Distance measurements and splitting of electromagnetic waves in water caused by dispersion.

Abstract

A half space is considered with the dispersion represented by a relation, between the electric field and the induction, which contains derivatives of fractional order and has a Fourier Transform similar to the empirical formula of Cole and Cole (1941), commonly used in experimental physics, and to the formula used by Jacquelin (1991) in studying the dispersion of energy in electric networks. The consequent index of refractionn contains a rational power of the imaginary frequencyif and is a polymorphic function off; this function, for each frequency, gives a set of different velocity fields whose number depends on the rational exponent ofif. Each electromagnetic wave leaving the surface of the half space, directed into the half space and with an assignedf, is split in a number of waves with different velocities; ifn is a function of the position the paths of the waves are different and reach a given depth at different points and times. In the case when the medium is water we assume for the dielectric parameter the expression most commonly used in the literature and containing a rational power ofif and the parameter values of Hasted (1973). It is found that the amplitude of each set of waves with the samef and direction at the source, due to the phase difference of the single components of the set and independently of the dissipation of energy, fluctuates with successive maxima whose amplitude decreases with increasing distance from the source; also the rate of decrease of the maxima is augmented by the dissipation. It is finally estimated that the measurement of a length of 1 km, using a frequency of 10 MHz, may be effected by an uncertainty of 38 ns or 1.22 m.