ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE HELMHOLTZ EQUATION CONCENTRATED NEAR THE AXIS OF A DEEP-WATER WAVEGUIDE IN A RANGE-INDEPENDENT MEDIUM

Abstract

To derive the integral representation for the axial wave describing the interference of near-axial waves in an arbitrary deep-water waveguide in a range-independent medium in long-range acoustic propagation in the ocean, it is necessary to construct the solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis in a narrow strip having a width of order ω-1/2, where ω is a cyclic frequency. In a range-independent (separable) case the desired solutions coincide with the principal term of the uniform asymptotic expansion as ω→∞ of normal modes when a mode number is a value of order of unity. In this paper, the solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis in a range-independent ocean and which decrease exponentially outside a strip containing the axis are constructed in the form admitting generalization to the case of a range-dependent medium. The solutions are represented as the product of exponentials and parabolic cylinder functions whose arguments are infinite series in powers of ω-1/2. Coefficients of these series are found from a recurrent system of partial differential equations up to terms allowing to get a residual (difference between the left-hand side of the Helmholtz equation and zero) of order ω-1/2. Numerical results are obtained for medium parameters corresponding to the Munk canonical sound-speed profile.