Wave motion in preloaded media. Analytical continuation of dispersion diagram

The simplest example of the preloaded media is an elastic cord anchored to a solid base by elastic elements. Wave motion in such a media are described by the Scalar Klein-Gordon equation. We apply to this equation the method of analytic continuation of dispersion diagram. Previously, this method was successfully applied to elastic waveguides, and to continuous layered waveguides. New analytic results were obtained. The idea of the method as follows. The solution of the problem is presented in terms of double Fourier integral on frequency and wavenumber variables. The latter are analytically continued into the complex plane. So, the integral is studied in a two-dimensional complex space. We show that the dispersion equation of the scalar Klein-Gordon equation is topologically equivalent to a tube analytically embedded in two-dimensional complex space. The Fourier integral is studied on this tube using Cauchy theorem. In the result, the basic properties of the scalar Klein-Gordon are obtained. The work is supported by the RFBR grant 19-29-06048.