Boundary effects on modal shape in deep ocean via non-integer order parabolic cylinder functions

This study investigates the effects of ocean boundaries on modal shapes in very-low-frequency (VLF, 1–10 Hz) sound propagation through the deep ocean. Utilizing a normal mode solution formulated in terms of parabolic cylinder functions (PCF), we demonstrate that boundary interactions induce a phase change reduction below −π at frequencies of several hertz. This reduction, in turn, forces a key transition in the solution, shifting the order of the PCF from integer to non-integer values. Analysis of the characteristic shape of the PCF versus its order reveals that these boundary-influenced modes exhibit an energy shift toward deeper regions and a weakened axial convergence of the underwater sound field.