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

136 Integral transform method
9.2.2 Cylindrically stratified media
The starting point in this case is again the wave equation in cylindrical coordinates, but
now we perform instead a Fourier transform in the axial and azimuthal directions:
⎧
1
∫ 2π ∫ +∞
⎪⎨ n = 0
˜b n = a n T n b(r,θ,z,ω) e −ikzz dzdθ , a n =
2π
(9.58)
0 −∞
1 ⎪⎩ n ≠ 0
π
b(r,θ,z,ω) = 1 ∫ { }
+∞ ∞∑
T n ˜b n e −ikzz dk z (9.59)
2π
n=0
−∞
∫ 2π ∫ +∞
ũ n = a n T n u(r,θ,z,ω) e −ikzz dzdθ (9.60)
0
−∞
u(r,θ,z,ω) = 1
2π
∫ +∞
−∞
{ ∞∑
n=0
T n ũ n
}
e −ikzz dk z (9.61)
in which again the boxed equations are those commonly used. Applying the stated Fourier
transforms in θ, z, and t to the wave equation, collecting terms, and applying factors
−i =− √ −1 to each of the vertical components so that u = [ ũ r ũ θ −iũ z ] and b =
[ ˜b r
˜b θ −i ˜b z ], we obtain the one-dimensional system of differential equations in the
radial direction,
⎧ ⎡
⎤
⎤
⎨ λ + 2µ 0 0
b +
⎩ ρω2 I + ⎣ 0 µ 0
0 0 µ
⎡
λ + 2µ −n(λ + µ) 0
∂r + ⎣n(λ + µ) µ 0⎦ 1
2 r
0 0 µ
⎦ ∂2
⎡
⎤ ⎡
⎤
λ + 2µ + n 2 µ −n(λ + 3µ) 0
⎢
− ⎣ −n(λ + 3µ) µ + n 2 ⎥
(λ + 2µ) 0 ⎦ 1 µ 0 0
r − ⎢
⎥
2 ⎣ 0 µ 0 ⎦ kz
2
0 0 n 2 µ 0 0 λ + 2µ
⎡ ⎤
⎡ ⎤ ⎫
0 0 1
0 0 0
+ ⎣ 0 0 0⎦ ∂
(λ + µ) k z
∂r − ⎣ 0 0 n ⎦ (λ + µ) k ⎬
z
r ⎭ u = 0 (9.62)
−1 0 0
1 n 0
In the absence of body loads (i.e., b = 0), this equation can be shown to admit solutions
of the form
u(r, k z ,ω) = H (1)
n a 1 + H (1)
n a 2 (a 1 , a 2 : 3× 1 vectors of arbitrary constants) (9.63)
⎧
⎪⎨
H (1)
n =
⎫
( (1)) ′
H αn n H(1) βn k z ( (1)) ′
H βn
k β r k β ⎪⎬
n H(1) αn ( (1)) ′
H βn n k z H (1)
βn
k α r
k β k β r
⎪⎩ − k z
H αn (1) 0 H (1) ⎪⎭
βn
k α
, H (2)
n =
⎧
∂
∂r
( (2)) ′
H αn n H(2) βn k z ( (2)) ′
H βn
k ⎪⎨
β r k β ⎪⎬
n H(2) αn ( (2)) ′
H βn n k z H (2)
βn ,
k α r
k β k β r
⎪⎩ − k z
H αn (2) 0 H (2) ⎪⎭
βn
k α
⎫
(9.64)