Resonances are intrinsic characteristics of an elastic object, which are completely independent of the source of excitation. The appropriate utilization of the information contained within the resonance spectra and the identification of the resonance frequencies of the object can be used as a potent tool for material characterization. In this paper, a new mathematical model for the wave diffraction from a cylindrical nanofiber encased in an elastic matrix is introduced. The new model is used to evaluate the scattered pressure field resulting from normal insonification on a single nanofibrous composite. It is shown that there are specific resonances, which arise from the surface/interface energy between the nanofiber and solid matrix. They can be used to determine the characteristics and properties of fibrous nanocomposites.