  Working in pressureless magnetohydrodynamics, we examine the consequences of
some peculiar dispersive properties of linear fast sausage modes (FSMs) in
one-dimensional cylindrical equilibria with a continuous radial density profile
($\rho_0(r)$). As recognized recently on solid mathematical grounds, cutoff
axial wavenumbers may be absent for FSMs when $\rho_0(r)$ varies sufficiently
slowly outside the nominal cylinder. Trapped modes may therefore exist for
arbitrary axial wavenumbers and density contrasts, their axial phase speeds in
the long-wavelength regime differing little from the external Alfv$\acute{\rm
e}$n speed. If these trapped modes indeed show up in the solutions to the
associated initial value problem (IVP), then FSMs have a much better chance to
be observed than expected with classical theory, and can be invoked to account
for a considerably broader range of periodicities than practiced. However, with
axial fundamentals in active region loops as an example, we show that this
long-wavelength expectation is not seen in our finite-difference solutions to
the IVP, the reason for which is then explored by superposing the necessary
eigenmodes to re-solve the IVP. At least for the parameters we examine, the
eigenfunctions of trapped modes are characterized by a spatial extent well
exceeding the observationally reasonable range of the spatial extent of initial
perturbations, meaning a negligible fraction of energy that a trapped mode can
receive. We conclude that the absence of cutoff wavenumbers for FSMs in the
examined equilibrium does not guarantee a distinct temporal behavior.
