
668 SECOND-ORDER HYPERBOLIC EQUATIONS WITH TWO SPACE VARIABLES
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
π
∞
X
n=0
∞
X
m=1
A
n
µ
2
nm
B
nm
λ
nm
Z
nm
(r)Z
nm
(ξ) cos[n(ϕ − η)] sin(λ
nm
t),
B
nm
= (k
2
2
R
2
2
+ µ
2
nm
R
2
2
− n
2
)Z
2
nm
(R
2
) −(k
2
1
R
2
1
+ µ
2
nm
R
2
1
− n
2
)Z
2
nm
(R
1
),
Z
nm
(r) =
µ
nm
J
′
n
(µ
nm
R
1
) − k
1
J
n
(µ
nm
R
1
)
Y
n
(µ
nm
r)
−
µ
nm
Y
′
n
(µ
nm
R
1
) −k
1
Y
n
(µ
nm
R
1
)
J
n
(µ
nm
r).
Here, A
0
= 1 and A
n
= 2 for n = 1, 2, . . . ; λ
nm
=
p
a
2
µ
2
nm
+ b; the J
n
(r) and Y
n
(r)
are Bessel functions; and the µ
nm
are positive roots of the transcendental equation
µJ
′
n
(µR
1
) −k
1
J
n
(µR
1
)
µY
′
n
(µR
2
) + k
2
Y
n
(µR
2
)
=
µY
′
n
(µR
1
) − k
1
Y
n
(µR
1
)
µJ
′