Electromagnetic scattering and radiation of waves by a thin toroidal tube with semi-transparent walls

The work deals with a scalar model of wave scattering and radiation by a thin toroidal tube with the varying longitudinal “conductivity”. The walls of the toroidal tube are assumed to be semi-transparent for the waves. The corresponding boundary conditions represent a direct scalar analog of the electromagnetic boundary conditions describing electromagnetic properties of carbon nanotubes (CNTs) recently discussed in the literature. The method of matching of asymptotic expansions is applied to study the model problem provided that the length of the tube is much greater than its thickness. The leading term of the formal asymptotics is developed and analyzed. The scattering amplitude is then considered and its resonant properties are discussed.