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Litian Wang Østfold University College Existence of extraordinary transonic states in monoclinic elastic media Litian Wang and Kent Ryne Østfold University College 1757 Halden Norway

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Litian Wang Østfold University College Main problems a)Existence of extraordinary transonic states associated with extraordinary zero-curvature slowness curve b)Existence of space of degeneracy c)Existence of generalized surface waves

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Litian Wang Østfold University College Surface geometry of slowness surface Cubic (Cu)Monoclinic

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Litian Wang Østfold University College Surface geometry of slowness surface Cubic (Cu)Monoclinic

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Litian Wang Østfold University College Zero-curvature transonic states E1E2E3E4 Barnett, Lothe & Gundersen m n

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Litian Wang Østfold University College Surface geometry of slowness surface Cubic (Cu)Monoclinic

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Litian Wang Østfold University College Problem 1 a)Can a slowness curve have zero- curvature locally? b)How flat a slowness curve can be?

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Litian Wang Østfold University College Degree of freedom Degree of freedom = 6

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Litian Wang Østfold University College Wave propagation in monoclinic media Elastic stiffness matrix:

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Litian Wang Østfold University College θ k

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Litian Wang Østfold University College Christoffel equation Where d 13 =c 13 +c 55, ∆ 15 =c 11 -c 55, ∆ 64 =c 66 -c 44, ∆ 53 =c 55 -c 33,

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Litian Wang Østfold University College Curvature in slowness plot Let Curvature k and its second derivative k’’ in the neighborhood of z- axis are given by θ k

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Litian Wang Østfold University College How to find the eigenvalue ? Where d 13 =c 13 +c 55, ∆ 15 =c 11 -c 55, ∆ 64 =c 66 -c 44, ∆ 53 =c 55 -c 33, θ k

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Litian Wang Østfold University College Perturbation method θ k

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Litian Wang Østfold University College Where θ k

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Litian Wang Østfold University College Results - 1 (a) Normal curvature of slowness curve along z-axis (b) Zero-Curvature along z-axis when d 13 2 = c 11 ∆ 35 or (c 13 +c 55 ) 2 =c 11 (c 33 -c 55 ) (See also Shuvalov et al) θ k

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Litian Wang Østfold University College (a) The second derivative of curvature: Results - 2 (b) Extraordinary zero-curvature along z-axis when (c 11 c 36 -d 13 c 16 ) 2 =c 11 2 c 55 ∆ 45 ) θ k

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Litian Wang Østfold University College

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Litian Wang Østfold University College

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Litian Wang Østfold University College Problem 2 a)Space of degeneracy in monoclinic media b)Generalized surface waves

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Litian Wang Østfold University College Degeneracy of the Stroh eigenvalues E1 zero-curvature transonic state:

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Litian Wang Østfold University College E4 zero-curvature transonic state: Degeneracy of the Stroh eigenvalues

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Litian Wang Østfold University College Result 3 Space of degeneracy vs zero-curvature slowness curve:

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Litian Wang Østfold University College Result 4 Space of degeneracy vs generalized surface waves Subsonic surface waves Supersonic surface waves

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Litian Wang Østfold University College Conclusions a)Existence of extraordinary zero-curvature slowness curve b)Existence of space of degeneracy c)Existence of supersonic surface wave along the space of degeneracy d)Existence of generalized subsonic surface wave along the space of degeneracy

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