Generalized Form of the Invariant Imbedding Method and Its Application to the Study of Back-Scattering in Shallow-Water Acoustics

A generalized form of the matrix-invariant imbedding method was developed to solve boundary-value problems for coupled systems of Helmholtz-type equations. Within this approach, a boundary-value problem solution can be obtained by solving evolutionary first-order imbedding equations for a matrix-valued function. The proposed method is applied to the solution of coupled equations for mode amplitudes describing the propagation of acoustic waves in a range-dependent shallow-water waveguide. The back-scattering of modes by bathymetry features is investigated, and the coefficients of the modal expansion of the wave reflected by an inhomogeneity in the bottom relief are computed. It is demonstrated that back-scattering is strongly connected with the modal interactions and that the back-scattered field consists of modes with numbers different from the number of the incident mode.

Angular Distribution of XPS Peaks by Layers of a Finite Thickness

On the basis of the invariant imbedding method the equations describing energy spectra of X-ray photoelectron emission by layers of a finite thickness are presented. The analytical decisions describing angular distributions of electrons emitted by layer were received in small-angle approximation and neglect the backscattering factor. Approbation of the received decisions is executed on the basis of comparison with results of Monte-Carlo simulation. The limitations of the theory neglecting with processes of an electron elastic scattering are shown.