The density wave theory of spiral structure in galaxies is studied as a dynamical problem: the gravitational instability of a galactic disk with respect to spiral modes. The asymptotic theory for tightly wound spirals is presented in detail, although the problem has been formulated in a more general form to open the way to future research work in other directions. Some preliminary results based on the general formulation are included; they confirm those based on the asymptotic theory in cases where the latter should be applicable. The fluid dynamical model is adopted; reference to the stellar dynamical model is made when necessary. Besides providing a coherent account of the present status of the theory, it is hoped that this article will further stimulate workers in continuum mechanics to play their important role in future research work in the astrophysics of galaxies.
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