By Don Dissanayake
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Extra resources for Acoustic Waves
A. (2008). Dynamics of anisotropic multilayers, Wave Motion, 45(5): 629-640. Auld, B. A. (1990). Acoustic Fields and Waves in Solids, Second Edition, Vol. I & II, Robert E. Krieger Publishing Company, Malabar, Florida. Brekhovskikh, L. M. (1980). Waves in Layered Media, Second Edition. Academic Press, New York. Chakraborty, A. & Gopalakrishnan, S. (2006). A spectral finite element model for wave propagation analysis in laminated composite plate, ASME Journal of Vibration and Acoustics, 128: 477-488.
This is the main advantage of introducing the dual local coordinates. It should be noticed that half of the components of vectors vˆ 1E and vˆ (EN + 1) are known, which are denoted by vectors vˆ 1K and vˆ (KN + 1) , respectively, while the remaining half are unknown, denoted by vectors vˆ U1 and vˆ U( N + 1) , respectively. Substituting Eqs. (23) and (24) into Eq. (22), we can derive A 12 a 12 + D12 d 12 = TK1 vˆ K1 = s01 , A( N + 1) N a ( N + 1) N + D( N + 1) N d( N + 1) N = TK( N + 1) vˆ (KN + 1) = s(0N + 1) (25) 34 Acoustic Waves where A 12 , D12 , A( N + 1) N , D( N + 1) N , TK1 ( TK( N + 1) ) are the coefficient matrices with components ( N + 1) N 12 extracted, in accordance with vˆ 1K and vˆ (KN + 1) , from Φ12 , Φ(+N + 1) N and TE ( TE − , Φ+ , Φ− and Tv ) respectively.
By virtue of Eqs. (20) and (21), the solutions to vˆ 12 (0) and vˆ ( N + 1) N (0) can be obtained as vˆ 12 (0) = Φ12 w 12 = ⎡⎣Φ12 − ⎧⎪ a 12 ⎫⎪ ⎤ Φ 12 + ⎦ ⎨ 12 ⎬ ⎪⎩d ⎭⎪ vˆ ( N + 1) N (0) = Φ( N + 1) N w( N + 1) N = ⎡⎣Φ(−N + 1) N ( N + 1) N ⎪⎧ a ⎪⎫ Φ(+N + 1) N ⎤⎦ ⎨ ( N + 1) N ⎬ d ⎪⎩ ⎭⎪ (23) (24) where the exponential functions disappear since the thickness coordinates on the surfaces are always zero in the corresponding local coordinates. This is the main advantage of introducing the dual local coordinates.
Acoustic Waves by Don Dissanayake