Introduction
This family is best read alongside the swimbladder-less fish and composite-scatterer literature that motivates explicit flesh-body and backbone terms (Gorska, Ona, and Korneliussen 2005; Stanton et al. 1998; Clay and Horne 1994).
The body-backbone fish model (BBFM) is a composite
scattering family for swimbladder-less targets whose flesh body and
backbone should remain explicit, separately parameterized contributors.
The point of the model is not to solve a fully coupled three-medium
boundary-value problem exactly. The point is to keep the two dominant
anatomical components acoustically visible in one coherent backscatter
calculation.
That makes BBFM conceptually closer to the hybrid logic
of KRM than to a single-region canonical modal
solution:
- the flesh body is treated as a weakly scattering fluid-like region,
- the backbone is treated as an elastic cylindrical structure, and
- the two terms are embedded into one body-fixed frame through a phase translation before their complex amplitudes are summed.
The result is a transparent composite model rather than a homogenized single-medium approximation.
Geometry and medium indexing
The family uses the shared package convention: medium 1
is the surrounding seawater, medium 2 is the flesh body,
and medium 3 is the backbone.
The exterior seawater wavenumber is:
k_1 = \frac{\omega}{c_1},
where c_1 is the seawater sound speed. The flesh density and sound-speed contrasts are therefore:
g_{21} = \frac{\rho_2}{\rho_1}, \qquad h_{21} = \frac{c_2}{c_1}.
For the backbone, the model keeps the absolute elastic properties explicit:
\rho_3, \qquad c_{L,3}, \qquad c_{T,3},
where c_{L,3} and c_{T,3} are the longitudinal and transverse wave speeds of the elastic backbone.
The important approximation enters immediately here: the backbone
term used here is not the exact solution for an elastic region
3 embedded inside flesh region 2. Instead, it
is a seawater-referenced elastic-cylinder surrogate that is then
positioned inside the same body frame as the flesh solve.
Flesh-body contribution
Weak-fluid assumption
The flesh component is treated as a weakly scattering fluid-like body. Its physics therefore follows the distorted-wave Born logic: the material contrasts relative to seawater are small enough that first-order scattering remains meaningful, while the body is still extended enough that phase accumulation along the body cannot be ignored.
Using the same contrast notation as the DWBA theory
page, the compressibility and density perturbations are:
\gamma_{\kappa,21} = \frac{\kappa_2 - \kappa_1}{\kappa_1}, \qquad \gamma_{\rho,21} = \frac{\rho_2 - \rho_1}{\rho_2}.
Schematic backscattering amplitude
At the volume-integral level, the flesh contribution may be written schematically as:
f_{\mathrm{bs}}^{(2)} = \frac{k_1^2}{4\pi} \iiint_{V_2} \left( \gamma_{\kappa,21} - \gamma_{\rho,21}\cos^2\beta \right) \exp\!\left(2 i \mathbf{k}_2\cdot \mathbf{r}\right) \, dV.
Here V_2 is the flesh-body volume, \mathbf{k}_2 is the distorted interior propagation vector, and \beta is the local angle between propagation direction and body tangent.
For the elongated axisymmetric bodies used in practice, this is
reduced to the usual one-dimensional DWBA body integral.
The important point for BBFM is not the exact quadrature
form, but the physical role of the term: it is the weak-fluid flesh-body
amplitude.
Backbone contribution
Elastic-cylinder surrogate
The backbone is represented as a finite elastic cylinder rather than
as another weak-fluid inclusion. Its local physics therefore follows the
same elastic cylinder modal logic used by ECMS.
The elastic interior supports both longitudinal and transverse waves, with wavenumbers:
k_{L,3} = \frac{\omega}{c_{L,3}}, \qquad k_{T,3} = \frac{\omega}{c_{T,3}}.
For each cylindrical modal order m, the elastic boundary conditions produce an order-dependent phase shift \eta_m. The finite-cylinder backscattering amplitude may then be written schematically as:
f_{\mathrm{bs}}^{(3)} = \frac{L_3}{\pi} \frac{\sin(k_1 L_3 \cos\theta_3)} {k_1 L_3 \cos\theta_3} \sum_{m=0}^{\infty} (-1)^m \epsilon_m \sin\eta_m e^{-i\eta_m}.
Here L_3 is backbone length, \theta_3 is the backbone incidence angle, \epsilon_m is the usual Neumann factor, and \eta_m collects the elastic-cylinder boundary-condition physics.
This term is what distinguishes BBFM from a pure
body-only weak-scattering model. The backbone is not just another
contrast perturbation. It is an explicit elastic structure with its own
internal wave-conversion physics.
Spatial placement in the body frame
The flesh and backbone amplitudes cannot be added meaningfully unless they are referred to the same spatial frame. The family uses the body-fixed coordinate system for that purpose.
If the representative backbone position is \mathbf{r}_c, then the backbone amplitude is translated into the body frame by the monostatic two-way phase factor:
\exp\!\left(2 i k_1 \hat{\mathbf{q}}_{\mathrm{bs}}\cdot\mathbf{r}_c\right).
where \hat{\mathbf{q}}_{\mathrm{bs}} is the backscatter direction.
In the axisymmetric body-frame convention used here, that projection becomes:
\hat{\mathbf{q}}_{\mathrm{bs}}\cdot\mathbf{r}_c = x_c\cos\theta + z_c\sin\theta.
with (x_c, z_c) the backbone centroid and \theta the stored body angle.
This is the step that embeds the backbone inside the same coordinate frame as the flesh solve. Without it, the model would implicitly assume that the flesh and backbone scatter from the same effective point.
Coherent composite amplitude
Once the flesh and backbone terms are available in a common frame, the total backscattering amplitude is:
f_{\mathrm{bs}}^{(\mathrm{BBFM})} = f_{\mathrm{bs}}^{(2)} + f_{\mathrm{bs}}^{(3)} \exp\!\left(2 i k_1 \hat{\mathbf{q}}_{\mathrm{bs}}\cdot\mathbf{r}_c\right).
This is the core BBFM statement. The family is coherent
because it adds the two complex amplitudes before squaring.
That distinction matters. If the model instead added cross-sections directly, all interference between flesh and backbone would be lost.
Cross-section and interference structure
The linear backscattering cross-section is:
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}^{(\mathrm{BBFM})}\right|^2,
and the target strength is (MacLennan, Fernandes, and Dalen 2002; Urick 1983; Simmonds and MacLennan 2005):
\mathrm{TS} = 10 \log_{10}\left(\sigma_{\mathrm{bs}}\right).
Expanding the squared magnitude makes the composite physics explicit:
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}^{(2)}\right|^2 + \left|f_{\mathrm{bs}}^{(3)}\right|^2 + 2\,\Re\!\left\{ f_{\mathrm{bs}}^{(2)} \overline{f_{\mathrm{bs}}^{(3)}} \exp\!\left( -2 i k_1 \hat{\mathbf{q}}_{\mathrm{bs}}\cdot\mathbf{r}_c \right) \right\}.
The third term is the interference term. It is frequency-dependent
and position-dependent, and it is the reason the composite
TS does not reduce to a simple sum of the flesh and
backbone TS curves.
What the family does and does not solve
Included physics
BBFM explicitly includes:
- a weak-fluid flesh-body contribution,
- an elastic backbone contribution,
- coherent interference between those two components through a shared body frame.
Excluded physics
BBFM does not yet solve:
- a true embedded elastic-cylinder-in-flesh transmission problem,
- repeated rescattering between flesh and backbone,
- shadowing or blockage of one component by the other,
- anatomical variability in backbone placement across an ensemble.
These omissions matter because they define exactly what
BBFM is: a component-resolved coherent model, not a full
coupled composite-wave solver.
Why this family is still useful
Even with those approximations, BBFM fills a real
modeling gap. A swimbladder-less fish often has flesh that behaves
broadly like a weakly scattering body and a backbone that is
acoustically much stiffer than the surrounding tissue.
Folding both of those into one effective region can hide the very
mechanism one is trying to study. BBFM therefore earns its
place not by being exact, but by keeping the dominant anatomical
contributors explicit while preserving coherent interference between
them.