  Three-dimensional numerical models for underwater sound propagation are
popular in computational ocean acoustics. For horizontally slowly varying
waveguide environments, an adiabatic mode-parabolic equation hybrid theory can
be used for simulation. This theory employs adiabatic modes in the vertical
direction, simplifying the solution of the sound pressure to the solution of
horizontal refractive index of vertical modes. The refractive equations in the
horizontal direction are further solved by a ``split-step" wide-angle parabolic
equation model, following the approach of the ``vertical modes and horizontal
parabolic equation". Existing three-dimensional sound propagation models mostly
use finite difference methods for discretization, but in recent years, the
academic community has proposed new types of sound propagation models based on
spectral methods. Spectral methods are numerical discretization methods based
on orthogonal polynomial approximation and weighted residual principles. They
offer advantages such as high computational accuracy and fast convergence. In
this study, a three-dimensional adiabatic mode-parabolic equation hybrid model
discretized using spectral methods is proposed. In the vertical direction, the
modal functions are solved using the Chebyshev spectral method. The medium
layering is handled using a domain decomposition strategy, and the leaky modes
under semi-infinite boundary conditions are addressed using an eigenvalue
transformation technique. In the horizontal direction, the perfectly matched
layer technique is utilized to handle unbounded computational domains, and the
perfectly matched layer and computational domain are segmented into multiple
layers. Numerical simulations show that the Chebyshev spectral method achieves
reliable results in the application of the adiabatic mode-parabolic equation
hybrid model.
