Bent cylinder modal series solution

acousticTS implementation

Unvalidated Experimental

Overview Theory

BCMS applies a curvature-dependent coherence correction to the finite-cylinder modal series (Stanton 1989, 1988).

The bent-cylinder modal-series solution is available through target_strength(..., model = "bcms"). It is intended for slender, weakly scattering cylinders whose cross-section is still well represented by the finite-cylinder modal-series family, but whose centerline curvature reduces backscattering coherence relative to the straight-cylinder case.

Use FCMS for a straight finite cylinder. Use BCMS when the target remains locally cylindrical but centerline curvature changes its along-axis coherence.

BCMS is marked unvalidated because there is not an external public software benchmark ladder wired into the package. The implementation checks documented here are the two defining identities of the model: the straight branch must reduce to FCMS, and the bent branch must match the Stanton (1989) coherence-corrected construction built from that same straight-cylinder solution.

Building straight and bent reference objects

The reference case used throughout this page is a weakly scattering liquid-filled cylinder with:

• length 10.5 mm
• radius 1 mm
• density contrast g = 1.0357
• sound-speed contrast h = 1.0279
• broadside incidence
• 12-400 kHz in 2 kHz steps

Two geometries are compared:

• a straight finite cylinder
• a bent cylinder with rho_c / L = 1.5

In acousticTS, the objects are built as:

library(acousticTS)

density_sw <- 1026.8
sound_speed_sw <- 1477.3

straight_shape <- cylinder(
  length_body = 10.5e-3,
  radius_body = 1e-3,
  n_segments = 401
)

straight_object <- fls_generate(
  shape = straight_shape,
  density_body = density_sw * 1.0357,
  sound_speed_body = sound_speed_sw * 1.0279,
  theta_body = pi / 2
)

bent_object <- brake(
  straight_object,
  radius_curvature = 1.5
)

straight_object

## FLS-object
##  Fluid-like scatterer 
##  ID:UID
## Body dimensions:
##  Length:0.011 m(n = 401 cylinders)
##  Mean radius:0.001 m
##  Max radius:0.001 m
## Shape parameters:
##  Defined shape:Cylinder
##  L/a ratio:10.5
##  Taper order:N/A
## Material properties:
##  Density: 1063.4568 kg m^-3 | Sound speed: 1518.5167 m s^-1
## Body orientation (relative to transducer face/axis):1.571 radians

bent_object

## FLS-object
##  Fluid-like scatterer 
##  ID:UID
## Body dimensions:
##  Length:0.01 m(n = 401 cylinders)
##  Mean radius:0.001 m
##  Max radius:0.001 m
## Shape parameters:
##  Defined shape:Cylinder
##  L/a ratio:10.5
##  Taper order:N/A
## Material properties:
##  Density: 1063.4568 kg m^-3 | Sound speed: 1518.5167 m s^-1
## Body orientation (relative to transducer face/axis):1.571 radians

The straight object carries the same geometric information that would be passed into FCMS. The bent object keeps the same local cylindrical radius and material contrasts, but adds the curvature ratio needed by the bent-cylinder coherence correction. That separation is important because it makes clear what BCMS is changing: the local cross-sectional scattering physics remain cylindrical, while the along-body coherence is modified by curvature.

Calculating TS-frequency spectra

The same BCMS call can be applied to either geometry:

frequency <- seq(12e3, 400e3, by = 2e3)

straight_object <- target_strength(
  object = straight_object,
  frequency = frequency,
  model = "bcms",
  density_sw = density_sw,
  sound_speed_sw = sound_speed_sw
)

bent_object <- target_strength(
  object = bent_object,
  frequency = frequency,
  model = "bcms",
  density_sw = density_sw,
  sound_speed_sw = sound_speed_sw
)

For a straight cylinder, the BCMS result should collapse onto the straight finite-cylinder modal-series result. For a bent cylinder, the result should show the same basic modal structure, but with the coherence reductions expected from curvature.

Extracting model results

Model results can be inspected visually or pulled directly from the stored object.

Plotting results

The geometry panel is useful because BCMS is only meaningful when the object is still locally cylinder-like. The spectrum panel then shows the corresponding acoustic effect of that curvature: the bent-cylinder response remains related to the straight-cylinder result, but the backscattered level is reduced by the coherence loss built into the model.

Accessing results

bcms_results <- extract(bent_object, "model")$BCMS
head(bcms_results)

##   frequency         ka                        f_bs     sigma_bs        TS
## 1     12000 0.05103786 -8.271248e-07-2.430843e-08i 6.847263e-13 -121.6448
## 2     14000 0.05954416 -1.124880e-06-3.854701e-08i 1.266842e-12 -118.9728
## 3     16000 0.06805047 -1.467840e-06-5.745290e-08i 2.157856e-12 -116.6598
## 4     18000 0.07655678 -1.855750e-06-8.167087e-08i 3.450479e-12 -114.6212
## 5     20000 0.08506309 -2.288324e-06-1.118381e-07i 5.248936e-12 -112.7993
## 6     22000 0.09356940 -2.765245e-06-1.485835e-07i 7.668657e-12 -111.1528

The stored data.frame contains the modeled frequency, the complex backscattering amplitude f_bs, the linear backscattering cross-section sigma_bs, and TS. For BCMS, it is often helpful to inspect these results alongside the corresponding straight-cylinder case, because the defining behavior of the model is the way curvature changes coherence relative to the straight-cylinder baseline.

Implementation checks

Straight-cylinder reduction

Reference relation	Max abs. \Delta TS (dB)	Mean abs. \Delta TS (dB)	f at max \Delta (kHz)
BCMS straight branch vs FCMS	0	0	12

With no curvature, BCMS must reduce to FCMS. The residual verifies that the curvature correction drops out in this limit.

Bent-cylinder coherence relation

Reference relation	Max abs. \Delta TS (dB)	Mean abs. \Delta TS (dB)	f at max \Delta (kHz)
BCMS bent branch vs Stanton (1989) coherence construction	0	0	12

This is the second defining identity. The bent-cylinder branch is expected to match the straight-cylinder modal-series result multiplied by the Fresnel-style coherence factor implied by Stanton (1989, Eq. 25-26). The check is therefore not asking whether BCMS matches itself. It is asking whether the implemented bent-cylinder branch reproduces the explicit coherence-corrected construction that defines the model.

Spectrum overlay

The overlay shows the straight branch on the FCMS baseline and the bent branch on the coherence-corrected reference.

Closing note

BCMS is a curvature correction to FCMS. Its two implementation checks establish the straight-cylinder limit and the bent-cylinder coherence relation. They do not replace an external benchmark ladder.

References

Stanton, T. K. 1988. “Sound Scattering by Cylinders of Finite Length. I. Fluid Cylinders.” The Journal of the Acoustical Society of America 83 (1): 55–63. https://doi.org/10.1121/1.396184.

Stanton, T. K. 1989. “Sound Scattering by Cylinders of Finite Length. III. Deformed Cylinders.” The Journal of the Acoustical Society of America 86 (2): 691–705. https://doi.org/10.1121/1.398193.