
760 SECOND-ORDER HYPERBOLIC EQUATIONS WITH THREE OR MORE SPA CE VARIABLES
8.4.3 Problems in Spherical Coordinates
A three-dimensional nonhomogeneous telegraph equation in the spherical coordinate sys-
tem is written as
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
1
r
2
∂
∂r
r
2
∂w
∂r
+
1
r
2
sin θ
∂
∂θ
sin θ
∂w
∂θ
+
1
r
2
sin
2
θ
∂
2
w
∂ϕ
2
− bw + Φ(r, θ, ϕ, t).
◮ Domain: 0 ≤ r ≤ R, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π. First boundary value problem.
A spherical domain is considered. The following conditions are prescribed:
w = f
0
(r, θ, ϕ) at t = 0 (initial condition),
∂
t
w = f
1
(r, θ, ϕ) at t = 0 (initial condition),
w = g(θ, ϕ, t) at r = R (boundary condition).
Solution:
w(r, θ, ϕ, t)=
∂
∂t
Z
2π
0
Z
π
0
Z
R
0
f
0
(ξ, η, ζ)G(r, θ, ϕ, ...