The Helmholtz equation in a locally perturbed half-space with non-absorbing boundary (Q1000560)

The authors construct the Green's function for the Helmholtz equation in the half-space \(x_3>0\) with the boundary condition \(-\frac{\partial u}{\partial x_3} +z u=f\), where \(u(x_1,x_2,x_3)\) is an unknown function, \(z\) is a given complex number, \(f\) is a known function. The main attention is paid to asymptotic investigation of the Green's function at infinity in order to establish the radiation condition which is somewhat different from the classic Sommerfeld's one. Similar problems are discussed for the exterior of the half-sphere in the half-space and for perturbed half-spaces. Here, the authors prove existence and uniqueness of the solution of the problems with the established radiation condition.