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An additional effect not considered in our earlier papers is found to enhance the rate of growth of spiral modes in disk-shaped galaxies. This enhanced growth appears to be attributable to the combined influence of differential rotation and tangential forces. For certain galaxy models, rapid temporal growths are possible (comparable to those found in numerical experiments), but in many real galaxies, these tendencies to extreme growth rates are probably reduced to lower levels by competing effects such as wave absorption at inner Lindblad resonance.

Discrete growing spiral modes are calculated, on the basis of principles of stellar dynamics, according to the recently developed asymptotic theory of density waves for spiral structure in galaxies. From the growth rates obtained, it appears likely that several spiral modes coexist in a given disk galaxy. General properties of these modes are described for two galaxy models, and some of their important observational implications are discussed.

The mechanisms for the maintenance and the excitation of trailing spiral modes of density waves in diskshaped galaxies, as proposed by Lin in 1969 and by Mark recently, are substantiated by an analysis of the gas-dynamical model of the galaxy. The self-excitation of the unstable mode in caused by waves propagating outwards from the corotation circle, which carry away angular momentum of a sign opposite to that contained in the wave system inside that circle. Specifically, a simple dispersion relationship is given as a definite integral, which allows the immediate determination of the pattern frequency and the amplification rate, once the basic galactic model is known.

A numerical scheme has been devised to calculate discrete spiral modes in a fluid-dynamical model of disk galaxies. The procedure is free from the usual restrictions imposed upon the asymptotic theory. A radiation condition has been formulated as one of the boundary conditions. For a test model galaxy with moderately low surface density, quantitative agreement is found between the asymptotic spiral modes and the corresponding ones obtained from the present scheme. The present scheme should find ready application to the study of open spiral modes.