  Black hole (BH) oscillations known as quasi-normal modes (QNMs) are one of
the most important gravitational wave (GW) sources. We propose that higher
perturbative order of QNMs, generated by nonlinear gravitational interaction
near the BHs, are detectable and worth searching for in observations and
simulations of binary BH mergers. We calculate the metric perturbations to
second-order and explicitly regularize the master equation at the horizon and
spatial infinity. We find that the second-order QNMs have frequencies twice the
first-order ones and the GW amplitude is up to ~10% that of the first-order
one. The QNM frequency would also shift blueward up to ~1%. This provides a new
test of general relativity as well as a possible distance indicator.
