TMM implementation

acousticTS implementation

Benchmarked Partially validated Experimental

TMM represents a target by the linear map from incident modal coefficients to scattered modal coefficients (Waterman 1969, 2009).

Build a supported canonical scatterer, solve it with target_strength(), and optionally retain the modal blocks for angular or orientation post-processing.

Why use `TMM` over `SPHMS` or `PSMS`?

On a canonical shape, TMM should reproduce the corresponding shape-specific family when both solve the same boundary-value problem. Its added value is the retained coefficient map, which supports bistatic scattering and orientation averaging without repeating the boundary solve.

In other words:

if the goal is only a sphere target-strength calculation, SPHMS is the more direct exact model,
if the goal is only a prolate-spheroid target-strength calculation, PSMS is the more direct exact model,
if the goal is only a finite-cylinder target-strength calculation, FCMS is the more direct cylinder model, and
if the goal is to stay inside a transition-matrix framework that can grow into broader scattering workflows while using one common post-processing layer across spheres, spheroids, and cylinders, TMM is the relevant model family.

The TMM cylinder branch is intentionally narrower than the sphere, oblate, and prolate branches. The default monostatic path is benchmark-matched to FCMS, but full retained-angle cylinder grids and bistatic summaries remain outside the validated public scope.

General-angle post-processing is available for sphere, oblate-spheroid, and prolate-spheroid branches. Spherical ESS objects also support the documented fluid-shell and elastic-shell boundaries. Cylinder storage is limited to monostatic reuse and orientation-averaged monostatic products.

Object generation

library(acousticTS)

density_sw <- 1026.8
sound_speed_sw <- 1477.3

sphere_shape <- sphere(radius_body = 0.01)
oblate_shape <- oblate_spheroid(
  length_body = 0.012,
  radius_body = 0.01,
  n_segments = 80
)
prolate_shape <- prolate_spheroid(
  length_body = 0.14,
  radius_body = 0.01,
  n_segments = 80
)
cylinder_shape <- cylinder(
  length_body = 0.07,
  radius_body = 0.01,
  n_segments = 80
)

sphere_object <- fls_generate(
  shape = sphere_shape,
  density_body = 1028.9,
  sound_speed_body = 1480.3,
  theta_body = pi / 2
)

oblate_object <- fls_generate(
  shape = oblate_shape,
  density_body = 1028.9,
  sound_speed_body = 1480.3,
  theta_body = pi / 2
)

prolate_object <- fls_generate(
  shape = prolate_shape,
  density_body = 1028.9,
  sound_speed_body = 1480.3,
  theta_body = pi / 2
)

cylinder_object <- fls_generate(
  shape = cylinder_shape,
  density_body = 1028.9,
  sound_speed_body = 1480.3,
  theta_body = pi / 2
)

The homogeneous-scatterer branches support:

Sphere
OblateSpheroid
ProlateSpheroid
Cylinder

Calculating a TS-frequency spectrum

sphere_object <- target_strength(
  object = sphere_object,
  frequency = seq(12e3, 120e3, by = 12e3),
  model = "tmm",
  boundary = "liquid_filled",
  density_sw = density_sw,
  sound_speed_sw = sound_speed_sw
)

prolate_object <- target_strength(
  object = prolate_object,
  frequency = c(12e3, 18e3, 38e3, 70e3, 100e3),
  model = "tmm",
  boundary = "liquid_filled",
  density_sw = density_sw,
  sound_speed_sw = sound_speed_sw
)

sphere_object@model$TMM

##    frequency                        f_bs     sigma_bs         TS n_max
## 1      12000 -6.343290e-06+1.231703e-09i 4.023733e-11 -103.95371     6
## 2      24000 -1.805437e-05+2.830355e-08i 3.259611e-10  -94.86834     8
## 3      36000 -2.075053e-05+1.169577e-07i 4.305981e-10  -93.65928     9
## 4      48000 -8.027899e-06+1.690861e-07i 6.447575e-11 -101.90604    10
## 5      60000  1.119724e-05-1.833465e-09i 1.253781e-10  -99.01778    11
## 6      72000  2.046997e-05-3.101531e-07i 4.191160e-10  -93.77666    11
## 7      84000  1.133235e-05-3.356291e-07i 1.285348e-10  -98.90979    12
## 8      96000 -8.152309e-06+1.022055e-07i 6.647058e-11 -101.77371    13
## 9     108000 -2.017964e-05+5.314642e-07i 4.075003e-10  -93.89872    14
## 10    120000 -1.347015e-05+3.698175e-07i 1.815818e-10  -97.40928    14

prolate_object@model$TMM

##   frequency                        f_bs     sigma_bs        TS n_max
## 1     12000 -4.439503e-05+3.114131e-08i 1.970920e-09 -87.05331     4
## 2     18000 -8.709975e-05+1.363906e-07i 7.586386e-09 -81.19965     5
## 3     38000 -1.380278e-04+9.507581e-07i 1.905259e-08 -77.20046    10
## 4     70000  1.420489e-04-1.952261e-06i 2.018169e-08 -76.95042    17
## 5    100000 -9.784203e-05+1.983609e-06i 9.576997e-09 -80.18771    24

For spheres, n_max is the spherical-wave cutoff. For prolates, it is inherited from the spheroidal retained system and describes the size of that solve, not a spherical-wave truncation.

Extracting model results

Model results can be extracted either visually or directly through extract().

Plotting results

Pre-rendered TMM example plots showing the sphere geometry and its stored target-strength spectrum.

Accessing results

tmm_results <- extract(sphere_object, "model")$TMM
head(tmm_results)

##   frequency                        f_bs     sigma_bs         TS n_max
## 1     12000 -6.343290e-06+1.231703e-09i 4.023733e-11 -103.95371     6
## 2     24000 -1.805437e-05+2.830355e-08i 3.259611e-10  -94.86834     8
## 3     36000 -2.075053e-05+1.169577e-07i 4.305981e-10  -93.65928     9
## 4     48000 -8.027899e-06+1.690861e-07i 6.447575e-11 -101.90604    10
## 5     60000  1.119724e-05-1.833465e-09i 1.253781e-10  -99.01778    11
## 6     72000  2.046997e-05-3.101531e-07i 4.191160e-10  -93.77666    11

For a canonical shape, compare the stored monostatic result with the matching shape-specific model before relying on post-processed angular products.

Explicit T-matrix storage

Explicit per-frequency modal-block storage is enabled with store_t_matrix = TRUE.

sphere_store <- target_strength(
  object = fls_generate(
    shape = sphere(radius_body = 0.01, n_segments = 80),
    g_body = 1,
    h_body = 1,
    theta_body = pi / 2
  ),
  frequency = 38e3,
  model = "tmm",
  boundary = "pressure_release",
  density_sw = density_sw,
  sound_speed_sw = sound_speed_sw,
  store_t_matrix = TRUE
)

length(sphere_store@model_parameters$TMM$parameters$t_matrix[[1]])

## [1] 10

For prolate spheroids, the stored blocks live in the same @model_parameters$TMM$parameters$t_matrix slot and can be reused through helpers such as tmm_scattering(), tmm_average_orientation(), tmm_bistatic_summary(), and tmm_products() without rebuilding the retained modal solve. Cylinders can also be stored, but the retained cylinder state is intentionally narrower: it reuses the exact geometry-matched cylindrical family only for exact monostatic evaluations and orientation-averaged monostatic products. Full general-angle cylinder grids and bistatic summaries are not exposed because a validated retained cylinder angular operator is missing.

Orientation averaging from stored blocks

One of the practical reasons to keep the result in a T-matrix framework is that the same stored blocks can be reused for other single-target summaries without rerunning the boundary solve. The simplest example is an orientation-averaged monostatic cross section.

prolate_store <- target_strength(
  object = fls_generate(
    shape = prolate_spheroid(
      length_body = 0.14,
      radius_body = 0.01,
      n_segments = 80
    ),
    density_body = 1028.9,
    sound_speed_body = 1480.3,
    theta_body = pi / 2
  ),
  frequency = seq(12e3, 100e3, 2e3),
  model = "tmm",
  boundary = "liquid_filled",
  density_sw = density_sw,
  sound_speed_sw = sound_speed_sw,
  store_t_matrix = TRUE
)

orientation_dist <- tmm_orientation_distribution(
  distribution = "uniform",
  lower = 0.45 * pi,
  upper = pi,
  n_theta = 7
)

orientation_avg <- tmm_average_orientation(
  object = prolate_store,
  distribution = orientation_dist
)

orientation_avg

##    frequency     sigma_bs         TS
## 1      12000 4.502163e-10  -93.46579
## 2      14000 6.702413e-10  -91.73769
## 3      16000 9.192786e-10  -90.36553
## 4      18000 1.170862e-09  -89.31494
## 5      20000 1.399276e-09  -88.54097
## 6      22000 1.548806e-09  -88.10003
## 7      24000 1.585990e-09  -87.99700
## 8      26000 1.469957e-09  -88.32695
## 9      28000 1.232136e-09  -89.09341
## 10     30000 9.347098e-10  -90.29323
## 11     32000 6.301750e-10  -92.00539
## 12     34000 3.607924e-10  -94.42743
## 13     36000 1.701901e-10  -97.69066
## 14     38000 8.946596e-11 -100.48342
## 15     40000 1.192773e-10  -99.23442
## 16     42000 2.704566e-10  -95.67902
## 17     44000 5.266602e-10  -92.78470
## 18     46000 8.330752e-10  -90.79316
## 19     48000 1.119975e-09  -89.50792
## 20     50000 1.305324e-09  -88.84282
## 21     52000 1.354505e-09  -88.68219
## 22     54000 1.285562e-09  -88.90907
## 23     56000 1.118337e-09  -89.51427
## 24     58000 8.827687e-10  -90.54153
## 25     60000 6.240222e-10  -92.04800
## 26     62000 3.913264e-10  -94.07461
## 27     64000 2.144002e-10  -96.68775
## 28     66000 1.343882e-10  -98.71639
## 29     68000 1.862644e-10  -97.29870
## 30     70000 3.663744e-10  -94.36075
## 31     72000 6.192308e-10  -92.08147
## 32     74000 8.730832e-10  -90.58944
## 33     76000 1.074125e-09  -89.68945
## 34     78000 1.193339e-09  -89.23236
## 35     80000 1.223384e-09  -89.12437
## 36     82000 1.161184e-09  -89.35099
## 37     84000 1.016937e-09  -89.92706
## 38     86000 8.210382e-10  -90.85637
## 39     88000 5.984286e-10  -92.22988
## 40     90000 3.856175e-10  -94.13843
## 41     92000 2.527349e-10  -95.97335
## 42     94000 2.403773e-10  -96.19107
## 43     96000 3.303472e-10  -94.81029
## 44     98000 4.884121e-10  -93.11214
## 45    100000 6.738344e-10  -91.71447

By default, tmm_average_orientation() assumes the exact monostatic receive direction for each supplied incident angle. So the helper is averaging the differential backscattering cross section over the chosen orientation distribution, not rebuilding the underlying T-matrix solve at every angle.

The distribution helper is there so the main orientation pathways stay explicit and standardized:

distribution = "uniform" creates a uniform distribution in theta_body over a bounded interval,
distribution = "normal" creates a normal distribution on [0, pi],
distribution = "truncated_normal" creates a normal distribution restricted to [lower, upper], and
distribution = "quadrature" or distribution = "pdf" lets the user supply their own angle grid and either direct weights or a density.

Because this helper is meant for monostatic orientation-averaged backscatter, its returned columns follow the same naming convention used elsewhere in the package: sigma_bs and TS.

Pre-rendered comparison of the stored monostatic TMM spectrum and the orientation-averaged TMM spectrum for the liquid-filled prolate spheroid example.

The orientation-averaged line can sit far below the stored monostatic line, and that is not a bug by itself. The stored monostatic spectrum in this example is the broadside response of one fixed orientation, while the orientation-averaged spectrum is a linear average of sigma_bs over a wide range of body angles. For a long prolate spheroid, broadside is typically much stronger than the more oblique and near-end-on views, so the average can be many decibels lower than the broadside-only curve.

Angular scattering slices from stored blocks

The same stored blocks can also be sent through plot(..., type = "scattering") to inspect a single angular slice of the far-field response at a chosen stored frequency.

Angular TMM scattering slice for a stored liquid-filled prolate spheroid at 70 kHz, plotted as differential scattering cross section in dB versus receive polar angle.

The angular slice shows how a stored operator distributes scattering over receive angle at one frequency.

All of the tmm_scattering()-style helpers use the body-fixed angular coordinates of the canonical axisymmetric target, not arbitrary world-frame directions. For the built-in prolate, oblate, and cylinder shapes in acousticTS, the symmetry axis is the body x-axis. So when these helpers are compared against an external BEM/FEM slice generated in a world x-y plane, the world directions need to be converted to the body-fixed (\theta, \phi) convention before the comparison is apples-to-apples.

Higher-level bistatic summaries

Once the stored blocks are available, the next natural step is to move beyond individual points and slices and ask for higher-level bistatic summaries. Within the documented package scope, that applies to the spherical and spheroidal stored branches. tmm_bistatic_summary() does that for one stored frequency by returning:

a forward-centered great-circle slice,
an orthogonal dorsal/ventral-style great-circle slice,
the peak-scattering direction on a 2D receive-angle grid,
the exact monostatic backscatter value,
a -3 dB backscatter-lobe width measured on the forward-centered slice, and
integrated scattering over coarse angular sectors measured relative to the forward direction.

summary_70 <- tmm_bistatic_summary(
  object = prolate_store,
  frequency = 70e3,
  n_theta = 21,
  n_phi = 41,
  n_psi = 41
)

summary_70$metrics

##   frequency forward_sigma_scat forward_sigma_scat_dB     sigma_bs        TS
## 1     70000        7.00483e-07             -61.54602 2.018169e-08 -76.95042
##   peak_theta_scatter peak_phi_scatter peak_psi_scatter peak_sigma_scat
## 1           1.570796         3.141593                0     7.00483e-07
##   peak_sigma_scat_dB backscatter_lobe_width
## 1          -61.54602               1.021018

summary_70$sector_integrals

##            sector  psi_min  psi_max integrated_sigma_scat
## 1  forward_sector 0.000000 1.047198          1.711232e-07
## 2  oblique_sector 1.047198 2.094395          1.659806e-08
## 3 backward_sector 2.094395 3.141593          1.427224e-08

These summaries reuse the stored blocks. The monostatic value in summary_70$metrics$TS should agree with the TS already stored at 70 kHz.

Single solved target, many post-process products

Use tmm_products() when several post-processing products are required from the same solved target.

products_70 <- tmm_products(
  object = prolate_store,
  frequency = 70e3,
  orientation = orientation_dist,
  bistatic_summary = TRUE,
  n_theta = 21,
  n_phi = 41,
  n_psi = 41
)

names(products_70)

## [1] "monostatic"          "orientation_average" "bistatic_summary"

products_70$bistatic_summary$metrics

##   frequency forward_sigma_scat forward_sigma_scat_dB     sigma_bs        TS
## 1     70000        7.00483e-07             -61.54602 2.018169e-08 -76.95042
##   peak_theta_scatter peak_phi_scatter peak_psi_scatter peak_sigma_scat
## 1           1.570796         3.141593                0     7.00483e-07
##   peak_sigma_scat_dB backscatter_lobe_width
## 1          -61.54602               1.021018

The stored operator can therefore supply monostatic, orientation-averaged, and bistatic products without repeating the target solve.

Bistatic scattering grids and polar maps

For two-dimensional receive-angle views, the stored blocks can be reused through tmm_scattering_grid() or displayed directly through plot(..., type = "scattering", polar = TRUE) or plot(..., type = "scattering", heatmap = TRUE). The implementation page uses pre-rendered figures for those heavier views so the article stays lightweight to build.

The pre-rendered gallery covers the branches that support angular grids: Sphere, OblateSpheroid, and ProlateSpheroid. Cylinder is omitted because its retained state does not expose a validated angular grid.

Shape gallery

These two views show the stored pressure-release sphere at 70 kHz. The heatmap uses the direct (\phi_\text{scatter}, \theta_\text{scatter}) grid, while the polar map wraps the same grid into a circular view.

Pre-rendered TMM scattering maps for the stored pressure-release sphere at 70 kHz. The left panel is a heatmap in phi-scatter and theta-scatter, and the right panel is the corresponding polar map.

These two views show the stored liquid-filled oblate spheroid at 70 kHz, which is a useful intermediate case because it still uses the spherical-coordinate retained branch but no longer has the exact spherical symmetry of the sphere.

Pre-rendered TMM scattering maps for the stored liquid-filled oblate spheroid at 70 kHz. The left panel is a heatmap in phi-scatter and theta-scatter, and the right panel is the corresponding polar map.

These two views show the stored liquid-filled prolate spheroid at 70 kHz, which is the geometry where the retained spheroidal angular operator is most directly useful.

Pre-rendered TMM scattering maps for the stored liquid-filled prolate spheroid at 70 kHz. The left panel is a heatmap in phi-scatter and theta-scatter, and the right panel is the corresponding polar map.

In each polar view, the radius is the scattering polar angle theta_scatter, not physical distance from the target. So the center corresponds to theta_scatter = 0, the outer ring corresponds to theta_scatter = pi, and the target itself is not drawn as a literal object at the origin.

Benchmark comparisons

The benchmark ladder is intentionally shape-specific and follows the exact-family references already used elsewhere in the package.

Sphere cases are compared against SPHMS
Oblate spheroids are checked through both the exact sphere limit and an external pressure-release BEMPP far-field slice for a genuinely nonspherical oblate case
Prolate spheroid cases are compared against PSMS
Cylinder cases are compared against FCMS for the default monostatic branch over the tested 12-200 kHz grids
Shell-sphere cases (ESS objects) are compared against the same published benchmark reference used for ESSMS

Benchmark tables

a (mm)	Boundary	\max \vert \Delta TS \vert ~ (\text{dB}) SPHMS	\vert \overline{\Delta TS} \vert ~ (\text{dB}) SPHMS	\max \vert \Delta TS \vert ~ (\text{dB}) BEM	\vert \overline{\Delta TS} \vert ~ (\text{dB}) BEM	\max \vert \Delta TS \vert ~ (\text{dB}) FEM	\vert \overline{\Delta TS} \vert ~ (\text{dB}) FEM
`5`	`fixed_rigid`	`1.66e-11`	`1.95e-12`	`—`	`—`	`—`	`—`
`5`	`pressure_release`	`1.76e-11`	`1.76e-12`	`—`	`—`	`—`	`—`
`5`	`liquid_filled`	`4.19e-11`	`6.17e-12`	`—`	`—`	`—`	`—`
`5`	`gas_filled`	`1.75e-11`	`1.75e-12`	`—`	`—`	`—`	`—`
`10`	`fixed_rigid`	`2.37e-10`	`2.75e-11`	`0.048`	`0.010`	`0.528`	`0.087`
`10`	`pressure_release`	`2.44e-10`	`2.91e-11`	`0.072`	`0.014`	`0.164`	`0.031`
`10`	`liquid_filled`	`1.15e-10`	`1.74e-11`	`3.873`	`0.046`	`3.599`	`0.025`
`10`	`gas_filled`	`2.45e-10`	`2.86e-11`	`0.030`	`0.004`	`0.007`	`0.0004`
`18`	`fixed_rigid`	`7.62e-10`	`8.32e-11`	`—`	`—`	`—`	`—`
`18`	`pressure_release`	`7.91e-10`	`8.49e-11`	`—`	`—`	`—`	`—`
`18`	`liquid_filled`	`4.96e-09`	`2.48e-10`	`—`	`—`	`—`	`—`
`18`	`gas_filled`	`8.46e-10`	`8.71e-11`	`—`	`—`	`—`	`—`

Boundary	\max \vert \Delta TS \vert ~ (\text{dB}) SPHMS	\vert \overline{\Delta TS} \vert ~ (\text{dB}) SPHMS	\max \vert \Delta TS \vert ~ (\text{dB}) BEM	\vert \overline{\Delta TS} \vert ~ (\text{dB}) BEM
`fixed_rigid`	`3.60e-10`	`4.97e-11`	`0.034`	`0.008`
`pressure_release`	`3.12e-10`	`5.06e-11`	`0.009`	`0.002`
`liquid_filled`	`1.09e-10`	`1.92e-11`	`31.14`	`0.497`
`gas_filled`	`3.12e-10`	`5.07e-11`	`0.023`	`0.009`

L (mm)	a (mm)	Boundary	\max \vert \Delta TS \vert ~ (\text{dB}) PSMS	\vert \overline{\Delta TS} \vert ~ (\text{dB}) PSMS	\max \vert \Delta TS \vert ~ (\text{dB}) BEM	\vert \overline{\Delta TS} \vert ~ (\text{dB}) BEM
`60`	`8`	`fixed_rigid`	`0.00000`	`0.00000`	`—`	`—`
`60`	`8`	`pressure_release`	`0.00000`	`0.00000`	`—`	`—`
`60`	`8`	`liquid_filled`	`0.00000`	`0.00000`	`—`	`—`
`60`	`8`	`gas_filled`	`0.00000`	`0.00000`	`—`	`—`
`140`	`10`	`fixed_rigid`	`0.00000`	`0.00000`	`16.66`	`0.068`
`140`	`10`	`pressure_release`	`0.00000`	`0.00000`	`0.105`	`0.010`
`140`	`10`	`liquid_filled`	`0.00000`	`0.00000`	`58.26`	`1.378`
`140`	`10`	`gas_filled`	`0.00000`	`0.00000`	`0.295`	`0.028`

L (mm)	a (mm)	Boundary	\max \vert \Delta TS \vert ~ (\text{dB}) FCMS	\vert \overline{\Delta TS} \vert ~ (\text{dB}) FCMS	\max \vert \Delta TS \vert ~ (\text{dB}) BEM	\vert \overline{\Delta TS} \vert ~ (\text{dB}) BEM
`70`	`10`	`fixed_rigid`	`0.00e+00`	`0.00e+00`	`63.4`	`15.2`
`70`	`10`	`pressure_release`	`0.00e+00`	`0.00e+00`	`38.9`	`12.4`
`70`	`10`	`liquid_filled`	`0.00e+00`	`0.00e+00`	`74.9`	`14.3`
`70`	`10`	`gas_filled`	`0.00e+00`	`0.00e+00`	`39.7`	`12.3`
`50`	`8`	`fixed_rigid`	`3.55e-15`	`5.08e-16`	`—`	`—`
`50`	`8`	`pressure_release`	`0.00e+00`	`0.00e+00`	`—`	`—`
`50`	`8`	`liquid_filled`	`0.00e+00`	`0.00e+00`	`—`	`—`
`50`	`8`	`gas_filled`	`0.00e+00`	`0.00e+00`	`—`	`—`

Boundary	\max \vert \Delta TS \vert ~ (\text{dB}) bistatic grid	\vert \overline{\Delta TS} \vert ~ (\text{dB}) bistatic grid	\max \vert \Delta TS \vert ~ (\text{dB}) published monostatic	\vert \overline{\Delta TS} \vert ~ (\text{dB}) published monostatic
`shelled_pressure_release`	`2.1e-14`	`1.9e-15`	`4.97e-03`	`2.43e-03`
`shelled_gas`	`2.1e-14`	`2.4e-15`	`5.00e-03`	`2.58e-03`
`shelled_liquid`	`8.9e-11`	`3.0e-12`	`4.94e-03`	`2.50e-03`
`elastic_shelled`	`0.012`	`0.001`	`—`	`—`

Diagnostics and internal validation

Exact-family comparisons are the first validation ladder, but they are not the whole story for TMM. For the newer branches, it is also useful to check whether the retained operator is internally self-consistent. The tmm_diagnostics() helper does that by reusing the stored blocks and reporting:

monostatic reconstruction residuals,
reciprocity residuals under incident/receive-angle exchange,
an optical-theorem residual for the spherical-coordinate branch, and
block-level conditioning summaries.

For the rigid and pressure-release prolate branches, the stored-block TMM angular field can be checked directly against the exact general-angle spheroidal modal-series solution used by PSMS. That is the fastest way to constrain off-axis disagreements, because it removes external meshing and solver conventions from the loop and asks only whether the retained prolate operator reproduces the exact scalar spheroidal field at the same incident and receive angles.

diag_70 <- tmm_diagnostics(
  prolate_store,
  n_theta = 31,
  n_phi = 61
)

diag_70$summary

Independent BEMPP cross-checks were available for the pressure-release sphere, oblate, and prolate branches. The cylinder situation is different because the retained-cylinder scope is intentionally monostatic-only. So the validation ladder for these branches combines:

exact-family checks where they exist (SPHMS, PSMS, FCMS),
limiting-case checks such as the oblate sphere limit, and
theorem-based diagnostics from the stored T-matrix blocks.

The table below shows representative diagnostic summaries for one stored object from each supported branch, where e is the residual.

Case	\mathscr{f}~(\text{kHz})	Max monostatic e	Max reciprocity e	Max optical-theorem e	Min block `rcond`	Max block transpose e
sphere, `fixed_rigid`	`38, 70`	`2.55e-16`	`2.12e-16`	`1.24e-03`	`3.11e-09`	`1.93e-16`
oblate spheroid, `liquid_filled`	`38, 70`	`2.80e-16`	`1.14e-04`	`1.34e-03`	`2.24e-16`	`3.48e-01`
prolate spheroid, `liquid_filled`	`38, 70`	`3.28e-16`	`2.64e-12`	`NA`	`3.35e-47`	`1.15e-13`
finite cylinder, `fixed_rigid`	`38, 70`	`0.00e+00`	`NA`	`NA`	`NA`	`NA`

Two points are worth emphasizing.

The sphere and prolate branches are numerically self-consistent to essentially machine precision on the tested grids.
The stored-cylinder branch is deliberately narrower than the sphere and prolate branches. It carries only the geometry-matched monostatic reuse path, so reciprocity and optical-theorem-style angular diagnostics are left as NA rather than being computed from an unvalidated cylinder grid operator.

Sphere-to-spheroid continuation

For spheroidal targets, one of the most useful internal checks is to start from the equal-volume sphere limit and then deform the target smoothly to the requested aspect ratio. The monostatic response should move smoothly along that path. Abrupt jumps, non-finite values, or strong zig-zag second differences at modest aspect ratio are a sign that the truncation or angular reconstruction is not under control.

This check is built into tmm_diagnostics() for prolate and oblate targets. The continuation is generated at constant volume, beginning from the exact sphere limit and stepping to the requested spheroidal aspect ratio.

The figure below shows the built-in continuation path for a pressure-release prolate spheroid with target aspect ratio 3.5.

Sphere-to-spheroid continuation path for a pressure-release prolate spheroid. The exact sphere limit is at aspect ratio 1, and the target spheroid is at aspect ratio 3.5.

This continuation check can reveal an unstable retained branch before an external mesh or far-field normalization is introduced.

Exact prolate angular validation

For the retained prolate branch, the most useful validation is not a paper-style sketch and not an external mesh-based comparison. It is a direct check against the exact general-angle spheroidal solution already available in the package for rigid and pressure-release scalar prolates. That is the right apples-to-apples comparison because it uses:

the same geometry,
the same incident and receive-angle definitions,
the same scalar boundary conditions, and
the same far-field normalization.

The retained prolate operator is checked by comparing tmm_scattering() with prolate_spheroid_fbs() at the same frequency and angles.

The figure below shows that check for a rigid and a pressure-release prolate spheroid with L = 70 mm, a = 10 mm, broadside incidence, and a 38 kHz equatorial receive-angle sweep. The line is the exact general-angle spheroidal solution and the open symbols are the stored-block TMM reconstruction.

Exact-vs-stored validation for rigid and pressure-release prolate spheroids at 38 kHz, shown as equatorial receive-angle sweeps under broadside incidence.

This directly tests the retained angular operator that powers tmm_scattering(), tmm_scattering_grid(), and the higher-level post-processing helpers.

That body-fixed angle convention also resolves the mismatch with the exploratory BEMPP prolate slice.

External comparison tables

For the pressure-release prolate case with L = 70 mm, a = 10 mm, and 38 kHz, the retained TMM branch and the external BEMPP far-field solution line up closely across the tested incidence angles:

World-frame incidence (deg)	Max abs. \Delta amplitude (dB)	Mean abs. \Delta amplitude (dB)	Note
`0`	`0.0069`	`0.0033`	pressure-release, BEMPP far-field
`45`	`0.0076`	`0.0031`	pressure-release, BEMPP far-field
`90`	`0.0078`	`0.0030`	pressure-release, BEMPP far-field

The same body-fixed/world-frame conversion is also important for the nonspherical oblate branch, which uses the spherical-coordinate retained operator. For a pressure-release oblate with polar semiaxis c = 6 mm and equatorial semiaxis a = 10 mm, the corrected BEMPP comparison is likewise tight on the tested cases:

Frequency (kHz)	World-frame incidence (deg)	Max abs. \Delta amplitude (dB)	Mean abs. \Delta amplitude (dB)	Validation note
`38`	`0`	`0.0091`	`0.0054`	pressure-release, BEMPP far-field
`38`	`45`	`0.0083`	`0.0053`	pressure-release, BEMPP far-field
`38`	`90`	`0.0079`	`0.0051`	pressure-release, BEMPP far-field
`70`	`90`	`0.0091`	`0.0047`	pressure-release, BEMPP far-field

By contrast, the same corrected BEMPP comparison also clarifies why the package does not expose a full retained-cylinder angular operator. Additional pressure-release cylinder checks were run at a second frequency and a second geometry, and the exploratory retained-cylinder angular reconstruction remained much less reliable than the sphere and prolate branches:

L (mm)	a (mm)	Frequency (kHz)	World-frame incidence (deg)	Stored-cylinder `n_max`	Max abs. \Delta amplitude (dB)	Mean abs. \Delta amplitude (dB)
`70`	`10`	`38`	`90`	`24`	`1.58`	`0.74`
`70`	`10`	`70`	`90`	`24`	`8.12`	`3.73`
`50`	`8`	`38`	`90`	`24`	`0.71`	`0.29`

The exact monostatic cylinder branch can also be compared to the same BEMPP slices at the single backscatter point:

L (mm)	a (mm)	Frequency (kHz)	World-frame incidence (deg)	Backscatter-point \Delta amplitude (dB)	Interpretation
`70`	`10`	`38`	`90`	`-0.128`	broadside is the best tested case here, but it stays above a `< 0.1 dB` target
`70`	`10`	`70`	`90`	`-0.063`	broadside falls below `0.1 dB` on this tested case
`50`	`8`	`38`	`90`	`-0.198`	the smaller broadside case stays above a `< 0.1 dB` target
`70`	`10`	`38`	`45`	`-5.104`	oblique monostatic agreement is not externally closed
`70`	`10`	`38`	`0`	`-272.091`	the end-on pressure-release cylinder remains strongly inconsistent with BEMPP

Those extra checks matter because they show the remaining cylinder issue is not just one broken benchmark file. A retained cylinder angular operator can sometimes be tuned to look better on a low-to-moderate-frequency case, but that improvement does not generalize cleanly once the frequency or geometry changes. So the external benchmark work supports a more precise conclusion: the retained prolate branch behaves correctly, while the cylinder family stops at exact monostatic reuse and orientation-averaged monostatic products until a separate validated retained cylinder angular operator is built.

External BEM validation figures

Each figure shows TMM minus the external reference (ΔTS, dB) across the bistatic grid at 12, 38, 70, 120 kHz. The small circle-and-arrow marker is a legend marking the incident direction, not data.

External BEM figures

Fixed rigid

Pressure release

Liquid filled

Gas filled

Fixed rigid

Pressure release

Gas filled

Fixed rigid

Pressure release

Liquid filled

Shelled pressure release

Shelled gas

Shelled liquid

Elastic shelled

Pre-rendered pressure-release cylinder comparison against BEMPP for the exploratory retained-angle cylinder TMM branch. The figure shows 38 kHz world-frame x-y slice comparisons at 0, 45, and 90 degree incidence, with BEMPP and exploratory retained-angle TMM overlays and the corresponding amplitude residual curves.

For the cylinder, the analogous full-angle figure is shown here only as a constraint on the exploratory retained-angle cylinder operator. It is not documenting a supported public cylinder workflow. The point of the figure is to show why the package refuses to provide full retained cylinder grids and bistatic summaries instead of pretending they are validated.

The external comparisons support the following conclusions:

the sphere has been compared with BEM/FEM across all four boundaries,
pressure-release oblate and prolate angular fields agree closely after the frame convention is matched,
some penetrable BEM comparisons contain large maxima near nulls and should be interpreted with the exact-family and mean-residual columns,
the exploratory retained-angle cylinder branch does not agree externally and is therefore not part of the documented public retained-cylinder workflow, and
even the exact monostatic pressure-release cylinder branch is only externally close near broadside on the tested cases, so cylinder remains the least externally constrained TMM shape family and emits a warning by default when used.

At this point, no direct paper-figure reproduction is claimed here for the prolate angular response. If a future paper-grounded reproduction is added, it needs to match the original paper’s normalization, axes, and geometry conventions exactly rather than only mimicking the general visual style.

Representative spectra

Two-panel representative TMM benchmarks: liquid-filled sphere against SPHMS over 12 to 120 kHz, and liquid-filled prolate spheroid against PSMS over 12, 18, 38, 70, and 100 kHz. The upper panels overlay the spectra and the lower panels show the TMM minus reference difference.

The sphere spectrum checks agreement with SPHMS. The prolate spectrum checks agreement with the spheroidal solution used by PSMS.

References

Waterman, P. C. 1969. “New Formulation of Acoustic Scattering.” The Journal of the Acoustical Society of America 45 (6): 1417–29. https://doi.org/10.1121/1.1911619.

Waterman, P. C. 2009. “T -Matrix Methods in Acoustic Scattering.” The Journal of the Acoustical Society of America 125 (1): 42–51. https://doi.org/10.1121/1.3035839.

acousticTS implementation

Why use TMM over SPHMS or PSMS?

Object generation

Calculating a TS-frequency spectrum

Extracting model results

Plotting results

Accessing results

Explicit T-matrix storage

Orientation averaging from stored blocks

Angular scattering slices from stored blocks

Higher-level bistatic summaries

Single solved target, many post-process products

Bistatic scattering grids and polar maps

Shape gallery

Benchmark comparisons

Benchmark tables

Diagnostics and internal validation

Sphere-to-spheroid continuation

Exact prolate angular validation

External comparison tables

External BEM validation figures

External BEM figures

Representative spectra

References

Why use `TMM` over `SPHMS` or `PSMS`?