Approximate Analytical Formulas for ELF Air-to-Undersea Propagation: Accurate Modeling of the Near–Intermediate-Field Transition

Extremely low frequency (ELF) electromagnetic (EM) waves, with frequencies below 300 Hz, exhibit superior penetration through seawater, enabling longer communication ranges and enhancing the flexibility of vehicles for ocean exploration. Despite their potential, ELF propagation in seawater is more complex due to interactions with the air-sea boundary and frequency-dependent attenuation caused by seawater. Here we present a new set of approximate formulas for the EM field components for cross-medium propagation from air-to undersea, due to a vertical electric dipole (VED). The formulas, derived through asymptotic and series expansions of Sommerfeld integrals (SIs), address the limitations of existing models by accurately characterizing EM field in the transition zone between near-field and intermediate-field regions and impose no restrictions on reception depth. Comparisons and analyses with related work and numerical results demonstrate the accuracy and effectiveness of our formulas. The formulas provide simplified calculations, clear physical interpretation, and broad applicability without distance or frequency restrictions. The results enhance the feasibility of ELF communication across air-sea interfaces.