Development of representation of elastic waves and investigation of their fundamental properties

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The purpose in this research is the following : (1) to develop a new representation of elastic waves and to apply this to inverse problems;(2) to investigate fundamental properties of elastic waves connected with the inverse problems.The above (1) is an attempt to develop a primitive idea obtained previously into a detailed form. Although we have encountered an unexpected serious difficulty in this attempt, we succeed finally in proving the conjectured conclusion, and moreover make relation clear between the obtained representation and the known ones. Also we obtain an algorithm for an inverse problem to measure pre-stress from data of surface waves (Rayleigh waves) expressed by the representation. This is based on a formula which shows how the pre-stress depends on the velocity of the Rayleigh wave.Concerning the above (2), we get new pieces of knowledge about the energy of waves. The one is verification that the energy propagates essentially along the characteristics. The other is precise estimation for influence of the dissipative terms to the energy decay. We show also the representation in (1) is useful to prove the Huygens principle for more extensive classes than previously known. Further, we have tried to prove the unique continuation property for the general elastic equation, as was planned, and get an idea for the proof. However, we do not accomplish this proof expected first.