  Quasinormal modes describe the return to equilibrium of a perturbed system,
in particular the ringdown phase of a black hole merger. But as
globally-defined quantities, the quasinormal spectrum can be highly sensitive
to global structure, including distant small perturbations to the potential. In
what sense are quasinormal modes a property of the resulting black hole? We
explore this question for the linearized perturbation equation with two
potentials having disjoint bounded support. We give a composition law for the
Wronskian that determines the quasinormal frequencies of the combined system.
We show that over short time scales the evolution is governed by the
quasinormal frequencies of the individual potentials, while the sensitivity to
global structure can be understood in terms of echoes. We introduce an echo
expansion of the Green's function and show that, as expected on general
grounds, at any finite time causality limits the number of echoes that can
contribute. We illustrate our results with the soluble example of a pair of
$\delta$-function potentials. We explicate the causal structure of the Green's
function, demonstrating under what conditions two very different quasinormal
spectra give rise to very similar ringdown waveforms.
