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Fast sausage modes in solar magnetic coronal loops are only fully containedin unrealistically short dense loops. Otherwise they are leaky, losing energyto their surrounds as outgoing waves. This causes any oscillation to decayexponentially in time. Simultaneous observations of both period and decay ratetherefore reveal the eigenfrequency of the observed mode, and potentiallyinsight into the tubes' nonuniform internal structure. In this article, aglobal spectral description of the oscillations is presented that results in animplicit matrix eigenvalue equation where the eigenvalues are associatedpredominantly with the diagonal terms of the matrix. The off-diagonal termsvanish identically if the tube is uniform. A linearized perturbation approach,applied with respect to a uniform reference model, is developed that makes theeigenvalues explicit. The implicit eigenvalue problem is easily solvednumerically though, and it is shown that knowledge of the real and imaginaryparts of the eigenfrequency is sufficient to determine the width and densitycontrast of a boundary layer over which the tubes' enhanced internal densitiesdrop to ambient values. Linearized density kernels are developed that showsensitivity only to the extreme outside of the loops for radial fundamentalmodes, especially for small density enhancements, with no sensitivity to thecore. Higher radial harmonics do show some internal sensitivity, but these willbe more difficult to observe. Only kink modes are sensitive to the tubecentres. {Variation in internal and external Alfv\'en speed along the loop isshown to have little effect on the fundamental dimensionless eigenfrequency,though the associated eigenfunction becomes more compact at the loop apex asstratification increases, or may even displace from the apex.

Sensitivity of Coronal Loop Sausage Mode Frequencies and Decay Rates to Radial and Longitudinal Density Inhomogeneities: A Spectral Approach