A spherical source model for low frequency volcanic earthquakes

A resonant magma chamber is modeled by a spherically symmetric whole space consisting of a small gas‐filled chamber surrounded by magma which is encased in country rock. Source boundary conditions are specified on the cavity boundary, simulating a possible sudden gas expansion in the magma. The simplicity of this model leads to an analytical closed form solution for the Fourier transform of radial displacement which predicts a peaked spectrum and an oscillating, decaying seismogram. For physically reasonable values of the rock and magma elastic properties and dimensions of the model, the dominant frequency of the signal is in the range of 1 to 5 Hz, near the values observed for low‐frequency volcanic earthquakes and volcanic tremor. Furthermore, the seismograms and spectra bear resemblance to those observed for low‐frequency earthquakes. The fundamental mode of oscillation is a pure radial “breathing” mode which has markedly different properties from single‐cavity solutions. The fundamental frequency is insensitive to the magma chamber boundary radius when both the inner and outer radii are greater than specified values. This may help to explain the apparent similarity of the dominant frequency of volcanic earthquakes over a wide range of physical situations. The presence of an inner effective “bubble” of a certain minimum size is necessary for the existence of the dominant low fundamental frequency spectral peak. Although highly idealized, we believe that this fundamental fluid‐solid cavity mode may provide an explanation for the observations of tremor and low‐frequency volcanic earthquakes.