  In addition to the usual linear gravitational waves in transverse-traceless
coordinates, higher-order gravitational field equations, as well as their
corresponding solutions, are explicitly obtained. It is found that higher-order
waves do not represent corrections to the first-order wave. In contrast, all
higher than second-order solutions do represent corrections to the second-order
wave, a property that makes the first-order gravitational wave to stand apart
from higher-order waves. Furthermore, although the first-order solution is
transverse and traceless, all higher-order solutions are not. As a consequence,
the whole solution is neither transverse nor traceless, a result that could
eventually have important consequences for quantum gravity, and in particular
for the definition of graviton itself. Some additional properties and features
of these higher-order gravitational waves are explored and discussed.
