  We study the effect of cubic and tidal interactions on the spectrum of
gravitational waves emitted in the inspiral phase of the merger of two
non-spinning objects. There are two independent parity-even cubic interaction
terms, which we take to be $I_1 = {R^{\alpha \beta}}_{\mu \nu} {R^{\mu
\nu}}_{\rho \sigma} {R^{\rho \sigma}}_{\alpha \beta}$ and $G_3 = I_1-2
R^{\alpha}\,_{\mu}\,^{\beta}\,_{\nu} R^{\mu}\,_{\rho}\,^{\nu}\,_{\sigma}
R^{\rho}\,_{\alpha}\,^{\sigma}\,_{\beta}$. The latter has vanishing pure
graviton amplitudes but modifies mixed scalar/graviton amplitudes which are
crucial for our study. Working in an effective field theory set-up, we compute
the modifications to the quadrupole moment due to $I_1$, $G_3$ and tidal
interactions, from which we obtain the power of gravitational waves radiated in
the process to first order in the perturbations and leading order in the
post-Minkowskian expansion. The $I_1$ predictions are novel, and we find that
our results for $G_3$ are related to the known quadrupole corrections arising
from tidal perturbations, although the physical origin of the $G_3$ coupling is
unrelated to the finite-size effects underlying tidal interactions. We show
this by recomputing such tidal corrections and by presenting an explicit field
redefinition. In the post-Newtonian expansion our results are complete at
leading order, which for the gravitational-wave flux is 5PN for $G_3$ and tidal
interactions, and 6PN for $I_1$. Finally, we compute the corresponding
modifications to the waveforms.
