Oscillating solutions of the matter density contrast in Horndeski's
  theory
  Evolution of the large scale structure of the Universe is described by galaxy
bias and the growth rate function of the matter density contrast. The behavior
of the growth rate function, in general, depends on the types of dark energy or
modified gravity theories, because they modify Poisson's equation for gravity
by changing the Newtonian constant in the quasi-static solutions. Whereas,
another mode of the matter density contrast "oscillating mode" has not been
studied well, although there is a possibility that it drastically changes the
gravitational law. In this paper, we will construct the way to investigate
oscillating solutions and show the concrete forms of the solutions by using the
WKB approximation.
