||A generalized equation for the resonance frequencies of a fluid-filled crack
Maeda, YutaKumagai, Hiroyuki
Geophysical Journal International
201 , 2017-04 , Oxford University Press
Although a model of the resonance of a rectangular fluid-filled crack (crack model) is one of the most frequently used source models of long-period seismic events at volcanoes, there has been no analytical solution for the resonance frequencies. We previously proposed an empirical expression for the resonance frequencies as a mathematical function of the crack length, aperture, and properties of the fluid and the surrounding elastic medium. However, the expression contained an empirical constant that had to be investigated numerically for each crack aspect ratio and oscillation mode, a requirement that prevented widespread use of the expression. In the present study, we examined the theoretical basis for the expression. We assumed that the ratio of the crack wall displacement to the fluid pressure near each crack edge varied as the square root of the distance from the edge. Using this assumption, we showed theoretically that the previously proposed empirical analytical expression was a good approximation (difference ≤2 per cent) to another more complete expression. This theoretical expression is a closed form of a mathematical function of the crack model parameters and oscillation mode number; there are no empirical constants to be determined numerically. The expression thus enabled us to analytically compute the resonance frequencies for arbitrary rectangular cracks, and the results were in good agreement (difference ≤5 per cent) with numerical solutions. Resonance frequencies of cracks can be very easily predicted using this expression. This predictive ability may enhance our quantitative understanding of the processes that generate long-period events at volcanoes.