Linear Models of Dissipation whose Q is almost Frequency Independent--II

Laboratory experiments and field observations indicate that the Q of many non-ferromagnetic inorganic solids is almost frequency independent in the range 10−2–107 c/s, although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression for a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stress-strain relation. Since the aim of this research is also to contribute to elucidating the dissipating mechanism in the Earth free modes, we shall treat the dissipation in the free, purely torsional, modes of a shell. The dissipation in a plane wave will also be treated. The theory is checked with the new values determined for the Q of spheroidal free modes of the Earth in the range between 10 and 5 min integrated with the Q of Rayleigh waves in the range between 5 and 0·6 min. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an aluminium rod in the range between 10−5 and 10−3 s. In both checks the theory represents the observed phenomena very satisfactorily. The time derivative which enters the stress-strain relation in both cases is of order 0·15. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data.