A dynamic source model is presented, in which a three-dimensional crack containing a viscous compressible fluid is excited into resonance by an impulsive pressure transient applied over a small area ΔS of the crack surface. The crack excitation depends critically on two dimensionless parameters called the crack stiffness, C = (b/μ)(L/d), and viscous damping loss, F = (12ηL)/(ρƒd2α), where b is the bulk modulus, η is the viscosity, ρƒ is the density of the fluid, μ is the rigidity, α is the compressional velocity of the solid, L is the crack length, and d is the crack thickness. The first parameter characterizes the ability of the crack to vibrate and shapes the spectral signature of the source, and the second quantifies the effect of fluid viscosity on the duration of resonance. Resonance is sustained by a very slow wave trapped in the fluid-filled crack. This guided wave, called the crack wave, is similar to the tube wave propagating in a fluid-filled borehole; it is inversely dispersive, showing a phase velocity that decreases with increasing wavelength, and its wave speed is always lower than the acoustic velocity of the fluid, decreasing rapidly as the crack stiffness increases. The source spectrum shows many sharp peaks characterizing the individual modes of vibration of the crack; the variation of spectral shape, both in the number and width of peaks, is surprisingly complex, reflecting the interference between the lateral and longitudinal modes of resonance, as well as nodes for these modes. The far-field spectrum is marked by narrow-band dominant and subdominant peaks that reflect the interaction of the various source modes. The frequency of the dominant spectral peak radiated by the source is independent of the radiation direction. The frequency, bandwidth, and spacing of the resonant peaks are strongly dependent on the crack stiffness, larger values of the stiffness factor shifting these peaks to lower frequencies and decreasing their bandwidth. The excitation of a particular mode depends on the position of the trigger and on the extent of the crack surface affected by the pressure transient. Fluid viscosity decreases the amplitudes of the main spectral peaks, smears out the finer structure of the spectrum, and greatly reduces the duration of the radiated signal. The energy loss by radiation is stronger for high frequencies, producing a seismic signature that is marked by a high-frequency content near the onset of the signal and dominated by a longer-period component of much longer duration in the signal coda. Such signature is in harmony with those displayed by long-period events observed on active volcanoes and in hydrofracture experiments. The very low velocity which is possible in a crack with high stiffness (C ≥ 100) also provides an attractive explanation for very long period tremor, such as type 2 tremor at Aso volcano, Japan, without the requirement of an unrealistically large magma container. The standing wave pattern set up on the crack surface by the sustained resonance in the fluid is observable in the near field of the crack, suggesting that the location and extent of the source may be estimated from the mapping of the pattern of nodes and antinodes seen in its vicinity. According to the model, the long-period event and harmonic tremor share the same source but differ in the boundary conditions for fluid flow and in the triggering mechanism setting up the resonance of the source, the former being viewed as the impulse response of the tremor generating system and the latter representing the excitation due to more complex forcing functions.
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