  We apply the framework of tempered fractional calculus to investigate the
spatial dispersion of elastic waves in a one-dimensional elastic bar
characterized by range-dependent nonlocal interactions. The measure of the
interaction is given by the attenuation kernel present in the constitutive
stress-strain relation of the bar, which follows from the Kr\"oner-Eringen's
model of nonlocal elasticity. We employ a fractional power-law attenuation
kernel and spatially temper it, to make the model physically valid and
mathematically consistent. The spatial dispersion relation is derived, but it
turns out to be difficult to solve, both analytically and numerically.
Consequently, we use numerical techniques to extract the real and imaginary
parts of the complex wavenumber for a wide range of frequency values. From the
dispersion plots, it is found that the phase velocity dispersion of elastic
waves in the tempered nonlocal elastic bar is similar to that from the
time-fractional Zener model. Further, we also examine the unusual attenuation
pattern obtained for the elastic wave propagation in the bar.
