  We study the gravitational wave (GW) emission of sources perturbed by
periodic dynamical forces which do not cause secular evolution in the orbital
elements. We construct a corresponding post-Newtonian waveform model and
provide estimates for the detectability of the resulting GW phase
perturbations, for both space-based and future ground-based detectors. We
validate our results by performing a set of Bayesian parameter recovery
experiments with post-Newtonian waveforms. We find that, in stark contrast to
the more commonly studied secular dephasing, periodic phase perturbations do
not suffer from degeneracies with any of the tested vacuum binary parameters.
We discuss the applications of our findings to a range of possible
astrophysical scenarios, finding that such periodic perturbations may be
detectable for massive black hole binaries embedded in circum-binary discs,
extreme mass-ratio inspirals in accretion discs, as well as stellar-mass
compact objects perturbed by tidal fields. We argue that modelling conservative
sub-orbital dynamics opens up a promising new avenue to detect environmental
effects in binary sources of GWs that should be included in state-of-the-art
waveform templates.
