  The behavior of the axial group speeds of trapped sausage modes plays an
important role in determining impulsively generated wave trains, which have
often been invoked to account for quasi-periodic signals with quasi-periods of
order seconds in a considerable number of coronal structures. We conduct a
comprehensive eigenmode analysis, both analytically and numerically, on the
dispersive properties of sausage modes in pressureless tubes with three
families of continuous radial density profiles. We find a rich variety of the
dependence on the axial wavenumber $k$ of the axial group speed $v_{\rm gr}$.
Depending on the density contrast and profile steepness as well as on the
detailed profile description, the $v_{\rm gr}-k$ curves either possess or do
not possess cutoff wavenumbers, and they can behave in either a monotonical or
non-monotonical manner. With time-dependent simulations, we further show that
this rich variety of the group speed characteristics heavily influences the
temporal evolution and Morlet spectra of impulsively generated wave trains. In
particular, the Morlet spectra can look substantially different from "crazy
tadpoles" found for the much-studied discontinuous density profiles. We
conclude that it is necessary to re-examine available high-cadence data to look
for the rich set of temporal and spectral features, which can be employed to
discriminate between the unknown forms of the density distributions transverse
to coronal structures.
