Radiation condition
Encyclopedia
Arnold Sommerfeld
defined the condition of radiation for a scalar field satisfying the Helmholtz equation
as
Mathematically, consider the inhomogeneous Helmholtz equation
where
is the dimension of the space, 
is a given function with compact support representing a bounded source of energy, and 
is a constant, called the wavenumber. A solution 
to this equation is called radiating if it satisfies the Sommerfeld radiation condition
uniformly in all directions
(above,
is the imaginary unit
and
is the Euclidean norm). Here, it is assumed that the time-harmonic field is 
If the time-harmonic field is instead 
one should replace 
with 
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
in three dimensions, so the function 
in the Helmholtz equation is 
where 
is the Dirac delta function
. This problem has an infinite number of solutions. All solutions have the form
where
is a constant, and
Of all these solutions, only
satisfies the Sommerfeld radiation condition and corresponds to a field radiating from 
The other solutions are unphysical. For example, 
can be interpreted as energy coming from infinity and sinking at 
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
defined the condition of radiation for a scalar field satisfying the Helmholtz equation
Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
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
Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
where
is the dimension of the space, is a given function with compact support representing a bounded source of energy, and is a constant, called the wavenumber. A solution to this equation is called radiating if it satisfies the Sommerfeld radiation condition
uniformly in all directions
(above,
is the imaginary unitImaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
and
is the Euclidean norm). Here, it is assumed that the time-harmonic field is If the time-harmonic field is instead one should replace with 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
in three dimensions, so the function in the Helmholtz equation is where is the Dirac delta functionDirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
. This problem has an infinite number of solutions. All solutions have the form
where
is a constant, and
Of all these solutions, only
satisfies the Sommerfeld radiation condition and corresponds to a field radiating from The other solutions are unphysical. For example, can be interpreted as energy coming from infinity and sinking at