Single-target transition matrix method (TMM)

Computes monostatic backscatter from a single axisymmetric target using a transition-matrix formulation. The current implementation targets smooth bodies of revolution and finite cylinders already represented in the package as a Sphere, OblateSpheroid, ProlateSpheroid, or Cylinder, and supports rigid, pressure-release, and homogeneous penetrable fluid/gas interiors.

Details

This implementation is intentionally scoped to single targets and the monostatic backscatter quantity used by target_strength().

For spheres and oblate spheroids, the current implementation uses a spherical-wave basis with an explicit projected boundary solve over the target surface. For prolate spheroids, it instead uses a spheroidal-coordinate transition-matrix-equivalent backend, which is the more natural coordinate system for that geometry and is consistent with the scalar spheroidal T-matrix literature for single-target scattering. For finite cylinders, the default monostatic branch uses a cylindrical-coordinate modal/T-matrix-equivalent backend so that the backscatter benchmark remains aligned with the exact finite-cylinder family. When store_t_matrix = TRUE, cylinders retain lightweight cylindrical-family state that supports exact monostatic reuse and orientation-averaged monostatic products, while general-angle cylinder bistatic post-processing remains outside the current validated scope. Because of that narrower validation status, cylinder calls emit a warning by default; see options(acousticTS.warn_tmm_cylinder = FALSE) to silence it in controlled test or benchmarking workflows.

The present solver is therefore a practical single-target acoustic T-matrix method motivated by the classic transition-matrix literature, but it is not yet a full implementation of the far-field two-part T-matrix workflow of Ganesh and Hawkins (2008, 2022), which assumes an external far-field solver as the first stage.

Usage

This model is accessed via:


target_strength(
  ...,
  model = "TMM",
  boundary,
  sound_speed_sw,
  density_sw,
  n_max,
  store_t_matrix
)

Arguments

boundary: Boundary condition at the target surface. One of "fixed_rigid", "pressure_release", "liquid_filled", or "gas_filled".
sound_speed_sw: Surrounding-medium sound speed ($m~s^{-1}$).
density_sw: Surrounding-medium density ($kg~m^{-3}$).
n_max: Optional truncation limit. For spheres and oblate spheroids, this is the maximum spherical-wave degree used in the truncated T-matrix solve. For the default monostatic cylinder branch, it is the cylindrical modal cutoff used in the geometry-matched backend. When left as NULL, a geometry-aware rule is used frequency-by-frequency. This argument is currently ignored for prolate spheroids, which use the spheroidal-coordinate branch.
store_t_matrix: Logical flag controlling whether the frequency-specific retained state is stored under object@model_parameters$TMM$parameters$t_matrix. The default is FALSE to avoid large object sizes. Explicit block retention is available for the spherical and spheroidal branches. For cylinders, the stored state keeps the geometry-matched cylindrical monostatic family available for exact monostatic reuse and orientation-averaged monostatic products; full general-angle cylinder bistatic post-processing is not yet provided.

Theory

For a single target, the incident and scattered fields are expanded in regular and outgoing modal bases, respectively:

$$ p^{inc} = \sum_{\nu} a_{\nu} \, \psi_{\nu}^{(1)}, \qquad p^{sca} = \sum_{\nu} f_{\nu} \, \psi_{\nu}^{(3)}, $$

where the transition matrix $\mathbf{T}$ maps incident coefficients to scattered coefficients:

$$ \mathbf{f} = \mathbf{T}\mathbf{a}. $$

For the axisymmetric single-target case used here, the azimuthal orders decouple and each block is recovered by enforcing the boundary conditions on the target surface for the retained modal basis functions. The backscatter amplitude is then obtained by evaluating the outgoing expansion in the monostatic receive direction opposite to the incident plane wave.

References

Waterman, P. C. (1969). New formulation of acoustic scattering. The Journal of the Acoustical Society of America, 45, 1417-1429.

Varadan, V. K., Varadan, V. V., Bringi, V. N., and Waterman, P. C. (1982). Computation of rigid body scattering by prolate spheroids using the T-matrix approach. The Journal of the Acoustical Society of America, 71, 22-25.

Hackman, R. H. (1984). An application of the spheroidal-coordinate-based transition matrix: The acoustic scattering from high aspect ratio solids. The Journal of the Acoustical Society of America, 76, 1058-1070. Waterman, P. C. (2009). T-matrix methods in acoustic scattering. The Journal of the Acoustical Society of America, 125, 42-51.

Ganesh, M., and Hawkins, S. C. (2008). A far-field based T-matrix method for three dimensional acoustic scattering. Wave Motion, 45, 1441-1460.

Ganesh, M., and Hawkins, S. C. (2022). A numerically stable T-matrix method for acoustic scattering by nonspherical particles with large aspect ratios and size parameters. The Journal of the Acoustical Society of America, 151, 1978-1988.