Attenuation of dispersed waves

Abstract

A measure of the absorption of elastic waves is the specific absorption coefficient 1/Q. In dispersive mediums, whether the dispersion is due to geometry, inhomogeneity, or both, measurements are often made outside the body and the measurements must be interpreted as to the distribution of values of 1/Q within the body. Two definitive experiments of this type are those performed using standing waves set up in a confined sample of the body and with waves that propagate through or on the surface of the body. Typical examples of these experiments involve the measurement of the damping coefficient of the free modes of vibration of the earth and the measurement of the attenuation factor of propagating surface waves on the earth. These two types of experiments can themselves be interpreted in terms of dimensionless attenuation factors. We call the dimensionless attenuation factors in the standing wave and propagating wave experiments 1/Q_T and 1/Q_x, defined as the logarithmic decrements π/QT and π/Q_x in each experiment. Then in a damped standing wave the amplitude will diminish with time t at a fixed point as exp (−πt/TQ_T), where T is the period. In a propagating monochromatic wave the amplitude will diminish with distance x as exp (−πx/cTQ_x), where c is the phase velocity.