  We use $N$-body simulations to investigate the excitation of bending waves in
a Milky Way-like disc-bulge-halo system. The dark matter halo consists of a
smooth component and a population of subhaloes while the disc is composed of
thin and thick components. Also considered is a control simulation where all of
the halo mass is smoothly distributed. We find that bending waves are more
vigorously excited in the thin disc than the thick one and that they are
strongest in the outer regions of the disc, especially at late times. By way of
a Fourier decomposition, we find that the complicated pattern of bending across
the disc can be described as a superposition of waves, which concentrate along
two branches in the radius-rotational frequency plane. These branches
correspond to vertical resonance curves as predicted by a WKB analysis. Bending
waves in the simulation with substructure have a higher amplitude than those in
the smooth-halo simulation, though the frequency-radius characteristics of the
waves in the two simulations are very similar. A cross correlation analysis of
vertical displacement and bulk vertical velocity suggests that the waves
oscillate largely as simple plane waves. We suggest that the wave-like features
in astrometric surveys such as the Second Data Release from \textit{Gaia} may
be due to long-lived waves of a dynamically active disc rather than, or in
addition to, perturbations from a recent satellite-disc encounter.
