Zero-range potentials in boundary-contact problems of acoustics

Abstract

The problems of acoustic waves diffraction by elastic plates or shells with edges, conjunctions, cracks, inclusions and stiffeners attract wide interest caused by the needs of shipbuilding, acoustics of the ocean and other sciences. Much attention in paid to problems that can be solved exactly. Then the analysis of physical effects is the most simple. Up to present practically all the explicit solutions have been already examined. To analyse new physical effects one needs to use more and more cumbrous approaches with large computer resources needed for realization. Nevertheless, the set of problems having explicit solution is not yet exhausted. One must apply new not traditional for that type of problems mathematical apparatus. In this paper the operator extension theory is suggested to formulate the models of asymptotically small inhomogeneities of plates. The main problem arising in that approach is in the choice of the parameters of the model adéquate to the inhomogeneity. The inverse problem for finding the parameters of the zero—range potential is solved for a particular problem of scattering by a narrow crack in fluid loaded elastic plate. the components of the zero—range potential model are compared to the models in simpler problems. One is that for the isolated plate and the corresponding component of the zero—range potential allows to propose general procedure for choosing the parameters of the model.