
648 SECOND-ORDER HYPERBOLIC EQUATIONS WITH TWO SPACE VARIABLES
Here, the µ
n
and β
m
are positive roots of the transcendental equations
µJ
1
(µ) −k
1
RJ
0
(µ) = 0,
tan(βl)
β
=
k
2
+ k
3
β
2
− k
2
k
3
.
◮ Domain: 0 ≤ r ≤ R, 0 ≤ z ≤ l. Mixed boundary value problems.
1
◦
. A circular cylinder of finite length is considered. The following conditions are pre-
scribed:
w = f
0
(r, z) at t = 0 (initial condition),
∂
t
w = f
1
(r, z) at t = 0 (initial condition),
w = g
1
(z, t) at r = R (boundary condition),
∂
z
w = g
2
(r, t) at z = 0 (boundary condition),
∂
z
w = g
3
(r, t) at z = l (boundary condition).
Solution:
w(r, z, t)=
∂
∂t
Z
l
0
Z
R
0
f
0
(ξ, η)G(r, z, ξ, η, t) dξ dη+
Z
l
0
Z
R
0
f
1
(ξ, η)G(r, z, ξ, η, t) dξ dη ...