
7.1. Wave Equation
∂
2
w
∂t
2
= a
2
∆
2
w 643
◮ Domain: R
1
≤ r ≤ R
2
, 0 ≤ ϕ ≤ 2π. Third boundary value problem.
An annular domain is considered. The following conditions are prescribed:
w = f
0
(r, ϕ) at t = 0 (initial condition),
∂
t
w = f
1
(r, ϕ) at t = 0 (initial condition),
∂
r
w − k
1
w = g
1
(ϕ, t) at r = R
1
(boundary condition),
∂
r
w + k
2
w = g
2
(ϕ, t) at r = R
2
(boundary condition).
The solution w(r, ϕ, t) is determined by the formula in the previous paragraph (for the
second boundary value problem) where
G(r, ϕ, ξ, η, t) =
1
πa
∞
X
n=0
∞
X
m=1
A
n
µ
nm
B
nm
Z
n
(µ
nm
r)Z
n
(µ
nm
ξ) cos[n(ϕ − η)] sin(µ
nm
at),
B
nm
= (k
2
2
R
2
2
+ µ
2
nm
R
2
2
− n
2
)Z
2
n
(µ
nm
R
2
) −(k
2
1
R
2
1
+ µ
2
nm
R
2
1
− n
2
)Z
2
n
(µ
nm
R
1
),
Z
n
(µ
nm
r) =
µ
nm