Sommerfeld Radiation Condition

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 \mbox{ in } \mathbb R^n

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

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