Three-minute oscillations are observed in the chromosphere above sunspot umbrae. One of the models used to explain these oscillations is that of a chromospheric acoustic resonator, where the cavity between the photosphere and transition region partially reflects slow magnetoacoustic waves to form resonances in the lower sunspot atmosphere. We present a phenomenological study that compares simulation results with observations. The ideal magnetohydrodynamic equations are used with a uniform vertical magnetic field and a temperature profile that models sunspot atmospheres above umbrae. The simulations are initialized with a single broadband pulse in the vertical velocity inside the convection zone underneath the photosphere. The frequencies in the spectrum of the broadband pulse that lie below the acoustic cutoff frequency are filtered out so that frequencies equal and above the acoustic cutoff frequency resonate inside the chromospheric cavity. The chromospheric cavity resonates with approximately three-minute oscillations and is a leaky resonator, so that these oscillations generate traveling waves that propagate upward into the corona. Thus, there is no requirement that a narrowband three-minute signal is present in the photosphere to explain the narrowband three-minute oscillations in the chromosphere and corona. The oscillations in the chromospheric cavity have larger relative amplitudes (normalized to the local sound speed) than those in the corona and reproduce the intensity fluctuations of observations. Different umbral temperature profiles lead to different peaks in the spectrum of the resonating chromospheric cavity, which can explain the frequency shift in sunspot oscillations over the solar cycle.

The thin flux tube approximation is considered for a straight, symmetrical, force-free, rigidly rotating flux tube. The derived set of equations describes tube, body sausage, and Alfvén wave modes and is valid for any values of ␤. The linear waves and instabilities of force-free flux tubes are considered. The comparison of approximate and exact solutions for an untwisted, nonrotating flux tube is performed. It is shown that the approximate and exact dispersion equations coincides, except the 20% discrepancy of sausage frequencies. An effective cross section is proposed to introduce the removal of this discrepancy. It makes the derived approximation correct for the force-free thin flux tube dynamics, except the detailed structure of radial eigenfunction. The dispersion of Alfvén torsional waves in a force-free tubes appears. The valve effect of one directional propagation of waves in rotating twisted tube is revealed. The current and rotational sausage instabilities of a force-free, thin flux tube are considered.

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Two competing theories of sunspot oscillations are discussed. It is pointed out that the normal mode (eigenoscillations) theory is in contradiction with a number of observations. The reasons for this are discussed. The revised filter theory of the three-minute sunspot oscillations is outlined. It is shown that the reason for the occurrence of the multipassband filter for the slow waves is the interference that appears from the multilayer structure of the sunspot atmosphere. In contrast with Zhugzhda and Locans (Sov. Astron. Lett. 7, 25 -27, 1981) it is shown that along with the Fabry -Perot chromospheric passband the cutoff frequency passband and a number of the high-frequency passbands occur. The effect of the nonlinearity of the sunspot oscillations in the upper chromosphere and the transition region is taken into account. The spectra of the distinct empirical models of the sunspot atmosphere are explored. An example of the interpretation of the sunspot oscillations based on the revised filter theory is presented. Only the filter theory can explain the complicated behavior of the oscillations across the sunspot. The observations provide evidence of the nonuniformity of the sunspot atmosphere.A long time has elapsed since the discovery of the three-minute oscillations. Numerous studies of the oscillations have been performed. The story of the discovery and exploration of sunspot oscillations can be found in the review of Bogdan and Judge (2006) devoted to the observational aspects of the sunspot oscillations. We know a lot about the three-minute Helioseismology, Asteroseismology, and MHD Connections