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 $f(R)=R+a_2{R}^2/2+\dots$. 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 $|a_2|\lesssim 1.2\times {10}^{18}{\mathrm{m}}^2$. 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 ${10}^6{M}_{\odot }$ black hole if $|a_2|\gtrsim {10}^{17}{\mathrm{m}}^2$, assuming that the weak-field metric of the black hole coincides with that of a point mass. However Eöt-Wash experiments provide the strictest bound $|a_2|\lesssim 2\times {10}^{-9}{\mathrm{m}}^2$. 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.

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