Elastic and sound orthonormal beams and localized fields in linear mediums: I. Basic equations

Basic definitions and general relations for elastic and sound fields, defined by a given set of orthonormal functions on a real manifold, are presented. The proposed mathematical formalism makes it possible to obtain families of orthonormal beams and localized fields in both isotropic and anisotropic linear elastic mediums as well as the similar sound fields in an ideal liquid. All these fields are described as superpositions of plane waves whose intensities and phases are specified by a set of orthonormal scalar functions on a two- or three-dimensional manifold. By way of illustration, the fields defined by the spherical harmonics are considered.