  The acoustic resonator is an important model for explaining the three-minute
oscillations in the chromosphere above sunspot umbrae. The steep temperature
gradients at the photosphere and transition region provide the cavity for the
acoustic resonator, which allows waves to be both partially transmitted and
partially reflected. In this paper, a new method of estimating the size and
temperature profile of the chromospheric cavity above a sunspot umbra is
developed. The magnetic field above umbrae is modelled numerically in 1.5D with
slow magnetoacoustic wave trains travelling along magnetic fieldlines.
Resonances are driven by applying the random noise of three different
colours---white, pink and brown---as small velocity perturbations to the upper
convection zone. Energy escapes the resonating cavity and generates wave trains
moving into the corona. Line of sight (LOS) integration is also performed to
determine the observable spectra through SDO/AIA. The numerical results show
that the gradient of the coronal spectra is directly correlated with the
chromosperic temperature configuration. As the chromospheric cavity size
increases, the spectral gradient becomes shallower. When LOS integrations is
performed, the resulting spectra demonstrate a broadband of excited frequencies
that is correlated with the chromospheric cavity size. The broadband of excited
frequencies becomes narrower as the chromospheric cavity size increases. These
two results provide a potentially useful diagnostic for the chromospheric
temperature profile by considering coronal velocity oscillations.
