TAILIEUCHUNG - Waves in fluids and solids Part 2

Tham khảo tài liệu 'waves in fluids and solids part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 14 Waves in Fluids and Solids tW I F rW -G . 63 These correspond to ones given in Ursin and Haugen 1996 for VTI media and in Aki and Richards 1980 for isotropic media except that they are normalized with respect to the vertical energy flux and not with respect to amplitude. 6. Periodically layered media Let us introduce the infinite periodically layered VTI medium with the period thickness N H Y h where h is the thickness of j layer in the sequence of N layers comprising the j-1 period. The dispersion equation for this N layered medium is given by Helbig 1984 det P - exp imH0 I 0 64 and the period propagator matrix P is specified by formula 15 . The equation 64 is known as the Floquet 1883 equation. The parameter 0 o pm is effective and generally complex vertical component of the slowness vector. For plane waves with horizontal slowness p the real part of 0 which satisfies equation 64 Re 0 q defines the vertical slowness of the envelope while the imaginary part Im 0 ỵ characterizes the attenuation due to scattering. Note that for propagating waves lim ỵ 0 . This indicates that there is no scattering in the low frequency limit. 0 0 The low and high frequency limits In the low-frequency asymptotic of the propagator matrix P has the following form P exp imHM m with M m -1 ZhtMk -jm Yhht MjM - MtMj o m 65 H k i 2H j t Therefore the dispersion equation 64 in the low-frequency limit has roots similar to those defined for a homogeneous VTI medium given by the averaged matrix M 0 -1 Z k M k . 66 H k 1 One can see from observing the elements of the matrices in equation 31 that equation 66 is equivalent to the Backus averaging. The propagator matrix P which defines the propagation of mode rip in the k h layer k 1 . N can be defined as P exp imhN Fn . exp m Fk . exp imhiFi 67 where Fk 0k n - m is a 4x4 matrix of rank one m and n are the left- and right hand side eigenvectors of matrix Mk with eigenvalue 0k . Substituting Fk into equation 67 results in Acoustic Waves in .

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