In this work, a nonlocal elasticity model of Klein-Gordon type, characterized by nonlocality in space and time, is developed for the investigation of wave propagation in isotropic elastic media [1, 2]. Nonlocal elasticity theory having a close link to the underlying microstructure has the advantage to capture effects at small scales [3]. Specifically, nonlocal elasticity is valid down to the Ångström-scale and it can be considered as a generalized continuum theory of Ångström-mechanics as it has been shown in [4]. For the first time in the framework of nonlocal elasticity theory, the proposed nonlocal elasticity model of Klein-Gordon type possessing one characteristic internal time scale parameter in addition to the characteristic internal length scale parameter describes spatial and temporal nonlocal effects at small scales. 

The dispersion relations of the considered isotropic nonlocal model of Klein-Gordon type are analytically determined predicting in addition to the acoustic modes (low-frequency modes), optic modes (high-frequency modes) as well as frequency band-gaps between the acoustic and optic modes. The ranges of the frequency band-gaps for longitudinal and transverse waves are determined. Moreover, the phase and group velocities are calculated for the acoustic and optic branches of longitudinal and transverse waves showing that all four modes exhibit normal dispersion with positive group velocity. 

The proposed nonlocal model of Klein-Gordon type possessing only 4 constitutive parameters (2 elastic constants, 1 length scale and 1 time scale) provides an appropriate framework for the modelling of accurate frequency band-gaps and overall physically realistic dispersive wave propagation.