Green's function for the Helmholtz equation of a scalar wave in an anisotropic halfspace

This paper derives a closed-form expression for Green’s function of the three-dimensional Helmholtz scalar wave equation in an anisotropic halfspace. The boundary conditions are of either the homogeneous Neumann or Dirichlet type, and the method of solution is through Fourier transformations of the relevant boundary integral equations. The paper shows that, with the use of invariant tensor notation, the solution for the anisotropic problem has the same form as the isotropic solution. This simplicity is maintained through the derivation of a generalized reflection operator that tells, given the real source in the anisotropic halfspace, how to find the virtual source on the other side of the boundary due to the imposition of the boundary. The reflection operator is a function of the unit normal of the boundary to the halfspace (like the isotropic case) and is also a function of the symmetric tensor that characterizes the anisotropy. As a practical application, this paper also finds Green’s function for the ...