Three-dimensional basic problems of statics, pseudo-oscillations, general dynamics and steady state oscillations of the thermoelasticity of isotropic bodies have been completely investigated by many authors. In particular, exterior steady state oscillation problems have been studied on the basis of Sommerfeld-Kupradze radiation conditions in the thermoelasticity, and the uniqueness theorems were proved with the help of the well-known Rellich's lemma, since the components of the displacement vector and the temperature in the isotropic case can be represented as a sum of metaharmonic functions . Unfortunately, the methods of investigation of thermoelastic steady state oscillation problems developed for the isotropic case are not applicable in the case of general anisotropy. This is stipulated by a very complicated form of the corresponding characteristic equation which plays a significant role in the study of far field behaviour of solutions to the oscillation equa- tions. We note that the basic and crack type boundary value problems (BVPs) for the pseudo-oscillation equations of the thermoelasticity theory in the anisotropic case are considered in [3,14]. To the best of the authors' knowledge the problems of thermoelastic steady oscillations for anisotropic bodies have not been treated in the scientific literature. In the present paper we will consider a wide class of basic and mixed type BVPs for the equations of thermoelastic steady state oscillations. We will formulate thermoelastic radiation conditions for an anisotropic medium (the generalized Sommerfeld-Kupradze type radiation conditions) and prove the uniqueness theorems in corresponding spaces. To derive these conditions we have essentially applied results of Vainberg. Further, using the potential method and the theory of pseudodifferential equations on manifolds we will prove existence theorems in various functional spaces and establish the smoothness properties of solutions.

Three-dimensional basic problems of statics, pseudo-oscillations, general dynamics and steady state oscillations of the thermoelasticity of isotropic bodies have been completely investigated by many authors. In particular, exterior steady state oscillation problems have been studied on the basis of Sommerfeld-Kupradze radiation conditions in the thermoelasticity, and the uniqueness theorems were proved with the help of the well-known Rellich's lemma, since the components of the displacement vector and the temperature in the isotropic case can be represented as a sum of metaharmonic functions . Unfortunately, the methods of investigation of thermoelastic steady state oscillation problems developed for the isotropic case are not applicable in the case of general anisotropy. This is stipulated by a very complicated form of the corresponding characteristic equation which plays a significant role in the study of far field behaviour of solutions to the oscillation equa- tions. We note that the basic and crack type boundary value problems (BVPs) for the pseudo-oscillation equations of the thermoelasticity theory in the anisotropic case are considered in [3,14]. To the best of the authors' knowledge the problems of thermoelastic steady oscillations for anisotropic bodies have not been treated in the scientific literature. In the present paper we will consider a wide class of basic and mixed type BVPs for the equations of thermoelastic steady state oscillations. We will formulate thermoelastic radiation conditions for an anisotropic medium (the generalized Sommerfeld-Kupradze type radiation conditions) and prove the uniqueness theorems in corresponding spaces. To derive these conditions we have essentially applied results of Vainberg. Further, using the potential method and the theory of pseudodifferential equations on manifolds we will prove existence theorems in various functional spaces and establish the smoothness properties of solutions.

À l’heure actuelle, la prospection magnétique est reconnue comme une méthode efficace pour répondre à des problématiques archéologiques en partie du fait de sa capacité à détecter les structures enfouies (bâtiments, fosses, canaux, etc,…). Elle permet de mettre en évidence et de localiser avec précision les structures et elle est donc une aide précieuse avant le début des opérations de fouilles. Notre travail rend compte de cartographies magnétiques effectuées sur trois sites archéologiques en Syrie (Al-Rawda, Sh’airat, Malhat Ed-Deru) en utilisant deux méthodes de cartographie magnétique à différents niveaux de résolution. La première méthode (à la perche) permet de couvrir de grandes surfaces rapidement et la deuxième méthode (à la luge), plus lente, permet de mieux imager les structures enfouies. Ces méthodes fournissent des plans urbanistiques des villes circulaires du troisième millénaire avant J.C. Les opérateurs de méthodes potentielles et la modélisation ont permis de caractériser la géométrie des structures aimantées. / Currently, magnetic surveying is recognized as an effective method to address archaeological issues in part because of its ability to detect buried structures (buildings, pits, channels, etc ...). It allows identifying and accurately locating structures and is therefore a valuable aid before the commencement of excavation. Our work reports on magnetic mapping done at three archaeological sites in Syria (Al-Rawda, Sh’airat, Malhat Ed-Deru) using two methods of magnetic mapping at different levels of resolution, The first method (Backpack) allows to cover large areas quickly and the second method (Sledge), slower, allows better imaging of buried structures. These methods have provided urban planning circular cities of the third millennium BC. The modeling method swapped to characterize the magnetic sources.