FINITE FREQUENCY PROBING CAPABILITIES

A two-dimensional medium in which the fields are described by the Helmholtz equation is considered. A linearized formulation of the problem is studied, which ultimately reduces to reconstructing the unknown right-hand side of the inhomogeneous Helmholtz equation in an infinite strip. The specified right-hand side in this work is taken as a sum of delta functions, which can be interpreted as the total conductivities of thin layers. The values of the solution of the Helmholtz equation and the normal derivative of the solution at the band boundary for several values of the parameter in the Helmholtz equation are used as information for solving the inverse problem. These data can be interpreted as the values of the electric and magnetic field strengths at the band boundary for a finite set of frequencies. Using the Fourier series expansion, an integral equation is obtained that relates the sought quantities to the data for solving the inverse problem. Using the Fourier transform, the conditions for the uniqueness of the solution of the inverse problem are established. Along with this, examples of the multivalued nature of the solution of the inverse problem in unexpected situations are given.

two-dimensional medium, thin layers, infinite strip, inverse problem for the Helmholtz equation, uniqueness theorems, examples of the ambiguity of the solution when reconstructing the medium