  In modified gravity theories, such as the Brans-Dicke theory, the background
evolution of the Universe and the perturbation around it are different from
that in general relativity. Therefore, the gravitational waveforms used to
study standard sirens in these theories should be modified. The modifications
of the waveforms can be classified into two categories: wave generation effects
and wave propagation effects. Hitherto, the waveforms used to study standard
sirens in the modified gravity theories incorporate only the wave propagation
effects and ignore the wave generation effects; while the waveforms focusing on
the wave generation effects, such as the post-Newtonian waveforms, do not
incorporate the wave propagation effects and cannot be directly applied to the
sources with non-negligible redshifts in the study of standard sirens. In this
work, we construct the consistent waveforms for standard sirens in the
Brans-Dicke theory. The wave generation effects include the emission of the
scalar breathing polarization $h_b$ and the corrections to the tensor
polarizations $h_+$ and $h_\times$; the wave propagation effect is the
modification of the luminosity distance for the gravitational waveforms. Using
the consistent waveforms, we analyze the parameter estimation biases due to the
ignorance of the wave generation effects. Considering the observations by the
Einstein Telescope, we find that the ratio of the theoretical bias to the
statistical error of the redshifted chirp mass is two orders of magnitude
larger than that of the source distance. For black hole-neutron star binary
systems like GW191219, the theoretical bias of the redshifted chirp mass can be
several times larger than the statistical error.
