  We investigate the linearized form of metric f(R)-gravity, assuming that f(R)
is analytic about R = 0 so it may be expanded as f(R) = R + a_2 R^2/2 + ... .
Gravitational radiation is modified, admitting an extra mode of oscillation,
that of the Ricci scalar. We derive an effective energy-momentum tensor for the
radiation. We also present weak-field metrics for simple sources. These are
distinct from the equivalent Kerr (or Schwarzschild) forms. We apply the
metrics to tests that could constrain f(R). We show that light deflection
experiments cannot distinguish f(R)-gravity from general relativity as both
have an effective post-Newtonian parameter \gamma = 1. We find that planetary
precession rates are enhanced relative to general relativity; from the orbit of
Mercury we derive the bound |a_2| < 1.2 \times 10^18 m^2. Gravitational wave
astronomy may be more useful: considering the phase of a gravitational waveform
we estimate deviations from general relativity could be measurable for an
extreme-mass-ratio inspiral about a 10^6 M_sol black hole if |a_2| > 10^17 m^2,
assuming that the weak-field metric of the black hole coincides with that of a
point mass. However Eot-Wash experiments provide the strictest bound |a_2| < 2
\times 10^-9 m^2. Although the astronomical bounds are weaker, they are still
of interest in the case that the effective form of f(R) is modified in
different regions, perhaps through the chameleon mechanism. Assuming the
laboratory bound is universal, we conclude that the propagating Ricci scalar
mode cannot be excited by astrophysical sources.
