  We study the spontaneous generation and evolution of bending waves in
$N$-body simulations of two isolated Milky Way-like galaxy models. The models
differ by their disc-to-halo mass ratios, and hence by their susceptibility to
the formation of a bar and spiral structure. Seeded from shot noise in the
particle distribution, bending waves rapidly form in both models and persist
for many billions of years. Waves at intermediate radii manifest as corrugated
structures in vertical position and velocity that are tightly wound,
morphologically leading, and dominated by the $m=1$ azimuthal Fourier
component. A spectral analysis of the waves suggests they are a superposition
of modes from two continuous branches in the Galactocentric radius-rotational
frequency plane. The lower-frequency branch is dominant and is responsible for
the corrugated, leading, and warped structure. Over time, power in this branch
migrates outward, lending credence to an inside-out formation scenario for the
warp. Our power spectra qualitatively agree with results from linear
perturbation theory and a WKB analysis, both of which include self-gravity.
Thus, we conclude that the waves in our simulations are self-gravitating and
not purely kinematic. These waves are reminiscent of the wave-like pattern
recently found in Galactic star counts from the Sloan Digital Sky Survey and
smoothly transition to a warp near the disc's edge. Velocity measurements from
\textit{Gaia} data will be instrumental in testing the true wave nature of the
corrugations. We also compile a list of "minimum requirements" needed to
observe bending waves in external galaxies.
