
6.2. Wave Equations with Axial or Central Symmetry 577
The solution w(x, t) is determined by the formula in the previous paragraph (for the
second boundary value problem) where
G(x, ξ, t) = exp
b(ξ − x)
2a
2
∞
X
n=1
y
n
(x)y
n
(ξ) sin
t
√
λ
n
B
n
√
λ
n
.
Here,
y
n
(x) = cos(µ
n
x) +
2a
2
k
1
+ b
2a
2
µ
n
sin(µ
n
x), λ
n
= a
2
µ
2
n
+
b
2
4a
2
− c,
B
n
=
2a
2
k
2
− b
4a
2
µ
2
n
4a
4
µ
2
n
+ (2a
2
k
1
+ b)
2
4a
4
µ
2
n
+ (2a
2
k
2
− b)
2
+
2a
2
k
1
+ b
4a
2
µ
2
n
+
l
2
+
l(2a
2
k
1
+ b)
2
8a
4
µ
2
n
,
where the µ
n
are positive roots of the transcendental equation
tan(µl)
µ
=
4a
4
(k
1
+ k
2
)
4a
4
µ
2
− (2a
2
k
1
+ b)(2a
2
k
2
− b)
.
6.2 Wave Equations with Axial or Central Symmetry
6.2.1 Equation of the Form
∂
2
w
∂t
2
= a
2
∂
2
w
∂r
2
+
1
r
∂w
∂r
This is the one-dimensional wave equation with