The accumulation of eigenvalues in a stability problem

When two waves propagating in a one-dimensional medium are locked together as a composite wave, a natural question arises as to whether the new wave is stable. An interesting and novel instability mechanism is exposed here in which a cascade of eigenvalues accumulates at a distinguished point in the unstable half plane. The underlying assumption is that the transition between the two waves occurs at an unstable, homogeneous steady state of the partial differential equations. This causes the individual waves to have an unstable continuous spectrum, but the instability of the full wave cannot be predicted from the configuration of these spectra alone.