A fully three-dimensional (3D) omnidirectional numerical coupled mode model of acoustic propagation is detailed. A combination of normal mode and finite element computational methods is applied to produce the numerical results. The technique is tested in a strongly range-dependent ocean environment modeled after the Hudson Canyon. Modeled sound from three source locations selected over different bathymetric depths is examined to determine capabilities and difficulties associated with varying numbers of propagating vertical modes across the horizontal domain, and variable amounts of mode coupling. Model results are compared to those from a unidirectional Cartesian 3D parabolic equation simulation, and from adiabatic (uncoupled) simulations to illustrate the capabilities of the techniques to study the influences of coupling, strong refraction, and reflection.