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. The default target_strength() result is monostatic backscatter, and retained T-matrix state can also be reused for angular post-processing where the active branch supports it.

For spheres and oblate spheroids, the current implementation uses a spherical-wave T-matrix: each retained azimuthal order is solved as an incident-to-outgoing modal coefficient map after projecting the boundary conditions on the target surface. For prolate spheroids, the implementation uses a geometry-matched spheroidal-basis T-matrix formulation, which is the natural coordinate system for that geometry and is consistent with the scalar spheroidal transition-matrix literature for single-target scattering. For finite cylinders, the default monostatic branch uses a cylindrical-coordinate modal T-matrix-style 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 sphere, oblate, prolate, and shell-sphere branches are therefore single-target acoustic T-matrix methods in the modal coefficient-map sense: they represent the target response as a map from incident modal amplitudes to scattered modal amplitudes and reuse that retained state for supported post-processing. The cylinder branch is more limited and should be treated as a guarded monostatic modal T-matrix-style branch until a validated general-angle cylindrical operator is available.

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. Each retained block is recovered in the basis used by the active geometry branch: spherical for spheres and oblates, spheroidal for prolates, and exact spherical modal coefficients for supported shell spheres. The backscatter amplitude is obtained by evaluating the outgoing expansion in the monostatic receive direction opposite to the incident plane wave. When the retained state is stored, the same coefficient map can be reused for supported angular, bistatic, diagnostic, and orientation-averaged products.

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.