2023ApJ...946...76K

Pulsar timing arrays (PTAs) are searching for nanohertz-frequency gravitational waves (GWs) through cross-correlation of pulse arrival times from a set of radio pulsars. PTAs have relied on a frequency-shift formula of the pulse, where planar GWs are usually assumed. Phase corrections due to the wave front curvature have been recently discussed. In this paper, frequency-shift and timing-residual formulae are derived for GWs with fully spherical wave fronts from a compact source such as a binary of supermassive black holes, where the differences in the GW amplitude and direction between Earth and the pulsar are examined in the quadrupole approximation. By using the new formulae, effects beyond the plane-wave approximation are discussed, and a galactic center PTA and nearby GW source candidates are also mentioned.