Direct resonance problem for Rayleigh seismic surface waves

Abstract

In this paper we study the direct resonance problem for Rayleigh seismic surface waves and obtain a constraint on the location of resonances and establish a forbidden domain as the main result. In order to obtain the main result we make a Pekeris-Markushevich transformation of the Rayleigh system with free surface boundary condition such that we get a matrix Schr\"odinger-type form of it. We obtain parity and analytical properties of its fundamental solutions, which are needed to prove the main theorem. We construct a function made up by Rayleigh determinants factors, which is proven to be entire, of exponential type and in the Cartwright class and leads to the constraint on the location of resonances.