Eighty years of Sommerfeld's radiation condition

Abstract

In 1912 Sommerfeld introduced his radiation condition to ensure the uniqueness of the solution of certain exterior boundary value problems in mathematical physics. In physical applications these problems generally describe wave propagation where an incident time-harmonic wave is scattered by an object, and the resulting diffracted or scattered waves need to be calculated. When formulated mathematically, these problems usually take the form of an exterior Dirichlet or Neumann problem for the Helmholtz partial differential equation. The Sommerfeld condition is applied at infinity and, when added to the statement of the boundary value problem, singles out only the solution which represents “outgoing” (rather than “incoming” or “standing”) waves in the physical applications. Since its introduction, the Sommerfeld radiation condition has become indispensable for these types of problems and has stimulated a considerable amount of mathematical research, especially in uniqueness theorems. The present note traces the motivation and reasoning that led Sommerfeld to the original formulation of his radiation condition and surveys the extensions and modifications this condition has undergone since then.