  Within Bogoliubov-de Gennes theory, a semiclassical approximation is used to
study quantum oscillations and to determine the Fermi surface area associated
with these oscillations in a model of a $\pi$-striped superconductor, where the
d-wave superconducting order parameter oscillates spatially with period 8 and
zero average value. This system has a non-zero density of particle-hole states
at the Fermi energy, which form Landau-like levels in the presence of a
magnetic field, B. The Fermi surface is reconstructed via Andreev-Bragg
scattering, and the semiclassical motion is along these Fermi surface sections
as well as between them via magnetic breakdown. Within the approximation,
oscillations periodic in 1/B are found in both the positions and widths of the
lowest Landau levels. The area corresponding to these quantum oscillations for
intermediate pairing interaction strength is similar to that reported for
experimental measurements in the cuprates. A comparison is made of this theory
to data for quantum oscillations in the specific heat measured by Riggs et al.
