  Fast sausage modes in solar magnetic coronal loops are only fully contained
in unrealistically short dense loops. Otherwise they are leaky, losing energy
to their surrounds as outgoing waves. This causes any oscillation to decay
exponentially in time. Simultaneous observations of both period and decay rate
therefore reveal the eigenfrequency of the observed mode, and potentially
insight into the tubes' nonuniform internal structure. In this article, a
global spectral description of the oscillations is presented that results in an
implicit matrix eigenvalue equation where the eigenvalues are associated
predominantly with the diagonal terms of the matrix. The off-diagonal terms
vanish identically if the tube is uniform. A linearized perturbation approach,
applied with respect to a uniform reference model, is developed that makes the
eigenvalues explicit. The implicit eigenvalue problem is easily solved
numerically though, and it is shown that knowledge of the real and imaginary
parts of the eigenfrequency is sufficient to determine the width and density
contrast of a boundary layer over which the tubes' enhanced internal densities
drop to ambient values. Linearized density kernels are developed that show
sensitivity only to the extreme outside of the loops for radial fundamental
modes, especially for small density enhancements, with no sensitivity to the
core. Higher radial harmonics do show some internal sensitivity, but these will
be more difficult to observe. Only kink modes are sensitive to the tube
centres. {Variation in internal and external Alfv\'en speed along the loop is
shown to have little effect on the fundamental dimensionless eigenfrequency,
though the associated eigenfunction becomes more compact at the loop apex as
stratification increases, or may even displace from the apex.

Solar and Stellar Astrophysics