Improved Underwater Acoustic Modeling: High-Order WKB Approximations for Adiabatic Normal Modes

The adiabatic normal mode (ANM) method efficiently handles weakly range-dependent underwater acoustic propagation. While the WKB approximation underpins solving for ANM modal coefficients, existing work primarily focuses on the first-order solution; the potential of higher-order WKB approximations remains unexplored. To address this gap, this paper systematically derives second- and third-order WKB solutions for ANM. Key steps include: (1) Expanding modal coefficients exponentially to formulate a standard WKB equation; (2) Applying the dominant balance principle to establish coupled equations satisfied by WKB coefficients at each order; (3) Solving coupled equations to obtain the higher-order solutions. Six representative shallow-water test cases were designed to evaluate these theories. Results demonstrate that source frequency critically governs the applicability: (1) At 200 Hz, higher-order solutions yield superior accuracy. The third- and second-order solutions reduce computational errors by 69.35% and 35.48%, respectively, compared to the first-order. (2) Below 200 Hz, accuracy diminishes with increasing order due to wavelength reduction violating the slowly-varying medium assumption, causing compensation terms to diverge. This work bridges the gap in applying higher-order WKB methods to underwater acoustics and provides crucial guidance for selecting the optimal WKB order.