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

132 Integral transform method
C n are the cylindrical Bessel functions of the first kind and order n. Observe that the
differentiation of the Bessel functions is carried out with respect to the product kr, and
not simply r. A brief summary of the fundamental properties of these functions is given
in the Appendix.
To the uninitiated reader, it may not be at all obvious that the set of integral transform
equations given above for u and ũ indeed constitute a direct and inverse Fourier–Bessel
transform pair. The proof of the inversion is based on the fact that for arbitrary functions
f j (r), the forward transform
⎧ ⎫
⎪⎨ F 1 ⎪⎬
F 2 =
⎪⎩ ⎪ ⎭
F 3
∫ ∞
0
⎧
⎪⎨ J ′ n
n kr J ⎫ ⎧ ⎫
n 0 ⎪⎬ ⎪⎨ f 1 ⎪⎬
n
r
kr
⎪⎩
J n J n ′ 0 f
⎪ ⎭ ⎪ 2 dr (9.36)
⎩ ⎪ ⎭
0 0 J n f 3
can be written in the fully diagonal form
⎧
1
⎪⎨ 2 (F ⎫ ⎧
⎫ ⎧
1 + F 2 ) ⎪⎬ ∫ ∞
⎪⎨ J n−1 0 0 ⎪⎬ ⎪⎨
1
1
2
⎪⎩
(F 2 − F 1 ) = r 0 J
⎪ n+1 0
⎭ 0 ⎪⎩
⎪ ⎭ ⎪ ⎩
F 3 0 0 J n
which is a self-reciprocating Hankel transform, i.e.,
⎧
1
⎪⎨ 2 ( f ⎫ ⎧
⎫ ⎧
1 + f 2 ) ⎪⎬ ∫ ∞
⎪⎨ J n−1 0 0 ⎪⎬ ⎪⎨
1
1
2
⎪⎩
( f 2 − f 1 ) = k 0 J
⎪ n+1 0
⎭ 0 ⎪⎩
⎪ ⎭ ⎪ ⎩
f 3 0 0 J n
2 ( f 1 + f 2 )
1
2 ( f 2 − f 1 )
f 3
⎫
⎪⎬
⎪ ⎭
dr (9.37)
2 (F 1 + F 2 )
1
2 (F 2 − F 1 )
F 3
⎫
⎪⎬
⎪ ⎭
dk (9.38)
Also, the orthogonality of the Fourier series in θ must be used to complete the proof of
inversion. This involves the identity
∫ 2π
0
T n T j dθ = πδ (nj) (1 + δ n0 δ j0 ) = δ (n) j /a n . (9.39)
Transformed wave equation
Applying a spatial Fourier–Bessel transform to the wave equation, or equivalently, assuming
that u and b are of the form
u(r,θ,z, t) = T n C n f(k, n, z, t), b(r,θ,z, t) = T n C n q(k, n, z, t) (9.40)
in which f = [ f 1 f 2 f 3] T , one obtains the three expressions (after lengthy and tedious
algebra, which requires also consideration of the differential equation for the Bessel
functions)
⎧
⎪⎨ −k ( kf 1 − ∂ ∂z f )
3
∇∇·u = T n C n 0
⎪ ⎩ (
−kf1 + ∂ ∂z f 3)
∂
∂z
⎫
⎪⎬
( ∂
2
⎪⎭ , ∇ · ∇u = T nC n
)
∂z f − 2 k2 f
(9.41)
ü = T n C n¨f (9.42)