
710 SECOND-ORDER HYPERBOLIC EQUATIONS WITH THREE OR MORE SPA CE VARIABLES
Solution:
w(r, ϕ, z, t) =
∂
∂t
Z
l
0
Z
2π
0
Z
R
2
R
1
f
0
(ξ, η, ζ)G(r, ϕ, z, ξ, η, ζ, t)ξ dξ dη dζ
+
Z
l
0
Z
2π
0
Z
R
2
R
1
f
1
(ξ, η, ζ)G(r, ϕ, z, ξ, η, ζ, t)ξ dξ dη dζ
− a
2
R
1
Z
t
0
Z
l
0
Z
2π
0
g
1
(η, ζ, τ)G(r, ϕ, z, R
1
, η, ζ, t − τ ) dη dζ dτ
+ a
2
R
2
Z
t
0
Z
l
0
Z
2π
0
g
2
(η, ζ, τ)G(r, ϕ, z, R
2
, η, ζ, t − τ ) dη dζ dτ
+ a
2
Z
t
0
Z
2π
0
Z
R
2
R
1
g
3
(ξ, η, τ )
∂
∂ζ
G(r, ϕ, z, ξ, η, ζ, t − τ)
ζ=0
ξ dξ dη dτ
− a
2
Z
t
0
Z
2π
0
Z
R
2
R
1
g
4
(ξ, η, τ )
∂
∂ζ
G(r, ϕ, z, ξ, η, ζ, t − τ)
ζ=l
ξ dξ dη dτ.
Here,
G(r, ϕ, z, ξ, η, ζ, t) =
2
π
2
a(R
2
2
−R
2
1
)
∞
X
k=1
1
k
sin
kπz
l
sin
kπζ
l
sin
kπat
l
+
2
πl
∞
X
n=0
∞
X
m=1
∞
X
k=1
A
n
µ
2
nm
Z
nm
(r)Z
nm
(ξ)
(µ
2
nm
R
2
2
−n
2
)Z
2
nm
(R
2
)−(µ
2
nm
R
2
1
−n
2
)Z
2
nm
(R
1
)
×cos[n(ϕ