Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation as
"the sources must be sources, not sinks of energy. The energy which is radiated from the sources must scatter to infinity; no energy may be radiated from infinity into ... the field."
Mathematically, consider the inhomogeneous Helmholtz equation
(
∇
2
+
k
2
)
u
=
−
f
in
R
n
where
n
=
2
,
3
is the dimension of the space,
f
is a given function with compact support representing a bounded source of energy, and
k
>
0
is a constant, called the wavenumber. A solution
u
to this equation is called radiating if it satisfies the Sommerfeld radiation condition
lim
|
x
|
→
∞
|
x
|
n
−
1
2
(
∂
∂
|
x
|
−
i
k
)
u
(
x
)
=
0
uniformly in all directions
x
^
=
x
|
x
|
(above,
i
is the imaginary unit and
|
⋅
|
is the Euclidean norm). Here, it is assumed that the time-harmonic field is
e
−
i
ω
t
u
.
If the time-harmonic field is instead
e
i
ω
t
u
,
one should replace
−
i
with
+
i
in the Sommerfeld radiation condition.
The Sommerfeld radiation condition is used to solve uniquely the Helmholtz equation. For example, consider the problem of radiation due to a point source
x
0
in three dimensions, so the function
f
in the Helmholtz equation is
f
(
x
)
=
δ
(
x
−
x
0
)
,
where
δ
is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form
u
=
c
u
+
+
(
1
−
c
)
u
−
where
c
is a constant, and
u
±
(
x
)
=
e
±
i
k
|
x
−
x
0
|
4
π
|
x
−
x
0
|
.
Of all these solutions, only
u
+
satisfies the Sommerfeld radiation condition and corresponds to a field radiating from
x
0
.
The other solutions are unphysical. For example,
u
−
can be interpreted as energy coming from infinity and sinking at
x
0
.
