Numerical Modeling of Underwater Acoustic Propagation

Numerical modelling of underwater acoustic propagation encompasses a suite of mathematical and computational techniques designed to predict how sound travels through complex marine environments. Central approaches include normal-mode methods, which decompose the acoustic field into discrete modal functions, and parabolic equation (PE) models, which approximate the Helmholtz equation for range-dependent scenarios. Finite difference schemes have long provided straightforward discretisations of governing equations but can suffer from reduced accuracy and slow convergence in highly variable media. In response, spectral methods—employing orthogonal polynomials or other basis functions—have gained prominence for their superior precision per grid point. Recent efforts extend these frameworks to three-dimensional settings, account for elastic and inhomogeneous seabed properties, and incorporate interactions with seasonal sea ice, heterogeneous bathymetry and internal waves. Such advances support applications ranging from naval sonar performance prediction and offshore infrastructure monitoring to marine mammal impact assessments and seabed mapping. By balancing computational efficiency with physical fidelity, these models enable realistic simulations of transmission loss, scattering, refraction and diffraction in environments as diverse as shallow continental shelves, deep ocean basins and polar marginal ice zones.

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Numerical Modeling of Underwater Acoustic Propagation publication trend

The graph below shows the total number of articles in numerical modeling of underwater acoustic propagation across all publications each year (not limited to Nature Index journals).

Technical terms

Normal mode: A solution technique that represents the acoustic field as a sum of depth-dependent eigenfunctions, each propagating with a distinct horizontal wavenumber.

Parabolic equation (PE): A one-way approximation of the Helmholtz equation, suited to range-dependent environments and efficient for long-range, narrow-angle propagation.

Finite difference method: A discretisation scheme that approximates derivatives by algebraic differences on a grid, valued for simplicity but requiring fine meshes for high accuracy.

Spectral method: A high-order numerical approach that expands the solution in global basis functions, such as Chebyshev or Legendre polynomials, achieving rapid convergence in smooth regions.

Transmission loss: The reduction in acoustic intensity between source and receiver, expressed in decibels, incorporating spreading, absorption and scattering effects.

References

1. A Chebyshev-Tau spectral method for normal modes of underwater sound propagation with a layered marine environment. Journal of Sound and Vibration (2021).
2. Acoustic recordings and modeling under seasonally varying sea ice. Scientific Reports (2019).
3. A three-dimensional finite difference model for ocean acoustic propagation and benchmarking for topographic effects. The Journal of the Acoustical Society of America (2021).
4. Coupled-Mode Parabolic Equations for the Modeling of Sound Propagation in a Shallow-Water Waveguide with Weak Elastic Bottom. Journal of Marine Science and Engineering (2022).