  We present a systematic procedure of finding the shock wave equation in
anisotropic spacetime of quadratic gravity with Lagrangian ${\cal L}=R+
\Lambda+\alpha R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}+\beta
R_{\mu\nu}R^{\mu\nu}+\gamma R^2+{\cal L}_{\rm matter}$. The general formula of
the butterfly velocity is derived. We show that the shock wave equation in the
planar, spherical or hyperbolic black hole spacetime of Einstein-Gauss-Bonnet
gravity is the same as that in Einstein gravity if space is isotropic. We
consider the modified AdS spacetime deformed by the leading correction of the
quadratic curvatures and find that the fourth order derivative shock wave
equation leads to two butterfly velocities if $4\alpha+\beta<0$. We also show
that the butterfly velocity in a D=4 planar black hole is not corrected by the
quadratic gravity if $ 4\alpha+\beta=0$, which includes the $ R^2$ gravity. In
general, the correction of butterfly velocity by the quadratic gravity may be
positive or negative, depends on the values of $\alpha$, $\beta$, $\gamma$ and
temperature. We also investigate the butterfly velocity in the Gauss-Bonnet
massive gravity.
