  We study propagation of waves and appearance of caustics in k-essence and
galileon theories. First we show that previously known solutions for travelling
waves in k-essence and galileon models correspond to very specific fine-tuned
initial conditions. On the contrary, as we demonstrate by the method of
characteristics, generic initial conditions leads to a wave in k-essence which
ends up with formation of caustics. Finally, we find that any wave solution in
pure k-essence is also a solution for a galileon theory with the same k-essence
term. Thus in the Horndeski theory with a k-essence term formation of caustics
is generic. We discuss physical consequences of the caustics formation and
possible ways to cure the problem.
