  Cosmic strings may have formed in the early universe due to the Kibble
mechanism. While string networks are usually modeled as being of Nambu-Goto
type, this description is understood to be a convenient approximation, which
neglects the typically expected presence of additional degrees of freedom on
the string worldsheet. Previous simulations of cosmic strings in expanding
universes have established beyond doubt the existence of a significant amount
of short-wavelength propagation modes (commonly called wiggles) on the strings,
and a wiggly string extension of the canonical velocity-dependent one-scale
model has been recently developed. Here we improve the physical interpretation
of this model, by studying the possible asymptotic scaling solutions of this
model, and in particular how they are affected by the expansion of the universe
and the available energy loss or transfer mechanisms -- e.g., the production of
loops and wiggles. In addition to the Nambu-Goto solution, to which the wiggly
model reduces in the appropriate limit, we find that there are also solutions
where the amount of wiggliness can grow as the network evolves or, for specific
expansion rates, become a constant. Our results show that full scaling of the
network, including the wiggliness, is much more likely in the matter era than
in the radiation era, which is in agreement with numerical simulation results.
