On coupled modes theory of two-dimensional wave motion in elastic waveguides with slowly varying parameters in curvilinear orthogonal coordinates

Coupled modes theory, as applied to 2-D wave propagation in range-dependent isotropic elastic waveguides in curvilinear orthogonal coordinates, is proposed. The model consists of one or more curvilinear layers with interfaces and outer boundaries departing slowly from the coordinate surfaces. The material parameters are supposed to be slowly dependent on the longitudinal coordinate. Four-dimensional vector-valued functions with the displacements, rotation, and divergence as the components are shown to be a convenient form of wave field description to extend the coupled modes theory from liquid to elastic waveguides. For this case, the cross section operator is defined and the generalized bi-orthogonality property of its eigenfunctions is established. Using the generalized bi-orthogonality, the coupling equations are derived. Wave propagation in a tube which narrows slowly to a conic rod and in a disk of radially dependent thickness are considered as the examples. The differences between the results obtained by the coupled modes theory and those the multiple-scales theory leads to are discussed. The theory modification for dealing with a cutoff phenomenon and the general algorithm of computation of the wave field excited by a point source are considered. (C) 1998 Acoustical Society of America.