A perturbative analysis of Quasi-Radial density waves in galactic disks
  The theoretical understanding of density waves in disk galaxies starts from
the classical WKB perturbative analysis of tight-winding perturbations, the key
assumption being that the potential due to the density wave is approximately
radial. The above has served as a valuable guide in aiding the understanding of
both simulated and observed galaxies, in spite of a number of caveats being
present. The observed spiral or bar patterns in real galaxies are frequently
only marginally consistent with the tight-winding assumption, often in fact,
outright inconsistent. Here we derive a complementary formulation to the
problem, by treating quasi-radial density waves under simplified assumptions in
the linear regime. We assume that the potential due to the density wave is
approximately tangential, and derive the corresponding dispersion relation of
the problem. We obtain an instability criterion for the onset of quasi-radial
density waves, which allows a clear understanding of the increased stability of
the higher order modes, which appear at progressively larger radii, as often
seen in real galaxies. The theory naturally yields a range of pattern speeds
for these arms which appears constrained by the condition
$\Omega_{p}<\Omega_{0} \pm \kappa /m$. For the central regions of galaxies
where solid body rotation curves might apply, we find weak bars in the
oscillatory regime with various pattern speeds, including counter rotating
ones, and a prediction for $\Omega_{p}$ to increase towards the centre, as seen
in the rapidly rotating bars within bars of some numerical simulations. We
complement this study with detailed numerical simulations of galactic disks and
careful Fourier analysis of the emergent perturbations, which support the
theory presented.
