
7.4. Telegraph Equation
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
∆
2
w − bw + Φ(x, y, t) 683
Here,
G(r, ϕ, ξ, η, t) = exp
−
1
2
kt
"
sin
t
p
b−k
2
/4
πR
2
p
b−k
2
/4
+
1
π
∞
X
n=0
∞
X
m=1
A
n
µ
2
nm
J
n
(µ
nm
r)J
n
(µ
nm
ξ)
(µ
2
nm
R
2
−n
2
)[J
n
(µ
nm
R)]
2
cos[n(ϕ−η)]
sin
t
√
λ
nm
√
λ
nm
#
,
λ
nm
= a
2
µ
2
nm
+b−
1
4
k
2
, A
0
= 1, A
n
= 2 (n = 1, 2, . . .),
where the J
n
(ξ) are Bessel functions and the µ
m
are positive roots of the transcendental
equation J
′
n
(µR) = 0.
◮ Domain: 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π. Third boundary value problem.
A circle is considered. The following conditions are prescribed:
w = f
0
(r, ϕ) at t = 0 (initial condition),
∂
t
w = f
1
(r, ϕ) at t = 0 (initial condition),
∂
r
w + sw = g(ϕ, t) at r = R (boundary condition).
The solution w(r, ϕ,