On The Direct And Inverse Acoustic Scattering Problems For Complex Scatterers

The research for the electromagnetic and acoustic waves scattering problems is one of the hot issues in mathematics,we consider some time-harmonic acoustic waves scattering problems in this paper.After factoring out the time term e-iwt the only non zero space component of the time-harmonic wave u(x)satisfies the Helmholtz equation.The scattering problems can be divided into the direct and inverse problems.The direct problems are the boundary conditions problems for the Helmholtz equation,and the inverse problems arc to reshape the scattcrcrs from the measurements(the scattered waves or their far field patterns).Since in the.real applications the scattcrcrs usually are complex,we consider some kinds of mixed scattering problems.In this paper,our main interest lies in the mixed scattering problems with cracks,multiply obstacles scattering and interior scattering problems with point sources and observation point in a cavity.Specifically:In the first chapter,we introduce the overview of the direct scattering problem.The Helmholtz equation,the Sommcrfeld radiation condition and boundary conditions are introduced in the first section.Then the review and the main research methods of the inverse scattering problems are given in the second section.In the second chapter,we introduce the basic theories and tools for the scattering problems.The first section introduces some spaces used to research the scattering prob-lems,including Holder spaces and Sobolev spaces.The second section introduces some special functions about the Helmholtz equation,such as Bessel functions,Neumann func-tions,Hankel functions and Legendre polynomials.Section III introduces the potentials and boundary integral operators,along with their main properties.Section IV introduces the main theory tools for scattering problems,such as Green functions,the uniqueness theorems,Fredholm theorem.Section V introduces the regularization theories and the Tikhonov regularization method.In the third chapter,the mixed scattering problem by an open arc and an impenetrable obstacle is considered.In this problem,the impedance condition is specified on the boundary of the obstacle,and a jumping boundary condition is set on the arc.The well-posedness of the direct problem is verified by using the boundary integral equation method and Fredholm theorem in the first section.Then the uniqueness of the inverse problem is deduced after established a mixed reciprocity relation in the section ?.In the section III and IV,we reconstruct the shape of the arc and the obstacle by using the linear sampling method and present some numerical examples,respectively.In the chapter IV,we consider the scattering problem from multiple impenetrable obstacles.In detail,we consider the direct and inverse scattering problem by two im-penetrable obstacles,with Neumann and impedance boundary conditions,respectively.At first,the well-posedness of the direct scattering problem is proved by the boundary integral equations method and Fredholm theorem,then the inverse scattering problem is considered by using the linear sampling method and some numerical examples are presented in the last section.In the chapter V,we consider the scattering problem by an open arc with mixed boundary conditions.The arc has Dirichlet and impedance conditions on two sides,respectively.The factorization method is used to prove the inverse scattering problem in the first section and numerical examples are given in the second section.In the chapter VI,we consider an interior scattering problem in a cavity with partial coated boundary.The interior scattering problem is in a cavity with mixed boundary conditions,Dirichlet and impedance conditions are specified on two parts of the boundary.We use the variational method to prove the well-posedness of the direct problem in the section one,and reconstruct the cavity using the factorization method in the section two,then give some numerical examples in section three.