In the present work, the elastodynamic field of scattering of an anti-plane high frequency elastic shear wave due to an embedded nano cylindrical cavity in an infinite elastic medium is obtained by considering the effects of couple-stresses. In the theories accounting the effects of couple-stresses in their formulations, a new characteristic length of material is introduced into the formulations, and so, these are capable to capture size effect at micro and nano scales. Also, in contrary to classical continuum theory which has difficulties in describing dispersion of wave at high frequencies, observed dispersive wave in experiments can be explained in the framework of these theories. In this work, the analytical expressions of elastodynamic fields around the cavity are obtained by considering equation of motion, dispersion relation and appropriate boundary conditions in the framework of two theories considering couple-stresses. Also, the dynamic stress concentration factor around the cavity within these theories is obtained, and, as a limiting case, the results of two cases of dynamic stress concentration factor in classical theory as well as static stress concentration factor in couple stress theories are recovered. In the framework of these theories, by several examples, the effects of frequency of incident wave and the ratio of couple stress characteristic length to the size of the cross section of the cavity on the displacement field, stress field and dynamic stress concentration factor around the cavity are studied, and the results are compared with the corresponding classical solutions.