Capabilities of sounding at a finite set of frequencies

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. This right-hand side is taken as a sum of delta functions, which can be interpreted as the total conductivities of thin layers. The information for solving the inverse problem comprises the values of the solution to the Helmholtz equation and the normal derivative of the solution at the strip boundary for several values of the parameter in the Helmholtz equation. These data can be interpreted as the values of the electric and magnetic field strengths at the strip boundary for a finite set of frequencies. Using the expansion in Fourier series, an integral equation is obtained that relates the sought-for quantities with the data for solving the inverse problem. Using the Fourier transform, conditions for the uniqueness of the solution to the inverse problem are established. In addition, examples of the multivaluedness of the solution to the inverse problem in unexpected situations are given.