  It has been widely recognized that gamma-ray burst (GRB) afterglows arise
from interactions between GRB outflow and circumburst medium, while their
evolution follows the behaviors of relativistic shock waves. Assuming the
distribution of circumburst medium follows a general power-law form, that is,
$n = A_{\ast} R^{-k}$, where $R$ denotes the distance from the burst, it is
obvious that the value of density-distribution index $k$ can affect the
behaviors of the afterglow. In this paper, we analyze the temporal and spectral
behaviors of GRB radio afterglows with arbitrary $k$-values. In the radio band,
a standard GRB afterglow produced by forward shock exhibits a late-time flux
peak, and the relative peak fluxes as well as peak times at different
frequencies show dependencies on $k$. Thus with multi-band radio peak
observations, one can determine the density profile of circumburst medium by
comparing the relations between peak flux/time and frequency at each observing
band. Also, the effects of trans-relativistic shock waves, as well as jets in
afterglows are discussed. By analyzing 31 long and 1 short GRBs with multi-band
data of radio afterglows, we find that nearly half of them can be explained
with uniform interstellar medium ($k=0$), $\sim 1/5$ can be constrained to
exhibiting stellar wind environment ($k=2$), while less than $\sim 1/3$ samples
show $0< k< 2$.

High Energy Astrophysical Phenomena