2
Wave equation
2.1
Solution in higher dimension through Spherical Means
Assume
u
is a classical solution to the initial value problem for
n
dimensional wave equation for
n
≥
2:
u
tt

Δ
u
= 0
for
x
∈
R
n
for
t >
0
with
u
(
x
,
0) =
φ
(
x
)
,
u
t
(
x
,
0) =
ψ
(
x
)
(10)
where
φ
∈
C
2
and
ψ
∈
C
1
. For
t >
0,
r >
0, we de±ne
U
(
x
;
r, t
) to be the spherical average over
the surface of an
n
dimensional ball
B
(
x
, r
) of radius
r
, centered at
x
, and denoted by
U
(
x
;
r, t
) =
1
A
r
Z
∂B
(
x
;
r
)
u
(
y
, t
)
d
y
≡ 
Z
∂B
(
x
;
r
)
u
(
y
, t
)
d
y
(11)
where
A
r
is the surface area of an
n
dimensional ball of radius
r
. Note
A
r
=
nα
(
n
)
r
n

1
, where
volume of the
n
dimensional sphere is
α
(
n
)
r
n
. It is to be noted that
lim
r
→
0
+
U
(
x
;
r, t
) =
u
(
x
, t
)
from continuity of
u
. We can similarly de±ne
G
(
x
;
r
) =

Z
∂B
(
x
;
r
)
φ
(
y
, t
)
d
y
(12)
H
(
x
;
r
) =

Z
∂B
(
x
;
r
)
ψ
(
y
, t
)
d
y
(13)
For ±xed
x
, we regard
U
as a function of
r
and
t
. We claim
Lemma 1
For fxed
x
,
U
(
x
;
r, t
)
is a solution o± the initial value problem:
U
tt

U
rr

n

1
r
U
r
= 0
for
r >
0
,
t >
0
and
U
(
x
;
r,
0) =
G
(
x
, r
)
,
U
t
(
x
;
r,
0) =
H
(
x
, r
) (14)
Proof.
For convenience, we depart from our usual convention and denote ‘surface area’ element
on the
n
dimensional ball as
dS
. Symbol
dS
y
will denote surface area element in the variable
y
.
We note that