  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 upon a frequency-shift formula of the
pulse, where planar GWs are usually assumed. Phase corrections due to the
wavefront curvature have been recently discussed. In this paper,
frequency-shift and timing-residual formulae are derived for GWs with fully
spherical wavefronts from a compact source such as a binary of supermassive
black holes, where the differences in the GW amplitude and direction between
the 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 as well as nearby GW source candidates are also
mentioned.
