On reducing the order of the Rayleigh system. Protect I. General theory
The Rayleigh system, together with the Love equation, results from separation of variables in equations of linear elastodynamics in media of three kinds: flat, spherically, or cylindrically layered. This system consists of two second-order ordinary differential equations where frequency and wavenumber enter as parameters. This work investigates possibilities to simplify the Rayleigh system through matrix transformations independent of wavenumber. The reduction of the Rayleigh system to two second-order systems (simplification) is possible under conditions found in this paper. Any Rayleigh system is specified by a $2 \times 2$ matrix with zero trace. Three elements of this matrix describe a curve in a three-dimensional space when depth varies and the frequency is fixed. Zero curve torsion is equivalent to the simplification condition. This fact permits to express the simplification condition explicitly and in a compact form. The condition which depends on frequency is expressed as one differential relation; the condition independent of frequency takes the form of two differential relationships.