  In this paper, we study the gravitational waveforms, polarizations and
radiation powers of the first relativistic triple systems PSR J0337 + 1715,
observed in 2014, by using the post-Newtonian approximations to their lowest
order. Although they cannot be observed either by current or next generation of
the detectors, they do provide useful information to test different theories of
gravity. In particular, we carry out the studies in three different theories,
general relativity (GR), Brans-Dicke (BD) gravity, and Einstein-aether ($\ae$)
theory. The tensor modes $h_{+}$ and $h_{\times}$ exist in all three theories
and have almost equal amplitudes. Their frequencies are all peaked at two
locations, $ f^{(+, \times)}_1 \simeq 0.0686656 \mu$Hz and $ f^{(+, \times)}_2
\simeq 14.2138 \mu$Hz, which are about twice of the outer and inner orbital
frequencies of the triple system, as predicted in GR. In $\ae$-theory, all the
six polarization modes are different from zero, but the breathing $h_b$ and
longitudinal $h_L$ modes are not independent and also peaked at two
frequencies, but at the frequencies, $ f^{(b, L, \ae)}_{1} \simeq 0.0457771
\mu$Hz and $f^{(b, L, \ae)}_{ 2} \simeq 7.09545 \mu$Hz. A similar phenomenon is
also observed in BD gravity, in which only the three modes $h_{+},\;
h_{\times}$ and $h_{b}$ exit. We also study the radiation powers, and find that
the quadrupole emission in each of the three theories has almost the same
amplitude, but the dipole emission can be as big as the quadrupole emission in
$\ae$-theory. This can provide a very promising window to obtain severe
constraints on $\ae$-theory by the multi-band gravitational wave astronomy.
