
730 SECOND-ORDER HYPERBOLIC EQUATIONS WITH THREE OR MORE SPA CE VARIABLES
Here,
G(r, ϕ, z, ξ, η, ζ, t) =
1
πR
2
l
∞
X
n=0
∞
X
m=1
∞
X
k=0
A
n
A
k
[J
′
n
(µ
nm
R)]
2
√
λ
nmk
J
n
(µ
nm
r)J
n
(µ
nm
ξ)
× cos[n(ϕ − η)] cos
kπz
l
cos
kπζ
l
sin
t
p
λ
nmk
,
λ
nmk
= a
2
µ
2
nm
+
a
2
k
2
π
2
l
2
+ b, A
n
=
(
1 for n = 0,
2 for n > 0,
where the J
n
(ξ) are Bessel functions (the prime denotes a derivative with respect to the
argument) and the µ
nm
are positive roots of the transcendental equation J
n
(µR) = 0.
2
◦
. A circular cylinder of finite length is considered. The following conditions are pre-
scribed:
w = f
0
(r, ϕ, z) at t = 0 (initial condition),
∂
t
w = f
1
(r, ϕ, z) at t = 0 (initial condition),
∂
r
w = g
1
(ϕ, z, t) at r = R (boundary ...