A Scattering Problem Of Crack With Mixed Boundary Conditions

Direct scattering of time-harmonic electromagnetic acoustic waves or tim e-harmonic waves is one of the main research topics in the field of mathematical physics. It is also the foundation for solving inverse scattering prob lems, and it has becom e a popular research topic. As a result, electromagnetic and acoustic scattering theory and related methodologies were widely studied. The followings are the contents of this thesis:In the first chapter, we gave a brief introduction to the background and studying status of electromagnetic and acoustic scattering problems, as well as m ain contents studied in this thesis.In the second chapter, basic methodologie s and relate theories on direct scattering problems are introduced.In the third chapter, we consider the scattering problem from a thin infinite cylindrical conductor. The thin cylindrical conductor can be seen as a curve segment(a crack) in a cross-section. Assuming that the crack has three parts with different boundary conditions, the direct scattering problem for the crack is formulated in to a boundary value problem of Helmholtz equation with m ixed boundary conditions. The application of Green’ s formula and Rellich’s lemma leads to a unique solution of the problem. An equivalent system of boundary integral equations is set up by using jump relations and the potential theory, and it makes clear that the solution of the direct scattering problem for the crack exists.The numerical solution of the scatteri ng problem of cracks m ust solve logarithm ic singular and hypersingular integr al equations where the discrete m ethods are sim ilar to those discrete m ethods for Sy mm integral equations. Ther efore we study the discrete methods for Symm integral equations. The detailed derivation is give and som e results are proved.In the fifth chapter, we summarize the mainly studied content, significance and the shortage of this paper, and forecast the future research.