  We theoretically investigate the impact of weak perturbations of a flat
potential on the density of a quasi-two-dimensional dipolar Bose gas. We use a
mean-field perturbative treatment of the potential defects and derive their
effects at first order in the mean-field stable regime. We first focus on
defects containing a single spatial frequency and study the wavevector
dependence of the density perturbation. A qualitative modification of the
wavenumber dependence with the interaction parameters and a sensitivity in the
excitation direction reveal the long-range and anisotropic dipolar effects.
These effects are found to be most important at intermediate wavenumbers and
can give rise to a local maximum in the density perturbation reminiscent of the
roton mode softening and local instabilities. The dependence on the gas and
interaction parameters is studied. The case of a flat potential perturbed with
white noise on a certain momentum range is then examined. Here it is found that
the strength perturbation becomes independent of the mean density when
sufficiently large. Our study touches upon experimentally relevant issues,
giving hints on how flat a uniform potential should be to achieve uniform
quasi-two-dimensional dipolar Bose gases.
