  This work studies wave propagation in the most general scalar-tensor
theories, particularly focusing on the causal structure realized in these
theories and also the shock formation process induced by nonlinear effects. For
these studies we use the Horndeski theory and its generalization to the two
scalar field case. We show that propagation speeds of gravitational wave and
scalar field wave in these theories may differ from the light speed depending
on background field configuration, and find that a Killing horizon becomes a
boundary of causal domain if the scalar fields share the symmetry of the
background spacetime. About the shock formation, we focus on transport of
discontinuity in second derivatives of the metric and scalar field in the
shift-symmetric Horndeski theory. We find that amplitude of the discontinuity
generically diverges within finite time, which corresponds to shock formation.
It turns out that the canonical scalar field and the scalar DBI model, among
other theories described by the Horndeski theory, are free from such shock
formation even when the background geometry and scalar field configuration are
nontrivial. We also observe that gravitational wave is protected against shock
formation when the background has some symmetries at least. This fact may
indicate that the gravitational wave in this theory is more well-behaved
compared to the scalar field, which typically suffers from shock formation.
