An approximate formula for the waveguide invariant in the shallow-water Pekeris model, analytical insights and...

This work provides an explicit, accurate, and physically interpretable formulation of the waveguide invariant (WI) for the shallow-water Pekeris model with a finite-impedance seabed, addressing limitations of existing approximations in the low-frequency regime. This is achieved by deriving an approximate closed-form expression for the intermodal WI based on the physically intuitive cycle-distance formula for modal group velocities. Its accuracy is established through comprehensive validation against full-wave KRAKEN simulations, showing close agreement with the benchmarks and a rapid decay of error beginning immediately above the modal cutoff. Three key analytical insights are derived. First, a rigorous lower bound-confirming that finite seabed impedance elevates the WI above its ideal baseline-is formally established and experimentally supported by large WI values from seabed-dominated, low-frequency data. Second, a compact closed-form expression is obtained for the limit as the grazing angle approaches zero, helping to explain the stability of far-field interference structures. Third, a continuous angular-dependent approximation for adjacent-mode WIs is presented. Experimental analysis further defines the framework's operational boundary, confirming its optimal use where seabed effects dominate over water-column stratification. Together, the derived formulation, analytical insights, and experimental evidence constitute a refined framework for understanding modal interference in shallow-water waveguides.