Abstract
It was shown in an earlier paper that, if
u
(
x
1
, x
2
, x
3
, t
) =
u
(
x, t
) satisfies the wave equation
u
tt
= ∆
u
in the exterior of some fixed sphere
r
= │
x
│ =
a
and vanishes for
t
≤
r
, then
ru
(
rξ
,
t
) ~
f
(
ξ
,
t
—
r
) as
r
→ ∞, provided that
ξ
is a fixed unit vector and
t
—
r
remains bounded. It was also shown that the 'radiation field' ’
f
(
ξ
,
s
) determines
u
(
x
,
t
) uniquely in
r ≥ a
. In the present paper it is assumed that the Laplace transform of
u
with respect to
t
exists. This is found to imply that the Laplace transform of
f
with respect to
s
also exists, and is an analytic function of
ξ
that is regular for all complex unit vectors
ξ
. From this it can be inferred that, if
f
itself vanishes for all
8
, and for all
ξ
in any open subset of the (real) unit sphere, then
f
≡ 0, and hence
u
≡ 0 in
r ≥ a
. Furthermore, an integral representation of the Laplace transform of
u
in terms of the Laplace transform of
f
is obtained, which generalizes Weyl’s integral representation of diverging spherical waves in terms of plane waves with complex propagation vectors.
Reference7 articles.
1. Behnke H . & Thullen P . 1934 Theorie der Funlctionen mehreren komplexer Verdnderlichen B erlin: E rgebn. M ath. Vol. 3 no. 3.
2. Bochner S. & M artin W. T. 1948 Several complex variables. P rinceton U niversity Press.
3. C ourant R . &H ilbert D. 1962 Methods of mathematical physics vol. 2. N ew Y ork: Interscience.
4. Proc. Roy;Friedlander F .;Soc. A,1962
5. Jo h n F . 1949 Commun. Pure A p p l. M ath. 2 209-253.
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