
6.2. Wave Equations with Axial or Central Symmetry 587
Here,
G(r, ξ, t) =
2ξ
R
2
∞
X
n=1
1
J
2
1
(µ
n
)
J
0
µ
n
r
R
J
0
µ
n
ξ
R
sin
t
√
λ
n
√
λ
n
, λ
n
=
a
2
µ
2
n
R
2
+ b,
where the µ
n
are positive zeros of the Bessel function, J
0
(µ) = 0. The numerical values
of the first ten µ
n
are specified in Section 3.2.1 (see the first boundary value problem for
0 ≤ r ≤ R).
◮ Domain: 0 ≤ r ≤ R. Second boundary value problem.
The following conditions are prescribed:
w = f
0
(r) at t = 0 (initial condition),
∂
t
w = f
1
(r) at t = 0 (initial condition),
∂
r
w = g(t) at r = R (boundary condition).
Solution:
w(r, t) =
∂
∂t
Z
R
0
f
0
(ξ)G(r, ξ, t) dξ +
Z
R
0
f
1
(ξ)G(r, ξ, t) dξ
+ a
2
Z
t
0
g(τ)G(r, R, t − τ) dτ +
Z
t
0
Z
R
0
Φ(ξ, τ )G(r, ξ, t