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

138 Integral transform method
(continued)
⎡
⎤
0 · −(λ + 3µ)
+ ⎣ · 0 · ⎦
1 ∂
R
2(λ + 2µ) · 0
2 sin φ ∂θ
⎡
⎤
⎡
⎤
0 · ·
+ ⎣ · 0 −(λ + 3µ) ⎦
cot φ 0 −(λ + 3µ) ·
∂
R
· λ + 3µ 0
2 sin φ ∂θ + ⎣ · 0 · ⎦ cot φ
R
· · 0
2
⎡ ⎤
⎡
⎤ ⎫
0 · ·
0 · ·
+ ⎣ · −λ · ⎦
1
· · µ
R 2 sin 2 φ + ⎣ · −2µ · ⎦ cot2 φ ⎬
R
· · −2µ
2 ⎭ u (9.69)
This rather complicated looking system of differential equations can also be solved by
means of transforms, except that because each spherical surface is finite in size, we employ
series solutions in place of the integrals of the previous sections. However, instead of
attempting to re-derive the relevant equations in the transformed domain for this case,
we shall simply rely on the results already obtained in Section 8.10. Thus, the solution of
this equation follows along the following formal steps (the matrices involved are defined
in Section 8.10, and tabulated in Section 10.4):
1) Express the source in terms of spherical harmonics:
∫ π
˜b mn (R,ω) = J −1 sin φ L n m
0
∫ 2π
0
T n b(R,φ,θ,ω) dθ dφ (9.70)
which admits the formal inversion
∞∑ m∑
b = T n L n ˜b m mn (9.71)
m=0 n=0
2) For each m, n, and ω, obtain the solution ũ mn (R) in the frequency–wavenumber domain
by solving the one-dimensional system of equations in the radial coordinate R.
3) Obtain the solution in the space domain from the series solution
u(R,φ,θ,ω) =
∞∑ m∑
T n L n mũmn (9.72)
m=0 n=0
which again admits the formal inversion
∫ π
ũ mn (R,ω) = J −1 sin φ L m n
0
∫ 2π
0
T n u(R,φ,θ,ω) dθ dφ (9.73)
The proof of these inversion formulas is obtained by considering the orthogonality properties
of Fourier series in the azimuth and of the spheroidal matrix in co-latitude, namely,
∫ 2π
0
T n T j dθ = πδ (nj) (1 + δ n0 δ j0 ) (9.74)