
8.1. Wave Equation
∂
2
w
∂t
2
= a
2
∆
3
w 705
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 condition),
w = g
2
(r, ϕ, t) at z = 0 (boundary condition),
w = g
3
(r, ϕ, t) at z = l (boundary condition).
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
ξf
1
(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 )
∂
∂ζ
G(r, ϕ, z, ξ, η, ζ, t − τ)
ζ=0
dξ dη dτ
− a
2
Z
t
0
Z
2π
0
Z
R
0
ξg
3
(ξ, η, τ )
∂
∂ζ
G(r, ϕ, z, ξ, η, ζ, t − τ)
ζ=l
dξ dη dτ.
Here,
G(r, ϕ, z, ξ, η, ζ, t) =
2
π
2
aR
2
∞
X
k