  We consider the propagation of quasiparticle excitations in a dipolar
Bose-Einstein condensate, and derive a nonlocal field theory of quasiparticle
scattering at a stepwise inhomogeneity of the sound speed, obtained by tuning
the contact coupling part of the interaction on one side of the barrier. To
solve this problem $ab$ $initio$, i.e., without prior assumptions on the form
of the solutions, we reformulate the dipolar Bogoliubov-de Gennes equation as a
singular integral equation. The latter is of a $novel$ $hypersingular$ type, in
having a kernel which is hypersingular at only two isolated points. Deriving
its solution, we show that the integral equation reveals a continuum of
evanescent channels at the sound barrier which is absent for a purely
contact-interaction condensate. We furthermore demonstrate that by performing a
discrete approximation for the kernel, one achieves an excellent solution
accuracy for already a moderate number of discretization steps. Finally, we
show that the non-monotonic nature of the system dispersion, corresponding to
the emergence of a roton minimum in the excitation spectrum, results in
peculiar features of the transmission and reflection at the sound barrier which
are nonexistent for contact interactions.
