
750 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
0
ξf
0
(ξ, η, ζ)G(r, ϕ, z, ξ, η, ζ, t) dξ dη dζ
+
Z
l
0
Z
2π
0
Z
R
0
ξ
f
1
(ξ, η, ζ) + kf
0
(ξ, η, ζ)
G(r, ϕ, z, ξ, η, ζ, t) dξ dη dζ
+ a
2
R
Z
t
0
Z
l
0
Z
2π
0
g
1
(η, ζ, τ)G(r, ϕ, z, R, η, ζ, t − τ ) dη dζ dτ
− a
2
Z
t
0
Z
2π
0
Z
R
0
ξg
2
(ξ, η, τ )G(r, ϕ, z, ξ, η, 0, t − τ ) dξ dη dτ
+ a
2
Z
t
0
Z
2π
0
Z
R
0
ξg
3
(ξ, η, τ )G(r, ϕ, z, ξ, η, l, t − τ ) dξ dη dτ
+
Z
t
0
Z
l
0
Z
2π
0
Z
R
0
ξΦ(ξ, η, ζ, τ )G(r, ϕ, z, ξ, η, ζ, t − τ ) dξ dη dζ dτ.
Here,
G(r, ϕ, z, ξ, η, ζ, t)=exp
−
1
2
kt
sin
t
√
c
πR
2
l
√
c
+
2
πR
2
l
∞
X
s=1
1
√
β
s
cos
sπx
l
cos
sπξ
l
sin
t
p
β
s
+
1
πl
∞
X
n=0
∞
X
m=1
∞
X
s=0
A
n
A
s
µ
2
nm
J
n
(µ
nm
r)J
n
(µ
nm
ξ)
(µ
2
nm
R
2
−n
2
)[J
n
(µ
nm
R)]
2
cos[n(