  Scalar, tensor waves induce oscillatory perturbations in Keplerian systems
that can be probed with measurements of pulsar timing residuals. In this paper,
we consider the imprint of coherent oscillations produced by ultralight axion
dark matter on the Roemer time delay. We use the angle-action formalism to
calculate the time evolution of the observed signal and its dependence on the
orbital parameters and the axion phase. We derive exact analytical expressions
for arbitrary binary pulsar mass ratio and eccentricity, alleviating the need
for long numerical integrations. We emphasize the similarity of the expected
signal-to-noise ratio with the response of a harmonic oscillator to an external
oscillatory driving. We validate our theoretical predictions with numerical
simulations. Our results furnish a useful benchmark for numerical codes and
analysis procedures and, hopefully, will motivate the search for such imprints
in real data.
