  We present an alternative method for constructing the exact and approximate
solutions of electromagnetic wave equations whose source terms are arbitrary
order multipoles on a curved spacetime. The developed method is based on the
higher-order Green's functions for wave equations which are defined as
distributions that satisfy wave equations with the corresponding order
covariant derivatives of the Dirac delta function as the source terms. The
constructed solution is applied to the study of various geometric effects on
the generation and propagation of electromagnetic wave tails to first order in
the Riemann tensor. Generally the received radiation tail occurs after a time
delay which represents geometrical backscattering by the central gravitational
source. It is shown that the truly nonlocal wave-propagation correction (the
tail term) takes a universal form which is independent of multipole order. In a
particular case, if the radiation pulse is generated by the source during a
finite time interval, the tail term after the primary pulse is entirely
determined by the energy-momentum vector of the gravitational field source: the
form of the tail term is independent of the multipole structure of the
gravitational source. We apply the results to a compact binary system and
conclude that under certain conditions the tail energy can be a noticeable
fraction of the primary pulse energy. We argue that the wave tails should be
carefully considered in energy calculations of such systems.
