Cylindrical Surface Wave: Revisiting the Classical Biot’s Problem

Abstract

The problem on a surface harmonic elastic wave propagating along the free surface of cylindrical cavity in the direction of cavity axis is considered. In the case of isotropic medium, this is the classical Biot ’ s problem of 1952. First, the Biot pioneer work is revisited: the analytical part of Biot ’ s findings is shown in the main fragments. The features are using two potentials and representation of solution by Macdonald functions of different indexes. Then the new direct generalization of Biot ’ s problem on the case of transversely isotropic medium within the framework of linear theory of elasticity is proposed. Transition to the transverse isotropy needs some novelty — necessity of using the more complex representations of displacements through two potentials. Finally, a generalization of Biot ’ s problem on the case of isotropic and transversely isotropic media in the framework of linearized theory of elasticity with allowance for initial stresses is stated. This part repeats briefly the results of A.N. Guz with co-authors of 1974. The main features are using the linearized theory of elasticity and one only potential. All three parts are shown as analytical study up to the level when the numerical methods have to be used.