This work presents models for the behaviour of both upstream- and downstream-travelling waves in screeching elliptical jets. Proper orthogonal decomposition is performed on experimental velocity data in both the major and minor axis planes, for an aspect ratio $AR=2$ converging elliptical jet operating at nozzle pressure ratios of $2.6$ and $3.4$ . From this decomposition, the radial and axial structure of the guided-jet mode (GJM) and the Kelvin–Helmholtz instability are educed. Linear-stability analysis (LSA) is performed using both the experimentally obtained mean flow, and one obtained using Reynolds-averaged Navier–Stokes (RANS) at matched conditions. It is shown that the wavenumber predicted by LSA for both waves are within the range of experimentally observed wavenumbers. Furthermore, the model accurately predicts the structure of these waves at multiple axial locations, using either the experimental or RANS mean flow. Most critically, it is demonstrated that the GJM is only predicted to be neutrally stable at the screech frequency for a relatively limited streamwise domain, the size and location of which is dependent on the nozzle pressure ratio. A comparison with the amplitude envelope for the GJM extracted from the experimental measurements indicates that the maximum fluctuations associated with the GJM are collocated with this region of the flow that is predicted to support the GJM. While there have been extensive discussions about the frequency dependence of the GJM, this is the first demonstration that its existence is highly dependent on streamwise position within the flow.