The Atkinson-Wilcox theorem in thermoelasticity

An incident disturbance propagates in a thermoelastic medium of the Blot ty pe and it is scattered by a bounded discontinuity of the medium. On the sur face of the scatterer any kind of boundary or transmission conditions, that secures well posedness, can hold. The scattered field consists of three ki nds of displacement and two kinds of thermal waves. With the exception of o ne of the displacement waves, namely the transverse elastic wave, all other four scattered waves exhibit exponential attenuation as a result of the co upling between the longitudinal elastic and the thermal disturbances. We sh ow that the displacement field can be expanded in three uniformly and absol utely convergent series in inverse powers of the distance between the obser vation point and the geometrical center of the scatterer. For the thermal w ave a corresponding expansion with two series holds true. Each one of these three elastic and two thermal series describes the corresponding scattered wave and their validity is extended up to the sphere that circumscribes th e scatterer. The leading coefficients in the two displacement series of the longitudinal type have only radial components which coincide with the corr esponding radial scattering amplitudes. For the transverse displacement ser ies the leading coefficient has only tangential components which coincide W ith the angular scattering amplitudes. An amazing result, which was not not iced before, is that the thermal scattering amplitudes, appearing as leadin g coefficients in the thermal expansions, are proportional to the correspon ding radial longitudinal amplitudes of the elastic expansions. In other wor ds, both scattering amplitudes of the two thermal waves carry no independen t information about the scattering process. Finally, an analytic algorithm is provided which leads to the reconstruction of all five series from the k nowledge of the three leading coefficients coming from the expansions for t he displacement field alone. Consequently, if the radial and the tangential scattering amplitudes of the displacement field are given in the far field , then the exact displacement and thermal fields can be recovered all the w ay down to the smallest sphere containing the scatterer. In an equivalent c omponent form we claim that the nine elastic and the two thermal expansions can be completely obtained once the two longitudinal and the two transvers e elastic scattering amplitudes are given.