On the Interaction of Spiral Density Waves with Stars near the Inner Lindblad Resonance in Galactic Disks

The interaction of a spiral wave with stars near the inner Lindblad resonance in a galactic disk has been investigated. The dispersion relation describing the behavior of the complex wave number of the spiral wave as a function of the distance to the resonance has been derived within the framework of a purely linear problem and in the leading orders of the epicyclic and WKB approximations. We also have improved the result of Mark (1971) concerning behavior of the amplitude of leading spiral wave near the resonance circle. We have studied the consequences following from the hypothesis that weak nonlinearity in a narrow resonance region changes the standard rule of bypassing the pole in the complex plane, known as the Landau-Lin bypass rule, to taking the corresponding principal value integral. By analogy with hydrodynamics, where such a problem arises when analyzing the resonant interaction of waves with shear flows, we expect that a small, but finite amplitude can lead to a modification of the bypass rule and, as a consequence, to the elimination of the effect of spiral wave absorption at the resonance and its reflection. We have shown that under some assumptions the presumed picture actually takes place, but the detailed situation looks quite unexpected: near the resonance the regions where stars cause wave attenuation alternate with the regions where the wave is amplified. At the same time, there is no wave absorption effect when integrated over the resonance region.