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

1.3 Coordinate systems and differential operators 9
Laplacian
∂
∂r + 1 ∂ 2
r 2 ∂θ + ∂2
2 ∂z 2
∇ 2 =∇· ∇= ∂2
∂r + 1 2 r
[ ∂
∇ 2 2 u r
u =∇· ∇u = ˆr
∂r + 1 ∂u r
2 r
[ ∂
+ ˆt
2 u θ
∂r + 1 ∂u θ
2 r
∂r + 1 ( ∂ 2 u r
r 2 ∂θ 2
∂r + 1 ( ∂ 2 u θ
r 2 ∂θ 2
[ ∂
+ ˆk
2 u z
∂r + 1 ∂u z
2 r ∂r + 1
∂ 2 u z
r 2 ∂θ 2
− u r − 2 ∂u ) ]
θ
+ ∂2 u r
∂θ ∂z 2
− u θ + 2 ∂u r
∂θ
]
+
∂2 u z
∂z 2
) ]
+ ∂2 u θ
∂z 2
(1.27)
(Note: ∂ 2 /∂θ 2 in ∇ 2 acts both on the components of u and on the basis vectors ˆr, ˆt.)
Waveequation
(λ + µ)∇∇ · u + µ∇ · ∇u + b = ρü (1.28)
Expansion of avector in Fourier seriesinthe azimuth
∞∑
( ) cos nθ
u = u n (1.29a)
sin nθ
n=0
∞∑
( ) −sin nθ
v = v n (1.29b)
cos nθ
n=0
∞∑
( ) cos nθ
w = w n (1.29c)
sin nθ
n=0
in which u ≡ u r , v ≡ u θ , w ≡ u z , and either the lower or the upper element in the parentheses
must be used, as may be necessary. Also, u n , v n , w n are the coefficients of the Fourier
series, which do not depend on θ, but only on r and z, that is, u n = u n (r, z), and so forth.
1.3.3 Spherical coordinates
Source–receiver distance R = √ x 2 + y 2 + z 2 (1.30a)
Range r = √ x 2 + y 2 (1.30b)
Azimuth tan θ = y/x, 0 ≤ θ ≤ 2π (1.30c)
Polar angle φ = arccos (z/R), 0 ≤ φ ≤ π (1.30d)
Direction cosines γ 1 = sin φ cos θ ⇒ cos θ =
γ 1
√
1 − γ 2 3
(1.30e)
γ 2 = sin φ sin θ ⇒ sin θ = √
(1.30f)
1 − γ3
2
√
γ 3 = cos φ ⇒ sin φ = 1 − γ3 2 (1.30g)
γ 2