  Highly-damped quasi-normal frequencies are very often of the form omega_n =
(offset) + i n (gap). We investigate the genericity of this phenomenon by
considering a model potential that is piecewise Eckart (piecewise
Poeschl-Teller), and developing an analytic "quantization condition" for the
highly-damped quasi-normal frequencies. We find that this omega_n = (offset) +
i n (gap) behaviour is generic but not universal, with the controlling feature
being whether or not the ratio of the rates of exponential falloff in the two
asymptotic directions is a rational number. These observations are of direct
relevance to any physical situation where highly-damped quasi-normal modes
(damped modes) are important --- in particular (but not limited to) to black
hole physics, both theoretical and observational.
