  The generalized Brans-Dicke (abbreviated as GBD) theory is obtained by
replacing the Ricci scalar $R$ in the original Brans-Dicke (BD) action with an
arbitrary function $f(R)$. Comparing with other theories, some interesting
properties have been found or some problems existing in other theories could be
solved in the GBD theory. For example, (1) the state parameter of geometrical
dark energy in the GBD model can cross over the phantom boundary $w=-1$,
without bearing the problems existing in the quintom model (e.g. the problem of
negative kinetic term and the fine-tuning problem, etc); (2) $f(R)$ theory is
equivalent to the BD theory with a potential (BDV) with the couple parameter
$\omega=0$, where the kinetic term of field in the equivalent BDV theory is
absent. While the scalar fields in the GBD own the non-disappeared kinetic
term, when one compares the GBD theory with the $f(R)$ theory. In this paper,
we continue to investigate the GBD theory. Using the method of the weak-field
approximation, we explore the linearized physics in the GBD theory. The
linearized equations of the gravitational field and two scalar fields are
given. We investigate their solutions in the linearized theory for a point
mass. It is shown that the problem of the $\gamma$ (parametrized post-Newtonian
parameter) value in the $f(R)$ theory could be solved in the GBD theory, where
the theoretical $\gamma$ value in the GBD theory can be consistent with the
observational results. At last, we study the gravitational waves physics in the
vacuum for the GBD theory. It is found that the gravitational radiation in the
GBD theory has new freedoms beyond the two standard modes in the general
relativity theory.
