nv = 8 for piezoelectric materials. Consequently, the solution to the state equation given in Eq. (16) can be rewritten as ⎡exp ( Λ − z ) ⎤ ⎧a ⎫ 0 Φ+ ] ⎢ ⎥⎨ ⎬ 0 exp ( Λ + z ) ⎦ ⎩d ⎭ ⎣ Φ u + ⎤ ⎡exp ( Λ − z ) ⎤ ⎧a ⎫ 0 ⎧ vˆ ( z) ⎫ ⎡ Φ = ⎨ u ⎬ = ⎢ u− ⎥⎨ ⎬ ⎥⎢ ˆ z v Φ Φ ( ) z 0 Λ exp ( ) σ+⎦⎣ + ⎩ σ ⎭ ⎣ σ− ⎦ ⎩d ⎭ vˆ ( z) = [ Φ − (17) Reverberation-Ray Matrix Analysis of Acoustic Waves in Multilayered Anisotropic Structures 31 where Φ u − and Φσ − are nv / 2 × na sub eigenvector matrices of Φ − corresponding to the generalized displacement and stress vectors, respectively; Φ u + and Φσ + are those nv / 2 × nd sub eigenvector matrices of Φ + .
Acoustic Waves by Don Dissanayake