
8.4. Telegraph Equation
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
∆
3
w − bw + Φ(x, y, z, t) 755
the J
n
(r) and Y
n
(r) are Bessel functions, and the µ
nm
are positive roots of the transcen-
dental equation
J
′
n
(µR
1
)Y
′
n
(µR
2
) − Y
′
n
(µR
1
)J
′
n
(µR
2
) = 0.
◮ Domain: R
1
≤ r ≤ R
2
, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l. Third boundary value problem.
A hollow circular cylinder of finite length is considered. The following conditions are
prescribed:
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 − s
1
w = g
1
(ϕ, z, t) at r = R
1
(boundary condition),
∂
r
w + s
2
w = g
2
(ϕ, z, t) at r = R
2
(boundary condition),
∂
z
w − s
3
w = g
3
(r, ϕ, t) at z = 0 (boundary condition),
∂
z
w + s
4
w =