  We perform a comprehensive study of gravitational waves in the context of the
higher-order quadratic scalar curvature gravity, which encompasses the ordinary
Einstein-Hilbert term in the action plus an $R^{2}$ contribution and a term of
the type $R\square R$. The main focus is on gravitational waves emitted by
binary systems such as binary black holes and binary pulsars in the
approximation of circular orbits and nonrelativistic motion. The waveform of
higher-order gravitational waves from binary black holes is constructed and
compared with the waveform predicted by standard general relativity; we
conclude that the merger occurs earlier in our model than what would be
expected from GR. The decreasing rate of the orbital period in binary pulsars
is used to constrain the coupling parameters of our higher-order $R^{2}$
gravity; this is done with Hulse-Taylor binary pulsar data leading to
$\kappa_{0}^{-1}\lesssim1.1\times10^{16}\,\text{m}^{2}$, where
$\kappa_{0}^{-1}$ is the coupling constant for the $R^{2}$ contribution.
