Formal Derivations of Mode Coupling Equations in Underwater Acoustics: How the Method of Multiple Scales Results in an Expansion over Eigenfunctions and the Vectorized WKBJ Solution for the Amplitudes

Abstract

In this study formal derivation of mode coupling equations in underwater acoustics is revisited. This derivation is based on the method of multiple scales from which modal expansion of the field emerges, and the vectorized WKBJ equation for the coefficients in this expansion are obtained in an automatic way. Asymptotic analysis accomplished in this work also establishes a connection between coupled mode parabolic equations in three-dimensional case and the generalized WKBJ solution that emerges as its two-dimensional counterpart. Despite the fact that similar mode coupling equations can be found in literature, in our study a new systematic and formalized approach to their derivation is proposed. A theorem that guarantees asymptotic conservation of the energy flux in the considered two-dimensional waveguide is also proven.