Spectra of seismic radiation from a tensile crack

A model for the average far‐field P and S wave amplitude spectra from a circular crack failing in tension is presented. The model explicitly avoids specifying the details of the rupture process; instead, the spectral amplitude is determined by applying physically realistic constraints to an idealized spectral shape for each wave type. The idealized spectral form is given by Ω 0 /(1 + (ω/ω c ,) 2 ) ψ/2 , where ω c is a parameter representing the corner frequency, Ω 0 determines the overall scaling of the spectra, and ψ determines the falloff at high frequencies. For fixed ψ, the complete specification of the body wave spectra, P and S waves, requires the determination of four unknowns: Ω 0 P , Ω 0 S , ω c P , and ω c S . The P and S spectral models are constrained by fixing the low‐frequency level of each to be equivalent to a point source. They are further constrained by equating the total radiated energy with the available elastic energy through a seismic efficiency parameter η. A final constraint between the corner frequencies of the P and S spectra is needed to determine the four free parameters. For the type of model considered here: an equidimensional fault with greatest displacement in the center, we expect the P to S corner frequency ratio to range between 1 and 1.73 depending on the details of the rupture. For ψ = 2, this model gives an average S / P spectral amplitude ratio of about 2.1 at very low frequencies and between 2.1 and 0.7 at very high frequencies, depending on the P to S corner frequency ratio. Applying the same criteria to a circular shear crack gives an average S / P spectral amplitude of about 7.1 at very low frequencies and between 7.1 and 2.4 at very high frequencies, again depending on the P to S corner frequency ratio. The tensional crack thus has lower average S / P spectral amplitudes than the shear crack, and such low S / P spectral amplitudes may be an identifying characteristic of tensile or tensile equivalent rupture. Using the shear and tensional models, we construct composite spectra that have significantly smaller S / P spectral amplitude ratios than the shear cracks alone, even when the tensional events radiate just a fraction of the total seismic energy. The tensile crack average spectral model may be useful as a first approximation for modeling seismic sources with a volumetric component in them, such as slip on a nonplanar fault, or magma injection.