Investigation of Wave Processes in Inhomogeneous Media with Imperfect Contact Conditions

Abstract

The author considers the problem of numerical modeling and optimization of wave processes in inhomogeneous domains with imperfect contact conditions on the basis of Schr&& odinger parabolic wave equation. The optimality criterion is formulated. The differential properties of the optimization problem are investigated. The numerical method is proposed for modeling and optimization of acoustic fields in inhomogeneous domains.Numerical and numerical-analytical methods of mathematical modeling of sound wave propagation have been developed in the last decades in ocean acoustics to solve distant sounding and acoustic monitoring problems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Of considerable interest, first of all for geophysics and ocean acoustics, are problems related to formation of acoustic fields with prescribed properties and analysis of the propagation of sound waves in inhomogeneous waveguides with regard for thin inclusions. The mathematical difficulties that arise in solving such problems can be overcome by approximating the Helmholtz wave equation with parabolic equations [3-7], which necessitates the development of numerical methods to solve direct and extremum problems for Schr&& odinger-type equations with complex-valued non-self-adjoint operator.In the paper, we will analyze numerical modeling and optimization of acoustic fields on the basis of parabolic wave equation in an axisymmetric inhomogeneous double-layer waveguide with piecewise continuous (piecewise constant) acoustic parameters and imperfect-contact conditions, which allows taking into account thin inclusion in the waveguide. We will propose a numerical method to solve the direct and optimization problems, investigate the differential properties of the quality functional, and analyze the stability of difference scheme for the direct and adjoint boundary-value problems with complex-valued non-self-adjoint operator.