# Litian Wang Østfold University College Existence of extraordinary transonic states in monoclinic elastic media Litian Wang and Kent Ryne Østfold University.

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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

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

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

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

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

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

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

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

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

Litian Wang Østfold University College θ k

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,

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

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

Litian Wang Østfold University College Perturbation method θ k

Litian Wang Østfold University College Where θ k

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

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

Litian Wang Østfold University College

Litian Wang Østfold University College

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

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

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

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

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

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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