  Gravitational waves travel through the distributions of matter and dark
energy during propagation. For this reason, gravitational waves emitted from
binary compact objects serve as a useful tool especially to probe the nature of
dark energy. The geometrical optics approximation is a conventional way of
investigating wave propagation. However, the approximation becomes less
accurate as the wavelength approaches the curvature radius of the background,
which can occur in generic situations. In this paper, we suggest a formulation
for higher-order corrections of the geometrical optics expansion, applied to
Horndeski theory which accommodates many dark energy models. At the level of
the background, assuming that the derivative of the scalar field is
non-vanishing and timelike, we choose the time slices to coincide with the
contours of the scalar field. This choice of the background time slices is
advantageous as the sound cones of both scalar and tensor gravitational waves
are upright with respect to the background time slices whenever the scalar
field behaves as a perfect fluid. We then analyze the equations of motion for
scalar and tensor components of gravitational waves at the leading and
next-to-leading order in the geometrical optics expansion, deriving the
evolution equations for their amplitudes under certain conditions. In
particular, for Generalized Brans-Dicke theories, we find a simple description
of equations for gravitational waves in terms of an effective metric.
