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.