The set valued unified model of dispersion and attenuation for wave propagation in dielectric (and anelastic) media

Abstract

Since the dispersion and attenuation properties of dielectric and anelastic media, in the frequency domain, are expressed by similar formulae, as shown experimentally by Cole and Cole (1941) and Bagley and Torvik (1983, 1986) respectively, we note that the same properties may be represented in the time domain by means of an equation of the same form; this is obtained by introducing derivatives of fractional order into the system functions of the media. The Laplace Transforms (LT) of such system functions contain fractional powers of the imaginary frequency and are, therefore, multivalued functions defined in the Riemann Sheets (RS) of the function. We determine the response of the medium (dielectric o anelastic) to a generic signal summing the time domain representation due to the branches of the solutions in the RSs of the LT. It is found that, if the initial conditions are equal in all the RSs, the solution is a sum of two exponentials with complex exponents, if the initial conditions are different in some of the RSs, then a transient for each of those RSs is added to the exponentials. In all cases a monochromatic wave is split into a set of waves with the same frequency and slightly different wavelengths which interfere and disperse. As a consequence a monochromatic electromagnetic wave with frequency around 1 MHz in water has a relevant dispersion and beats generating a tunnel effect. In the atmosphere of the Earth the dispersion of a monochromatic wave with frequency around 1 GHz, like those used in tracking artificial satellites, has a negligible effect on the accuracy of the determination of the position of the satellites and the positioning of the bench marks on the Earth. We also find the split eigenfunctions of the free modes of infinite plates and shells made of dielectric and anelastic media.