
8.1. Wave Equation
∂
2
w
∂t
2
= a
2
∆
3
w 715
where
G(r, θ, ϕ, ξ, η, ζ, t) =
π
8a
√
rξ
∞
X
n=0
∞
X
m=1
n
X
k=0
A
k
B
nmk
Z
n+1/2
(λ
nm
r)Z
n+1/2
(λ
nm
ξ)
× P
k
n
(cos θ)P
k
n
(cos η) cos[k(ϕ − ζ)] sin (λ
nm
at).
Here,
Z
n+1/2
(λ
nm
r) = J
n+1/2
(λ
nm
R
1
)Y
n+1/2
(λ
nm
r) −Y
n+1/2
(λ
nm
R
1
)J
n+1/2
(λ
nm
r),
A
k
=
(
1 for k = 0,
2 for k 6= 0,
B
nmk
=
λ
nm
(2n + 1)(n −k)! J
2
n+1/2
(λ
nm
R
2
)
(n + k)!
J
2
n+1/2
(λ
nm
R
1
) −J
2
n+1/2
(λ
nm
R
2
)
,
where the J
n+1/2
(r) are Bessel functions, the P
k
n
(µ) are associated Legendre functions
expressed in terms of the L egendre polynomials P
n
(µ) as
P
k
n
(µ) = (1 − µ
2
)
k/2
d
k
dµ
k
P
n
(µ), P
n
(µ) =
1
n! 2
n
d
n
dµ
n
(µ
2
− 1)
n
,
and the λ
nm
are positive roots of the transcendental equation Z
n+1/2
(λR
2
) = 0.
◮ Domain: R
1
≤ r ≤ R
2
, 0