Uniqueness and existence of an outgoing solution of Helmholtz problem using Green's formula

In this article, first we present a new approach based on Green’s formula, to describe the uniqueness and existence of a solution of the Helmholtz equation. By imposing at infinity the outgoing wave condition or also called Sommerfeld radiation condition, we show how it is possible to define in a natural way an outgoing solution of the Helmholtz equation based on physical arguments. Then, we resolve the exterior problem, given by the scattering of time-harmonic acoustic wave by sound-soft obstacle, which leads to find a radiating solution ∈ 2 ℝ 3 \ ∩ ℝ 3 \ to the Helmholtz equation