This displacement models a triggering earthquake. The reservoirs represent volatile dissolved in the magma. We assume that the vapor pressure of the volatile in the reservoirs is given by Henry's Law, that is, it varies lineary with the concentration of dissolved volatile. The valve allows flow of the volatile from the reservoir into the chamber when the vapor pressure of the reservoir is higher than the pressure in the chamber, and flow from the chamber into the reservoir when it is lower. When the system is perturbed from its equilibrium by the sudden displacement of a piston, it begins to oscillate.
Seismic attenuation occurs.
I have performed many numerical integrations of this system, experimenting with various choices for the numerous parameters. In all cases considered a large pressure increase was correlated with strong attenuation. Derive the partial differential equations that are the continuum mechanical version of the system, solve them numerically for an incident plane compressional wave, and derive an expression for the rate of attenuation of that wave and the effective Q of the medium. Modify the continuum model so that it has more realistic physics. Develop a sets of parameters that together comprise a good model of a magma chamber, and use them to make quantitative predictions about the properties of the attenuation that might be observed in realistic circumstances.