  Gravitationally bound hierarchies containing three or more components are
very common in our Universe. In this paper we study {\em periodic}
gravitational wave (GW) form, their polarizations, response function, its
Fourier transform, and energy loss rate of a triple system through three
different channels of radiation, the scalar, vector and tensor modes, in
Einstein-aether theory of gravity. In the weak-field approximations and with
the recently obtained constraints of the theory, we first analyze the energy
loss rate of a binary system, and find that the dipole contributions from the
scalar and vector modes could be of the order of
${\cal{O}}\left(c_{14}\right){\cal{O}}\left(G_Nm/d\right)^2$, where $c_{14} \;
(\equiv c_{1} + c_{4})$ is constrained to $c_{14} \lesssim
{\cal{O}}\left(10^{-5}\right)$ by current observations, where $c_i$'s are the
four coupling constants of the theory. On the other hand, the "strong-field"
effects for a binary system of neutron stars are about six orders lower than
that of GR. So, in this paper we ignore these "strong-field" effects and first
develop the general formulas to the lowest post-Newtonian order, by taking the
coupling of the aether field with matter into account. Within this
approximation, we find that the scalar breather mode and the scalar
longitudinal mode are all suppressed by a factor of
${\cal{O}}\left(c_{14}\right)$ with respect to the transverse-traceless modes
($h_{+}$ and $h_{\times}$), while the vectorial modes $(h_{X}$ and $h_{Y}$) are
suppressed by a factor of $c_{13} \lesssim {\cal{O}}\left(10^{-15}\right)$.
Applying the general formulas to a triple system with periodic orbits, we find
that the corresponding GW form, response function, and its Fourier transform
depend sensitively on both the configuration of the triple system and their
orientations with respect to the detectors.
