An interpretation of absolutely and convectively unstable waves using series solutions

Integral solutions are found for linear wave equations, which, depending on the parameters, exhibit either absolute or convective instability. Series solutions are constructed to examine the instability behavior on a bounded domain. Solutions with non-real wave-numbers can be interpreted as a superposition of eigenmodes with jump-periodic boundary conditions. We show that an initial disturbance can be represented by not only periodic modes, but also spatially growing (or decaying) modes, and arbitrary modes using a Galerkin approach. For the examples presented here, growth in any such series solution matches the growth predictions obtained from the long-time asymptotic behavior of the integral solutions.