  The frequency dependence of the longitudinal group speeds of trapped sausage
waves plays an important role in determining impulsively generated wave trains,
which have often been invoked to account for quasi-periodic signals in coronal
loops. We examine how the group speeds ($v_{\rm gr}$) depend on angular
frequency ($\omega$) for sausage modes in pressureless coronal tubes with
continuous transverse density distributions by solving the dispersion relation
pertinent to the case where the density inhomogeneity of arbitrary form takes
place in a transition layer of arbitrary thickness. We find that in addition to
the transverse lengthscale $l$ and density contrast $\rho_{\rm i}/\rho_{\rm
e}$, the group speed behavior depends also on the detailed form of the density
inhomogeneity. For parabolic profiles, $v_{\rm gr}$ always decreases with
$\omega$ first before increasing again, as happens for the much studied top-hat
profiles. For linear profiles, however, the behavior of the $\omega-v_{\rm gr}$
curves is more complex. When $\rho_{\rm i}/\rho_{\rm e} \lesssim 6$, the curves
become monotonical for large values of $l$. On the other hand, for higher
density contrasts, a local maximum $v_{\rm gr}^{\rm max}$ exists in addition to
a local minimum $v_{\rm gr}^{\rm min}$ when coronal tubes are diffuse. With
time-dependent computations, we show that the different behavior of group speed
curves, the characteristic speeds $v_{\rm gr}^{\rm min}$ and $v_{\rm gr}^{\rm
max}$ in particular, is reflected in the temporal evolution and Morlet spectra
of impulsively generated wave trains. We conclude that the observed
quasi-periodic wave trains not only can be employed to probe such key
parameters as density contrasts and profile steepness, but also have the
potential to discriminate between the unknown forms of the transverse density
distribution.
