The modeling of acoustic wave propagation in three-dimensional environments is a subject of extensive study in last two decades. There exist large variety of substantially different approaches to these problems [1]. However, it is widely accepted that most direct methods of solution of acoustic wave equation in 3D-waveguides (e.g. using finite differences) are impractical due to computational complexity. This is the main reason why these equations are usually transformed into frequency domain where various parabolic approximations are applied [1-3] (since generally parabolic equations are more attractive in terms of efficiency). Most 3D parabolic equations however suffer from the inaccurate treatment of the inhomogeneities of interfaces - i.e. surfaces in the waveguides where media parameters have discontinuities. This problem emerges from the conventional derivation of the parabolic equations [1,3] (which is based on operator square root approximation) where in fact the absence of any horizontal inhomogeneities is assumed. Hence, such derivations do not lead explicitly to any consistent boundary conditions. This problem seems to severely affect even 2D parabolic equations theory where it is still subject of intensive study. Another common approach to 3D acoustic simulation is based on the normal modes theory [1]. Within this method at any horizontal point (x, y) acoustic field is represented as a linear combination of (local) acoustic spectral problem solutions - normal modes. The coefficients of this decomposition may depend on horizontal variables and must be calculated to obtain the acoustic field. This may be accomplished either by derivation of some equations governing the horizontal variations of these coefficients or by direct matching of sound field values along the boundaries of several regions in horizontal plane (x,y) in which the medium is assumed to be horizontally homogeneous [1,4]. The latter method (besides the limitations and inconveniences produced by such domain partitioning) leads to a computationally intensive procedure of the solution of very large system of linear equations. This is why in our opinion possible applications of such method are limited to construction of benchmark problems solutions (like in [4]). Approximate equations for the coefficients of the modal expansion of acoustic field seem to be much more fruitful in terms of efficiency and range of problems they can treat. Most of existing achievements within this approach were limited so far to the simulations of sound propagation in the environments with 3D-inhomogeneous sound speed profiles [5]. However the most interesting practical problems of shallow water acoustics usually involve 3D bottom relief (i.e. interface) inhomogeneities. A computationally efficient method for the solution of such problems is developed in [6] where the parabolic equations for modal coefficients accounting bottom relief and density inhomogenities were introduced. In this work we recall the derivation of these mode parabolic equations (MPEs) following [6] and address some issues arising in the applications of this method to the practical problems of shallow water acoustics. These equations are derived using the generalized multiple scale method [7]. The distinctive feature of this derivation is accurate and "uniform" treatment of both sound-speed and interface inhomogeneities.
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