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The frequency dependence of the longitudinal group speeds of trapped sausagewaves plays an important role in determining impulsively generated wave trains,which have often been invoked to account for quasi-periodic signals in coronalloops. We examine how the group speeds ($v_{\rm gr}$) depend on angularfrequency ($\omega$) for sausage modes in pressureless coronal tubes withcontinuous transverse density distributions by solving the dispersion relationpertinent to the case where the density inhomogeneity of arbitrary form takesplace in a transition layer of arbitrary thickness. We find that in addition tothe transverse lengthscale $l$ and density contrast $\rho_{\rm i}/\rho_{\rme}$, the group speed behavior depends also on the detailed form of the densityinhomogeneity. For parabolic profiles, $v_{\rm gr}$ always decreases with$\omega$ first before increasing again, as happens for the much studied top-hatprofiles. 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 curvesbecome monotonical for large values of $l$. On the other hand, for higherdensity contrasts, a local maximum $v_{\rm gr}^{\rm max}$ exists in addition toa local minimum $v_{\rm gr}^{\rm min}$ when coronal tubes are diffuse. Withtime-dependent computations, we show that the different behavior of group speedcurves, the characteristic speeds $v_{\rm gr}^{\rm min}$ and $v_{\rm gr}^{\rmmax}$ in particular, is reflected in the temporal evolution and Morlet spectraof impulsively generated wave trains. We conclude that the observedquasi-periodic wave trains not only can be employed to probe such keyparameters as density contrasts and profile steepness, but also have thepotential to discriminate between the unknown forms of the transverse densitydistribution.

Impulsively generated sausage waves in coronal tubes with transversally continuous structuring