  Propagation of elastic waves along the axis of cylindrical shells is of great
current interest due to their ubiquitous presence and technological importance.
Geometric imperfections and spatial variations of properties are inevitable in
such structures. Here we report the existence of branched flows of flexural
waves in such waveguides. The location of high amplitude motion, away from the
launch location, scales as a power law with respect to the variance and
linearly with respect to the correlation length of the spatial variation in the
bending stiffness. These scaling laws are then theoretically derived from the
ray equations. Numerical integration of the ray equations also exhibit this
behaviour-consistent with finite element numerical simulations as well as the
theoretically derived scaling. There appears to be a universality for the
exponents in the scaling with respect to similar observations in the past for
other types of waves, as well as flexural and hence dispersive waves in elastic
plates.
