  We investigate the operator-norm resolvent asymptotics of a high-contrast
composite, consisting of a "stiff" material, with annular "soft" inclusions (a
"stiff-soft-stiff" setup). This setup is derived from two models with very
different effective wave propagation behaviors. Our analysis is based on an
operator-framework proposed by Cherednichenko, Ershova, and Kiselev in
[Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency
Conversion, and Time Dispersive Properties. I. Commun. Math. Phys. 375, p.
1833-1884]. Then, as a first step towards studying wave propagation on the
stiff-soft-stiff composite, we use the effective description to derive
analogous "dispersion functions".
