# On parabolic-equation model verification for underwater acoustics

## TL;DR

This paper verifies the accuracy of the 3-D parabolic equation method in underwater acoustics by comparing it against exact reference solutions for media with specific lateral sound-speed variations, emphasizing mode coupling and computational techniques.

## Contribution

It introduces a validation approach for the 3-D PE method using exact solutions for media with azimuthal variations, highlighting the role of wavenumber integration and mode coupling verification.

## Key findings

- Wavenumber integration yields accurate field solutions.
- Exact solutions are available for specific media, aiding verification.
- The method effectively compares PE solutions with Helmholtz solutions.

## Abstract

Modeling of sound propagation in media with azimuthal variations of the material parameters typically necessitates approximations. Useful methods, such as the 3-D parabolic equation (PE) method, need verification by correct reference solutions for well-defined test examples. The present paper utilizes reference solutions for media with particular types of lateral sound-speed variation, as a complement to common wedge and canyon examples with bathymetry variation. Notably, the adiabatic approximation is exact for the particular media under study, implying, however, that verification of mode coupling remains. Wavenumber integration, for computation of modal expansion coefficients, produces accurate solutions for the field and its spatial derivatives in the 3-D case. Explicit expressions are typically available for the wavenumber integrands in terms of Airy and exponential functions. For a related 2-D case with azimuthal symmetry, Hankel functions appear instead of exponentials, and the wavenumber integration drops out. The PE verification focuses on comparing the two sides of the PE at insertion of appropriately scaled Helmholtz-equation solutions.