Analysis and formation of acoustic fields in inhomogeneous waveguides
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ANALYSIS AND FORMATION OF ACOUSTIC FIELDS IN INHOMOGENEOUS WAVEGUIDES A. V. Gladkii,a† V. V. Skopetskii,a‡ and D. A. Harrisona
UDC 517.9:519.6
The problem of numerical modeling and formation of acoustic fields with definite properties in an axisymmetric inhomogeneous underwater waveguide is considered. A numerical method to solve && a boundary-value and extremal problems for a parabolic Schrodinger-type wave equation with a complex nonself-adjoint operator is proposed and investigated. && Keywords: acoustic field, Schrodinger-type equation, boundary-value problem, extremum problem, difference scheme, stability. INTRODUCTION Mathematical simulation of a wide class of problems for acoustic energy propagating in two- and three-dimensional inhomogeneous underwater waveguides involves the development of mathematical models of wave processes and numerical-analytic methods to solve boundary-value problems for the Helmholtz equation [1–5]. Severe mathematical difficulties arise in the analysis of such problems since the solution is complex-valued, the differential operator in spatial variables is nonself-adjoint and sign-indefinite, the sonic velocity in a waveguide depends on spatial coordinates, there are impedance boundaries, etc. Numerical methods approximating the Helmholtz wave equation by Schr&&odinger parabolic equations [3–8] are intensively developed nowadays. Of interest is also the formation of acoustic fields with preset properties in inhomogeneous waveguides with regard for attenuation in the medium. To solve a class of extremum problems for acoustic fields described by the Helmholtz equation, numerical-analytic methods of solving direct problems of sound propagation can be used [9–11]. The studies [5, 12, 13] address difference schemes to analyze the formation of sound fields based on Schr&&odinger-type wave parabolic equations. The present paper considers the numerical solution of the direct and extremum problems for a sound field formed in an axisymmetric inhomogeneous waveguide with impedance boundary. The propagation of acoustic energy is assumed to be described by a parabolic wave equation that takes into account a wide range of variation in the sonic velocity with distance. PROBLEM FORMULATION Within the framework of parabolic approximation, we will use the following initial–boundary-value problem to describe the acoustic field in an axisymmetric waveguide G = {r 0 < r < R , 0 < z < H , r 0 > 0} (with soft upper and impedance lower boundaries), where ( r, z ) are cylindrical coordinates and the axis z is directed vertically downwards: 2ik 0
¶p ¶ æ 1 ¶p ö + ç ÷ + 2k 02 ( n( r, z ) - 1+ in ( r, z ) + m ( r, z )) p = 0, ( r, z ) Î G , ¶r ¶z çè n( r, z ) ¶z ÷ø æ ¶p ö p | z = 0 = 0, ç + ap ÷ = 0, è ¶z ø z= H p |r = r0 = u( z ).
(1) (2) (3)
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 62–71, March–April 2009. Original article submitted November 6, 2
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