We aim to provide more insights into the applicability of the much-studied discrete leaky modes (DLMs) in classic analyses to solar coronal seismology. Under linear ideal pressureless magnetohydrodynamics (MHD), we examined 2D axial fundamental kink motions that arise when localized velocity exciters impact some symmetric slab equilibria. Continuous structuring is allowed. A 1D initial value problem (IVP) is formulated in conjunction with an eigenvalue problem (EVP) for laterally open systems, with no strict boundary conditions (BCs) at infinity. The IVP is solved by eigenfunction expansion, allowing a clear distinction between the contributions from proper eigenmodes and improper continuum eigenmodes. Example solutions are offered for parameters typical of active region loops. Our solutions show that the system evolves toward
long periodicities due to proper eigenmodes
(on the order of the axial time), whereas the interference of the improper continuum may lead to short periodicities initially (on the order of the lateral time). Specializing to the slab axis, we demonstrate that the proper contribution strengthens with the density contrast, but may occasionally be stronger for less steep density profiles. Short periodicities are not guaranteed in the improper contribution, the details of the initial exciter being key. When identifiable, these periodicities tend to agree with the oscillation frequencies expected for DLMs, despite the differences in the BCs between our EVP and classic analyses. The eigenfunction expansion approach enables all qualitative features to be interpreted as the interplay between the initial exciter and some response function, the latter being determined solely by the equilibria. Classic theories for DLMs can find seismological applications, with time-dependent studies
offering additional ways for constraining initial exciters.
