Absolute and convective instabilities of semi-bounded spatially developing flows

We analyse the absolute and convective instabilities of, and spatially amplifying waves in, semi-bounded spatially developing flows and media by applying the Laplace transform in time to the corresponding initial-value linear stability problem and treating the resulting boundary-value problem on $\mathbb R^+$ for a vector equation as a dynamical system. The analysis is an extension of our recently developed linear stability theory for spatially developing open flows and media with algebraically decaying tails and for fronts to flows in a semi-infinite domain. We derive the global normal-mode dispersion relations for different domains of frequency and treat absolute instability, convectively unstable wave packets and signalling. It is shown that when the limit state at infinity, i.e. the associated uniform state, is stable, the inhomogeneous flow is either stable or absolutely unstable. The inhomogeneous flow is absolutely stable but convectively unstable if and only if the flow is globally stable and the associated uniform state is convectively unstable. In such a case signalling in the inhomogeneous flow is identical with signalling in the associated uniform state.