Eigen oscillation of a fluid sphere and source mechanism of harmonic volcanic tremor

The eigen oscillation of a fluid sphere embedded in an infinite elastic medium is analyzed to understand the source mechanism of volcanic tremor that vibrates nearly monotonically and attenuates slowly. The dimensionless eigen frequencies of the sinusoidal oscillation are calculated in a complex form with the attenuation factor in its imaginary part for various combinations of the three parameters: the contrasts of P-wave velocity, density and rigidity between the fluid and the country rock. Eigen oscillations consist of a high attenuation made with a rapidly decaying pulsive wave at a low frequency and infinite number of regular modes with slowly decaying vibrations. For regular modes, the frequency of oscillation obtained from the real part of an eigen value is distributed in approximately regular intervals while the attenuation factor from the imaginary part is almost constant independent of the mode. Each eigen frequency of regular and high attenuation modes is degenerated with two independent eigen functions describing different distributions of displacement, velocity and stress. The theory is applied to harmonic volcanic tremor observed at Kusatsu-Shirane Volcano, central Japan. Observed spectral peaks of the tremor are explained by the eigen frequencies and attenuation factors of several lowest regular modes if the spherical fluid oscillator has a radius of dozens of meters and a P-wave velocity of about several hundred meters per second.