The analytical mathematical model is presented that describes one of the possible mechanisms for the occurrence of long-period seismic events that are often recorded near active volcanic centers. The feeding system of the volcano is modeled in the simplest form of a cylindrical channel filled with a compressible magmatic melt with the rheology of a Maxwell body. It is shown that such a magmatic body can experience harmonic damped oscillations, the damping coefficient of which is determined by the relaxation time of the magmatic melt. These fluctuations may appear as a response to a density perturbation caused by the influx of denser magma from deep layers or a change in pressure in the supply system of the volcano. The dependence of the natural oscillatory frequency on the physical characteristics of the magmatic melt and the geometric dimensions of the feed channel is shown. When the compressibility of the magmatic melt is taken into account, density perturbations depend on the size of the feeding system and are characterized by periodic oscillations, which are most pronounced near the channel axis. Oscillations are also experienced by the flow velocity component directed along the radius of the cylinder. The source mechanism of the long-period seismic events is discussed. The model is used to describe long-period oscillations recorded near Santiaguito (Guatemala).

A simple analytical model representing one of the possible mechanisms for the occurrence of low-frequency oscillations in a feeding system of volcano has been developed. The model is presented for a cylindrical chamber filled with magma with Maxwell rheology. It is shown that damped harmonic oscillations in the magma flow velocity can occur in the volcanic chamber. These damped harmonic oscillations can appear as a reaction to remote seismic events or seismic events in the volcanic feeding system. The dependence of the oscillation frequency on the physical characteristics of the magmatic melt and the geometric dimensions of the volcano chamber is shown. The occurrence of magmatic oscillations can be observed near the surface as a volcanic tremor. The model is applied to the measurements result of low-frequency oscillations for the magma chamber of the Elbrus volcanic center.

Within the simplified one-dimensional aerodynamic model, periodic temperature oscillations are described for a column of compressible atmosphere unlimited from above. When obtaining the model, two conditions for the equilibrium of the air column are used: the classical condition of hydrostatics and the second - the absence of movement due to changes in the density of the atmosphere over time. The obtained analytical and numerical solutions satisfy two boundary conditions near a solid surface: the value of a quantity (pressure, density, or temperature) and its gradient. Solutions for adiabatic temperature, pressure, and density show a non-stationary pulsating solution, in which the column temperature periodically increases indefinitely at the moment of pulsation and then decreases. The physical interpretation of these solutions is discussed and compared with the data of paleoclimatic studies. The results of the study improve the understanding of ongoing periodic processes and pulsations in the atmosphere and can be used for scientific and educational purposes.