  We explore the gravitational waves (GWs) within the framework of the $f(R)$
gravity model represented by $f(R)=R^{1+\delta}/R^\delta_c$ in the weak field
approximation. In this scenario, gravitational waves exhibit an additional
polarization mode beyond the standard transverse-traceless (TT) tensor modes.
We show that the polarization characteristics of these waves are connected to
the scalaron mass and the effective potential derived from the function $f(R)$.
Furthermore, the study of the speed of gravitational waves ($c_g$) within the
Horndeski theory, particularly using the $f(R)$ model, reveals an intriguing
feature about the equality of the speed of gravitational waves to that that of
electromagnetic waves. This equivalence arises due to the modification
introduced in the Ricci scalar within the $f(R)$ model.
