We consider the gasless solid fuel combustion model of the Self-Propagating High-Temperature Synthesis process in which combustion waves are employed to synthesize desired materials. Specifically, we consider the combustion of a solid sample in which combustion occurs on the surface of a cylinder of radius R. We consider solution behavior as R is increased. If R is sufficiently small, planar pulsating waves are observed. As R is increased transitions to more complex, nonplanar waves occur. The study of different wave types is important since the mode of propagation determines the structure of the product. We describe a variety of different waves, including (i) spin waves, (ii) counterpropagating (CP) waves of various types, (iii) alternating spin CP waves (ASCP), (iv) modulated spin waves, (v) bound states of asymmetric spin waves, (vi) modulated asymmetric spin waves, (vii) asymmetric ASCP waves, and others.