Sommerfeld radiation condition
In mathematics, the Sommerfeld radiation condition is a concept from theory of differential equations and scattering theory used for choosing a particular solution to the Helmholtz equation. It was introduced by Arnold Sommerfeld in 1912 and is closely related to the limiting absorption principle (1905) and the limiting amplitude principle (1948). Formulation. Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation as Mathematically, consider the inhomogeneous Helmholtz equation where formula_2 is the dimension of the space, formula_3 is a given function with compact support representing a bounded source of energy, and  formula_4 is a constant, called the "wavenumber". A solution formula_5 to this equation is called "radiating" if it satisfies the Sommerfeld radiation condition uniformly in all directions (above, formula_8 is the imaginary unit and formula_9 is the Euclidean norm). Here, it is assumed that the time-harmonic field is formula_10 If the time-harmonic field is instead formula_11 one should replace formula_12 with formula_13 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 formula_14 in three dimensions, so the function  formula_3 in the Helmholtz equation is formula_16 where formula_17 is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form where formula_19 is a constant, and Of all these solutions, only formula_21 satisfies the Sommerfeld radiation condition and corresponds to a field radiating from formula_22 The other solutions are unphysical . For example, formula_23 can be interpreted as energy coming from infinity and sinking at formula_22