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 : (\nabla^2 + k^2) u = -f \text{ in } \mathbb 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_ - ik \right) u(x) = 0 uniformly in all directions :\hat{x} = \frac{x} (above, i is the
imaginary unit and |\cdot| is the
Euclidean norm). Here, it is assumed that the time-harmonic field is e^{-i\omega t}u. If the time-harmonic field is instead e^{i\omega 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)=\delta(x-x_0), where \delta is the
Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form :u = cu_+ + (1-c) u_- \, where c is a constant, and : u_{\pm}(x) = \frac{e^{\pm ik|x-x_0|}}{4\pi |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. ==See also==