Regularity and Asymptotic Behaviour of Infinite Systems in Acoustic Waveguides with Stepped Bottom

Abstract The theoretical basis for modelling sound propagation in marine waveguides essentially leads to analysis for the boundary value problem of the Helmholtz equation and the ocean bottom is generally assumed to be an uneven interface composed of different media. This paper presents an analytical solution to the classical acoustic problem of sound propagation by a point source in a waveguide which comprises a cylindrical seamount or a cavity. The velocity potential is constructed for each part of the waveguide as a series of normal modes. Conditions of continuity of acoustic field have led to an infinite system of linear algebraic equations in terms of the unknown coefficients of normal modes. It is shown for the first time that the derived infinite system is quasi-regular and has a unique bounded solution. Asymptotical behavior of unknowns in the system is established with the help of the law of singularity of particle velocity at the edge of the waveguide. The derived asymptotic solution of the unknowns permits the use of the method of improved reduction for the determination of the coefficients of normal modes. Examples of numerical implementation of the proposed theory are presented by varying significant parameters of geophysical waveguides.