tailieunhanh - Waves in fluids and solids Part 3

Tham khảo tài liệu 'waves in fluids and solids part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Acoustic Waves in Layered Media - From Theory to Seismic Applications 39 located on the same branch. The largest root corresponds to the minimum point C1 on the upper branch of the curve 123 . If E E E we have off-axis triplications. The coordinates EC 1 and EC 2 are given by equations EC1 2Q sin 21 3 ó -d EC2 2Q sin 21 3 5 ó -d 131 where d 1 2 g 3 g2 - e Q 1 - e2 - 2g 3 g2 71 1 - g 2 - e2 1 - e2 - g2 w arccos------------ ------ ------------- Q3 2 132 The case for the qSV-wave vertical on-axis incipient triplication can be obtained by setting u 1 0 0 with condition 120 being simplified to E EC2 or a or v 2 0 Tygel et al. 2007 . The case for the qSV-wave horizontal on-axis incipient triplication can be obtained by setting u -1 0 ft I2 with condition 118 being simplified to E E. or a - 1 2s - 0 2 1 - 02 . If Ea1 E min EB1 EC2 we have both on-axis triplications. If e 0 or s 0 then we have the following equality E ECC and therefore both on-axis triplications are incipient. Extension of qSV-wave triplications for multilayered case From the ray theory it follows that for any vertically heterogeneous medium including horizontally layered medium kinematically effective vertical slowness is always the average of the vertical slownesses from the individual layers. We have to stress that our approach is based on the high-frequency limit of the wave propagation not on the low-frequency one which results in effective medium averaging. Since the wave propagates through the layered medium with the same horizontal slowness p the effective vertical slowness has very simple form q q 133 where denotes the arithmetic thickness averaging m mh7 h with h i 1 N being the thickness of layer i in the stack of N layers. With notation 133 equations 112 are valid for the multilayered case. Similar approach is used in Stovas 2009 for a vertically heterogeneous isotropic medium. If a layered VTI medium results in more then one caustic there is no any kinematically effective VTI medium .