A generalized equation for the longitudinal and transverse resonance frequencies of a fluid-filled crack

Although the resonance of a rectangular fluid-filled crack (Chouet, 1986, JGR) is one of the most frequently used source models of long-period seismic events at volcanoes, there has been no analytical solution for the resonant frequencies. Maeda and Kumagai (2013, GRL) proposed an empirical expression for the resonant frequencies: fm = (m - 1)a/2L[1 + 2ɛmC]1/2, (1) where m is the mode number, a is the sound velocity of fluid, L is the crack length, ɛm is a constant, C is the crack stiffness, and fm is the resonant frequency. The constant ɛm depends on the mode number m and crack aspect ratio W/L, where W is the crack width. However ɛm was given to only a few number of m and W/L in Maeda and Kumagai (2013). Therefore the expression was not widely applicable. In the present study, we investigated an analytical expression for ɛm. To achieve this, we examined the theoretical background for equation (1) assuming 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 equation (1) was a good approximation (difference ≤ 2%) to another more complete expression. This theoretical expression enabled us to compute the resonant frequencies for arbitrary W/L, and the results were in good agreement (difference ≤ 10%) with numerical solutions. We also showed empirically that ɛm is proportional to 1/m. By combining these results, we obtained an analytical expression for the resonant frequencies applicable to arbitrary W/L and m. Using this expression, resonant frequencies of cracks can be very easily predicted. This predictive ability may enhance our quantitative understanding of the processes that generate long-period events at volcanoes.