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.
