The Atkinson-Wilcox theorem in thermoelasticity

An incident disturbance propagates in a thermoelastic medium of the Biot type and it is scattered by a bounded discontinuity of the medium. On the surface of the scatterer any kind of boundary or transmission conditions, that secures well posedness, can hold. The scattered field consists of three kinds of displacement and two kinds of thermal waves. With the exception of one of the displacement waves, namely the transverse elastic wave, all other four scattered waves exhibit exponential attenuation as a result of the coupling between the longitudinal elastic and the thermal disturbances. We show that the displacement field can be expanded in three uniformly and absolutely convergent series in inverse powers of the distance between the observation point and the geometrical center of the scatterer. For the thermal wave 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 the scatterer. The leading coefficients in the two displacement series of the longitudinal type have only radial components which coincide with the corresponding radial scattering amplitudes. For the transverse displacement series the leading coefficient has only tangential components which coincide with the angular scattering amplitudes. An amazing result, which was not noticed before, is that the thermal scattering amplitudes, appearing as leading coefficients in the thermal expansions, are proportional to the corresponding radial longitudinal amplitudes of the elastic expansions. In other words, both scattering amplitudes of the two thermal waves carry no independent information about the scattering process. Finally, an analytic algorithm is provided which leads to the reconstruction of all five series from the knowledge of the three leading coefficients coming from the expansions for the 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 way down to the smallest sphere containing the scatterer. In an equivalent component form we claim that the nine elastic and the two thermal expansions can be completely obtained once the two longitudinal and the two transverse elastic scattering amplitudes are given.