  The last decade has seen considerable interest in the quasi-normal
frequencies [QNFs] of black holes (and even wormholes), both asymptotically
flat and with cosmological horizons. There is wide agreement that the QNFs are
often of the form omega_n = (offset) + i n (gap), though some authors have
encountered situations where this behaviour seems to fail. To get a better
understanding of the general situation we consider a semi-analytic model based
on a piecewise Eckart (Poeschl-Teller) potential, allowing for different
heights and different rates of exponential falloff in the two asymptotic
directions. This model is sufficiently general to capture and display key
features of the black hole QNFs while simultaneously being analytically
tractable, at least for asymptotically large imaginary parts of the QNFs. We
shall derive an appropriate "quantization condition" for the asymptotic QNFs,
and extract as much analytic information as possible. In particular, we shall
explicitly verify that the (offset)+ i n (gap) behaviour is common but not
universal, with this behaviour failing unless the ratio of rates of exponential
falloff on the two sides of the potential is a rational number. (This is
"common but not universal" in the sense that the rational numbers are dense in
the reals.) We argue that this behaviour is likely to persist for black holes
with cosmological horizons.
