Nonspectral singularities of Green's function for the Helmholtz equation in the exterior of an arbitrary convex polygon

For the case of the exterior of an arbitrary convex polygon, an asymptotic expression is obtained at the physical level of rigor for the nonspectral singularities closest to the axis $\operatorname{Im}k=0$ of Green's function for the Helmholtz equation $(\Delta+k^2)q=0$ (with Neumann boundary conditions). The validity of this asymptotic expression is verified in the limiting case of a segment by analyzing the exact solution obtained by separation of variables. A geometrical interpretation of the asymptotic equations for the eigenfunctions of the Laplace operator in terms of geometrical optics is proposed.