Slow pressure modes in thin accretion discs
  Thin accretion discs around massive compact objects can support slow pressure
modes of oscillations in the linear regime that have azimuthal wavenumber
$m=1$. We consider finite, flat discs composed of barotropic fluid for various
surface density profiles and demonstrate--through WKB analysis and numerical
solution of the eigenvalue problem--that these modes are stable and have
spatial scales comparable to the size of the disc. We show that the eigenvalue
equation can be mapped to a Schr\"odinger-like equation. Analysis of this
equation shows that all eigenmodes have discrete spectra. We find that all the
models we have considered support negative frequency eigenmodes; however, the
positive eigenfrequency modes are only present in power law discs, albeit for
physically uninteresting values of the power law index $\beta$ and barotropic
index $\gamma$.
