  Recent numerical studies of the coupled Einstein-Klein-Gordon system in a
cavity have provided compelling evidence that {\it confined} scalar fields
generically collapse to form black holes. Motivated by this intriguing
discovery, we here use analytical tools in order to study the characteristic
resonance spectra of the confined fields. These discrete resonant frequencies
are expected to dominate the late-time dynamics of the coupled
black-hole-field-cage system. We consider caged Reissner-Nordstr\"om black
holes whose confining mirrors are placed in the near-horizon region
$x_{\text{m}}\equiv (r_{\text{m}}-r_+)/r_+\ll\tau\equiv (r_+-r_-)/r_+$ (here
$r_{\text{m}}$ is the radius of the confining mirror and $r_{\pm}$ are the
radii of the black-hole horizons). We obtain a simple analytical expression for
the fundamental quasinormal resonances of the coupled black-hole-field-cage
system: $\omega_n=-i2\pi T_{\text{BH}}\cdot n[1+O(x^n_{\text{m}}/\tau^n)]$,
where $T_{\text{BH}}$ is the temperature of the caged black hole and
$n=1,2,3,...$ is the resonance parameter.
