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The lack of observed sausage perturbations in solar active region loops is customarily attributed to the relevance of cutoff axial wavenumbers and the consequent absence of trapped modes (called ``evanescent eigenmodes'' here). However, some recent eigenvalue problem studies yield that cutoff wavenumbers may disappear for those equilibria where the external density varies sufficiently slowly, thereby casting doubt on the rarity of candidate sausage perturbations. We examine the responses of straight, transversely structured, coronal slabs to small-amplitude sausage-type perturbations that excite axial fundamentals by solving the pertinent initial value problem with eigensolutions for a closed domain. The density variation in the slab exterior is dictated by some steepness parameter $μ$, and cutoff wavenumbers are theoretically expected to be present (absent) when $μ\ge 2$ ($μ< 2$). However, our numerical results show no qualitative difference in the system evolution when $μ$ varies, despite the differences in the modal behavior. Only oscillatory eigenmodes are permitted when $μ\ge 2$. Our discrete eigenspectrum becomes increasingly closely spaced when the domain broadens, and an oscillatory continuum results for a truly open system. Oscillatory eigenmodes remain allowed and dominate the system evolution when $μ<2$. We show that the irrelevance of cutoff wavenumbers does not mean that all fast waves are evanescent. Rather, it means that an increasing number of evanescent eigenmodes emerge when the domain size increases. We conclude that sausage perturbations remain difficult to detect even for the waveguide formulated here.