  We model the resonance mechanism underpinning generation of A1 and A2 screech
tones in an under-expanded supersonic jet. Starting from the resonance model
recently proposed by \cite{mancinelli2019screech}, where the
upstream-travelling wave is a neutrally-stable guided jet mode, we here present
a more complete linear-stability-based model for screech prediction. We study
temperature and shear-layer thickness effects and show that, in order to
accurately describe the experimental data, the effect of the finite thickness
of the shear layer must be incorporated in the jet-dynamics model. We then
present an improved resonance model for screech-frequency predictions in which
both downstream- and upstream-travelling waves may have complex wavenumber and
frequency. This resonance model requires knowledge of the reflection
coefficients at the upstream and downstream locations of the resonance loop. We
explore the effect of the reflection coefficients on the resonance model and
propose an approach for their identification. The complex-mode model identifies
limited regions of frequency-flow parameter space for which the resonance loop
is amplified in time, a necessary condition for the resonance to be sustained.
This model provides an improved description of the experimental measurements.
