Mathematicians have known for many years that two dimensional wave propagation is very different from one (plane) or three (spherical) dimensional propagation. Plane and spherical pulses can be propagated without change of form--an impossibility for cylindrical waves.The object of the present paper is to clarify the physical picture involved. It is shown that for any initially given axially symmetric disturbance, confined between two concentric circular cylinders, if energy is subsequently transmitted outwards then it is also always subsequently transmitted inwards. The only possible case of no inward transmission is that of no transmission. Thus for a two dimensional pulse propagation every part of the pulse is being continually split at each instant, part traveling inwards and part outwards. Only the very simplest problem is solved in this paper. However the process of tail (coda) formation is now clear for this case.