
770 SECOND-ORDER HYPERBOLIC EQUATIONS WITH THREE OR MORE SPA CE VARIABLES
8.6.2 N o n homogeneous Wave Equation
∂
2
w
∂t
2
= a
2
∆
n
w + Φ(x
1
, . . . , x
n
, t)
◮ Domain: −∞ < x
k
< ∞ ; k = 1, . . . , n. Cauchy problem.
Initial conditions are prescribed:
w = f (x) at t = 0,
∂
t
w = g(x) at t = 0.
Solution:
w(x, t) =
1
a
n−1
(n − 2)!
∂
n−1
∂t
n−1
Z
at
0
a
2
t
2
− r
2
n−3
2
rT
r
[f(x)] dr
+
1
a
n−1
(n − 2)!
∂
n−2
∂t
n−2
Z
at
0
a
2
t
2
− r
2
n−3
2
rT
r
[g(x)] dr
+
1
a
n−1
(n − 2)!
∂
n−2
∂t
n−2
Z
at
0
dτ
Z
aτ
0
a
2
τ
2
− r
2
n−3
2
rT
r
[Φ(x, t −τ)] dr.
Here, T
r
[f(x)] is the average of f over the spherical surface of radius r with center at x:
T
r
[f(x)] ≡
1
σ
n
r
n−1
Z
|x−y|=r
f(y) dS
y
, σ
n
=
2π
n/2
Γ(n/2)
,
where σ
n
r
n−1
is the area of the surface of