Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

10.2 Stiffness matrix method in Cartesian coordinates 145
Table 10.1. Stiffness matrices, Cartesian coordinates, plane strain
√
√
kp = k 2 − (ω/α) 2 , ks = k 2 − (ω/β) 2
Note 1: In the first Riemann sheet, Im(kp) ≥ 0, Im(kp) ≥ 0, and Re(ks) ≥ 0, Im(ks) ≥ 0,
whether or not the medium has material damping (attenuation), that is, the products
kp, ks are complex numbers in the first quadrant, no matter what the sign of k should
be. Thus, the numbers p, s themselves are in the first quadrant if k > 0, and in the
third if k < 0. If material (hysteretic) damping is present, the complex wavenumbers
for P and S waves are of the form
ω
α c
= ω α
(
1 − i ξ P sgn(ω)
)
,
ω
β c
= ω β
(
)
1 − i ξ S sgn(ω)
in which α c ,β c are the complex wave velocities. This is based on the (very close)
approximation
√ √
µc µ
(
)
√
β c =
ρ = 1 µ
1 + 2i ξ S sgn(ω) ≈
ρ
1 − i ξ S sgn(ω) ρ
and a similar expression for α c . These approximations greatly simplify exponential
terms such as
(
exp − iωx ) (
= exp − iωx ) (
exp − ξ )
Sωx
, ξ S = 1/(2Q S ), Q S = quality factor
β c β
β
Note 2: Given k, all SH and SV-P elements exhibit complex-conjugate symmetry with
respect to ±ω. Given ω, the SV-P elements are checkerboard symmetric and
antisymmetric with respect to ±k.
1) SH waves
Layer
Half-space
{ }
coth ksh −1/ sinh ksh
K = ksµ
K = ksµ k > 0, ω>0
−1/ sinh ksh coth ksh
{ }
coth kh −1/ sinh kh
K = kµ
K = kµ k > 0, ω= 0
−1/ sinh kh coth kh
{ }
cot S −1/ sin
K = ρβω
S
, S = ωh K = i ωρβ k = 0, ω>0
−1/ sin S cot S β
K = µ { } 1 −1
K = 0 k = 0, ω= 0
h −1 1
2) SV-P waves
Note: The horizontal–vertical coupling terms in the matrices for SV-P waves given here
have opposite sign from those in Kausel and Röesset (footnote 6). This is the result of
a deliberate change here in the sign of the imaginary factor applied to the vertical
components. Also, we replace r in said reference by p, to avoid confusion with the
radius or range.
{ }
K11 K
K =
12
K 21 K 22
(continued )