  In this paper we analyze the implications of gravitational waves (GWs) as
standard sirens on the modified gravity models by using the third-generation
gravitational wave detector, i.e., the Einstein Telescope. Two viable models in
$f(R)$ theories within the Palatini formalism are considered in our analysis
($f_{1}(\mathcal{R})=\mathcal{R}-\frac{\beta}{\mathcal{R}^{n}}$ and
$f_{2}(\mathcal{R})=\mathcal{R}+\alpha\ln{\mathcal{R}}-\beta$), with the
combination of simulated GW data and the latest electromagnetic (EM)
observational data (including the recently released Pantheon type Ia supernovae
sample, the cosmic chronometer data, and baryon acoustic oscillation distance
measurements). Our analysis reveals that the standard sirens GWs, which provide
an independent and complementary alternative to current experiments, could
effectively eliminate the degeneracies among parameters in the two modified
gravity models. In addition, we thoroughly investigate the nature of
geometrical dark energy in the modified gravity theories with the assistance of
$Om(z)$ and statefinder diagnostic analysis. The present analysis makes it
clear-cut that the simplest cosmological constant model is still the most
preferred by the current data. However, the combination of future naturally
improved GW data most recent EM observations will reveal the consistency or
acknowledge the tension between the $\Lambda$CDM model and modified gravity
theories.
