We have derived the asymptotic expressions for high-frequency spectral amplitude of acceleration due to unsteadily propagating two-dimensional semi-infinite cracks both in antiplane strain and in plane strain. Property of generalized functions is employed in the derivation, by which mathematical analysis of high-frequency radiation will become rather simple.
It is shown, as Madariaga showed for rather simple crack models, that the abrupt change of crack tip velocity highly contributes to the generation of high-frequency waves. In the present paper near-source geometrical attenuation of high-frequency acceleration is specifically studied. We employ, as a measure of the intensity of acceleration, the envelope-amplitude of high-frequency spectral-amplitude of acceleration. If the crack tip velocity makes an abrupt change during its propagation, the envelope-amplitude is frequency-independent in the high-frequency range. In the calculation the crack surface is divided into segments, and the crack tip velocity is assumed to take a constant value in each segment.
It is shown that the intensity of acceleration is larger when the mean value and/or the variance of the distribution of crack tip velocity are higher. It is also shown that the near-source geometrical attenuation of envelope-amplitude is in harmony with the observed attenuation of peak acceleration associated with earthquake ruptures. This may suggest that seismic high-frequency waves are attributed to the abrupt changes of earthquake rupture velocity.
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