  Transverse wrinkles are known to appear in thin rectangular elastic sheets
when stretched in the long direction. Numerically computed bifurcation diagrams
for extremely thin, highly stretched films indicate entire orbits of wrinkling
solutions, cf. Healey, et. al. [J. Nonlinear Sci., 23 (2013), pp.~777--805].
These correspond to arbitrary phase shifts of the wrinkled pattern in the
transverse direction. While such behavior is normally associated with problems
in the presence of a continuous symmetry group, an unloaded rectangular sheet
possesses only a finite symmetry group. In order to understand this phenomenon,
we consider a simpler problem more amenable to analysis -- a finite-length beam
on a nonlinear softening foundation under axial compression. We first obtain
asymptotic results via amplitude equations, that are valid as a certain
non-dimensional beam length becomes sufficiently large. We deduce that any two
phase-shifts of a solution differ from one another by exponentially small terms
in that length. We validate this observation with numerical computations,
indicating the presence of solution orbits for sufficiently long beams. We
refer to this as "hidden asymptotic symmetry".
