  In dark-energy models where a scalar field is nonminimally coupled to the
spacetime geometry, gravitational waves are expected to be supplemented with a
scalar mode. Such scalar waves may interact with the standard tensor waves,
thereby affecting their observed amplitude and polarization. Understanding the
role of scalar waves is thus essential in order to design reliable
gravitational-wave probes of dark energy and gravity beyond general relativity.
In this article, we thoroughly investigate the propagation of scalar and tensor
waves in the subset of Horndeski theories in which tensor waves propagate at
the speed of light. We work at linear order in scalar and metric perturbations,
in the eikonal regime, and for arbitrary scalar and spacetime backgrounds. We
diagonalize the system of equations of motion and identify the physical tensor
mode, which differs from the metric perturbation. We find that interactions
between scalar and tensor waves generally depend on the scalar propagation
speed. If the scalar waves are luminal or quasiluminal, then interactions are
negligible. In the subluminal case, scalar-tensor interactions are effectively
suppressed due to the incoherence of the wave's phases.
