  We develop a perturbative approach to study the excitation of disturbances in
the extended atomic hydrogen (HI) discs of galaxies produced by passing dark
matter subhaloes. The shallow gravitational potential of the dark matter
subhaloes (compared to the primary halo) allows us to use the epicyclic
approximation, the equations of which we solve by modal analysis, i.e.,
assuming a disc is composed of N radial rings with M modes. We show that
properties of dark matter subhaloes can be inferred from the profile and
amplitude of the modal energy of the disc. Namely, we find that the overall
amplitude of the response gives the mass of the dark sub-halo. Motivated by
this modal analysis, we then show that the density response shows similar
features. Finally, we show that our results agree with those from full
hydrodynamic simulation. We find a simple scaling relation between the
satellite mass and Fourier amplitudes of the resultant surface density of the
gas disc where the effective Fourier amplitude (essentially a sum over the low
order modes) scales as $m_{s}^{1/2}$, where $m_{s}$ is the satellite mass. The
utility of this relation is that it can be readily applied to an observed HI
map to deduce the satellite mass without recourse to full numerical simulation.
This will greatly aid us in analyzing large samples of spiral galaxies to
constrain the population of dwarf satellites in the Local Volume.
