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

10.4 Stiffness matrix method for layered spheres 177
Table 10.7. Spheroidal, azimuthal and spherical Bessel matrices
P
⎧⎪ n ⎫
m 0 0
⎨ dPm
n n Pm
n
L n m = 0
⎪⎬
dφ sin φ , Pm n = Pn m (φ) = associated Legendre function
⎪ ⎩
n Pm
n dPm
n 0
⎪⎭
sin φ dφ
or
T n = diag[ cos nθ cos nθ − sin nθ ]
T n = diag[ sin nθ sin nθ cos nθ ]
⎧
dh Pm
m(m + 1) h Sm
0
dz P
z S
⎪⎨
H m =
⎪⎩
h Pm
z P
1
z S
d(z S h Sm )
dz S
0
0 0 h Sm
⎫⎪ ⎬
⎪ ⎭
, F m =
⎧
⎨
⎩
f 11 f 12 0
f 21 f 22 0
0 0 f 33
⎫
⎬
⎭
[
f 11 =−k P (λ + 2µ) h Pm − 2µ (
2h P,m+1 + m(m − 1) h )]
Pm
z P z P
[
f 22 =−k S µ h Sm − 2µ (
h S,m+1 − (m + 1) (m − 1) h )]
Sm
z S z S
f 12 =−k S m(m + 1) 2µ (
h S,m+1 − (m − 1) h )
Sm
z S z S
(
2µ
f 21 =−k P h P,m+1 − (m − 1) h )
Pm
z P z P
(
f 33 =−k S µ h S,m+1 − (m − 1) h )
Sm
z S
H m, F m → h Pm = j m(z P), h Sm = j m(z S) (use spherical Bessel functions)
H (1)
m , F (1)
m → h Pm = h (1)
m (z P), h Sm = h (1)
m (z S)
H (2)
m , F (2)
m → h Pm = h (2)
m (z P), h Sm = h (2)
m (z S)
(use 1st spherical Hankel functions)
(use 2nd spherical Hankel functions)
z P = k P R ≡ P , z S = k S R ≡ S , k P = ω α , k S = ω β
¯p(R,φ,θ,ω) = T n L n m
= T n L n m
(
)
F (1)
m a 1 + F (2)
m a 2
{ } { }
F (1)
m F (2) a 1
m
a 2
(10.146)
with subscripted matrices T, H, F defined in Table 10.7, and the overbar being a reminder
that this is a particular solution for given m, n. Also, the numbers in parenthesis refer
to the kind of spherical Bessel functions used to construct the matrices: (1) refers to h (1)
m ,
and (2) to h (2)
m . Alternatively, spherical Bessel functions of the first and second kind can be