Rayleigh boundary value problem in the matrix Sturm-Liouville form
We examine a boundary value problem for Rayleigh, or P-SV, vibrations occurring in elastic layered media of four types: flat layered half-space, spherically layered sphere, cylindrically layered circular cylinder, and flat centrally symmetric disk. In the previous research the authors, together with their colleagues, represented the equations governing the vibrations in the matrix Sturm-Liouville form with a nonsymmetric potential. Boundary conditions were prescribed at the surface of the body; it was also assumed that displacements approached zero at infinity for a half-space and were limited at the axis of a cylinder, at the center of a sphere, or at the center of a disk. In this paper we represent the boundary condition at the free surface as a linear combination of the vector solution and its first derivative. We prove that there exists a relationship between the matrix coefficient of this linear combination and the potential entering the Sturm-Liouville equation. This relationship implies that the determinant of the difference between two certain matrix functions remains constant at any argument value. The first function is the matrix impedance of the solution. The second function is such that the sum of its derivative and its square is the potential entering the Sturm-Liouville equation. This difference is more smooth than the impedance and the matrix potential. We prove that its elements are found from Volterra's integral equation of the second kind with a kernel depending on the matrix potential. Equations governing vibrations in bodies of different symmetry have a common structure; hence they can be treated as cases of the general matrix boundary value problem.