Slow waves trapped in a fluid‐filled infinite crack: Implication for volcanic tremor

The dynamics and seismic radiation of fluid‐filled cracks have been studied by numerous authors, as models for tremor and for long‐period events observed at volcanoes. One of the most intriguing results of the recent models is the existence of a very slow wave propagating along the crack boundary. In order to better understand this slow wave, which has so far only been studied numerically, we studied analytically normal modes trapped in a liquid layer sandwiched between two solid half‐space. A slow wave, similar to the tube wave found by Biot, exists for all wavelengths. In the short wavelength limit, this wave approaches the Stoneley wave for the liquid‐solid interface. Unlike the tube wave, however, as the wavelength increases to infinity, both the phase and group velocities approach zero, in inverse proportion to the square root of wavelength. The phase velocity and amplitude of this slow wave are in good agreement with those obtained by the numerical studies on the dynamics of fluid‐filled cracks by two dimensional and three‐dimensional finite difference methods. In the past the size of a magma body has been estimated from volcanic tremor periods and the acoustic velocity in the fluid. These estimates should be drastically reduced if the slow wave dominates the tremor. For example, the extremely long‐period volcanic tremor, with periods up to 7s, observed at Mount Aso may be generated by a fluid‐filled crack of modest size, a magma body 0.5 m thick and 0.5 km long.