The propagation of spiral density waves in a differentially rotating, self-gravitating, magnetoactive and highly flattened disk is investigated by using the asymptotic theory for tightly wound spirals developed by Lin and his collaborators. We adopt the continuum fluid model as the primary basis, and our treatment will be largely analytical. The disk plasma is studied in the frozen field approximation and inhomogeneous magnetic fields in the plane of the disk are considered in detail. In a differentially rotating disk with strong magnetic fields, the field lines will be considerably distorted and the mutual influence of magnetic fields and differential rotation is by no means obvious. In this paper we present a new asymptotic dispersion relation for tightly wound spiral density waves with magnetic fields along the spiral arms B(theta)(r). The effects of the magnetic fields lead to such terms like k2(a2 + V(A)2), where k is the wave number, a represents the speed of sound, V(A) = (B2/4pirho)1/2 is the Alfven speed, B denotes the field strength, and rho is the plasma density. These terms depict the well-known magnetoacoustic waves and could have been anticipated without a detailed computation. However the interaction of magnetic fields and differential rotation may give rise to other significant terms which are not so easy to foresee. We also present a more exact local dispersion relation by using the WKB approximation and study the effects of magnetic fields on the growth rate through the parameters Q and J defined in the literature. Although the effects of the magnetic fields are rather insignificant for applications to Galactic dynamics, the effects of the magnetic fields are important for applications to star formation and problems related to the solar nebula.
