A sharpened formulation of Sommerfeld's radiation condition for Green's functions of the Helmholtz equation

Abstract

It is shown that in order to ensure that a solution of a boundary-value problem of the Helmholtz equation, obtained with help of a Green's function, satisfies the ordinary radiation condition, the corresponding Green's function must fulfil a sharpened form of this condition. This is in general always the case because, according to a theorem proved below, every solution of the Helmholtz equation satisfying the ordinary radiation condition fulfils at the same time the sharpened form.