Introduction
The viscous-elastic spherical model describes a layered resonant target whose gas core is surrounded first by an elastic shell and then by a viscous biological layer before the whole target is embedded in seawater. In the mesopelagic-fish setting of Khodabandeloo et al. (2021), those layers represent gas, a mechanically stiffer shell-like inclusion, and soft tissue whose viscosity damps and broadens the gas-driven resonance (Khodabandeloo et al. 2021; Feuillade and Nero 1998).
This family is still spherical, so the angular dependence remains
separable. What makes it more intricate than SPHMS,
ESSMS, or a simple bubble model is that four regions must
be matched simultaneously and two of those regions support more than one
wave family.
The notation here follows the shared package convention introduced in
the acoustic
scattering primer and notation
guide: medium 1 is the surrounding seawater, medium
2 is the viscous outer layer, medium 3 is the
elastic shell, and medium 4 is the gas core.
Layered geometry and medium indexing
Nested spherical regions
The target consists of three nested interfaces:
R_2 > R_3 > R_4,
Here R_2 is the outer radius of the viscous layer, R_3 is the outer radius of the elastic shell, and R_4 is the radius of the gas core.
The model therefore contains three interfaces:
- seawater-viscous at r = R_2,
- viscous-shell at r = R_3,
- shell-gas at r = R_4.
Neutral-buoyancy relation
In the motivating mesopelagic-fish formulation, the outer radius of the viscous layer can be related to a neutral-buoyancy balance. Written in the present medium indexing, that relation is:
R_2 = R_4 \left( 1 + \frac{\rho_1 - \rho_4}{\rho_2 - \rho_1} \right)^{1/3},
where \rho_1, \rho_2, and \rho_4 are the densities of seawater, the viscous layer, and the gas core, respectively. This is not part of the modal solution itself; it is a physical closure used when the outer viscous radius is estimated from a buoyancy argument rather than specified independently.
Governing fields in the four regions
Exterior seawater
In the surrounding fluid, the scattered and incident acoustic pressures satisfy the exterior Helmholtz problem:
\nabla^2 p_1 + k_1^2 p_1 = 0, \qquad k_1 = \frac{\omega}{c_1},
with c_1 the seawater sound speed and \omega the angular frequency.
Viscous layer
The outer tissue layer is modeled as a viscous compressible medium that supports a damped compressional branch and a damped shear branch. Using \eta_2 for shear viscosity and \zeta_2 for bulk viscosity, define the effective compressional viscous combination:
f_2 = \zeta_2 + \frac{4}{3}\eta_2.
The corresponding complex viscous-layer wavenumbers are:
k_{L,2} = \frac{\omega}{c_2} \left( 1 - i\omega \frac{f_2}{\rho_2 c_2^2} \right)^{-1/2},
and:
k_{T,2} = (1+i)\sqrt{\frac{\omega \rho_2}{2\eta_2}},
where c_2 and \rho_2 are the compressional sound speed and density of the viscous layer.
The essential point is that the outer tissue is not treated as a simple fluid contrast. Its viscous properties give the compressional and shear branches complex wavenumbers, which is how attenuation and resonance broadening enter the model.
Elastic shell
The shell is treated as a homogeneous isotropic elastic medium. Its displacement field \mathbf{u}_3 satisfies the Navier equation:
(\lambda_3 + 2\mu_3)\nabla(\nabla\cdot\mathbf{u}_3) - \mu_3 \nabla\times(\nabla\times\mathbf{u}_3) + \rho_3 \omega^2 \mathbf{u}_3 = 0,
where \lambda_3 and \mu_3 are Lamé parameters and \rho_3 is shell density.
The shell therefore supports longitudinal and transverse elastic waves with wavenumbers:
k_{L,3} = \omega \sqrt{\frac{\rho_3}{\lambda_3 + 2\mu_3}}, \qquad k_{T,3} = \omega \sqrt{\frac{\rho_3}{\mu_3}}.
Spherical modal representation
Because every interface is spherical, the field equations separate by angular order. The angular dependence is therefore carried by the Legendre polynomials P_m(\cos\theta), and each region can be expanded mode by mode.
Exterior acoustic field
For an incident plane wave aligned with the polar axis, the exterior total pressure is written as:
p_1(r,\theta) = \sum_{m=0}^{\infty} \left[ A_m^{(\mathrm{inc})} j_m(k_1 r) + A_m^{(\mathrm{sca})} h_m^{(1)}(k_1 r) \right] P_m(\cos\theta),
where j_m is the spherical Bessel function, h_m^{(1)} is the outgoing spherical Hankel function, and A_m^{(\mathrm{sca})} is the unknown scattered coefficient of order m.
Viscous-layer potentials
The viscous layer occupies an annulus R_3 < r < R_2, so both regular and singular radial solutions are admissible there. A convenient representation is to decompose the motion into compressional and shear potentials:
\Phi_2(r,\theta) = \sum_{m=0}^{\infty} \left[ B_m j_m(k_{L,2} r) + C_m y_m(k_{L,2} r) \right] P_m(\cos\theta)
\Psi_2(r,\theta) = \sum_{m=0}^{\infty} \left[ D_m j_m(k_{T,2} r) + E_m y_m(k_{T,2} r) \right] P_m(\cos\theta),
where y_m is the spherical Neumann function. The viscous-layer displacement and stresses are reconstructed from these potentials.
Elastic-shell potentials
The shell also occupies an annulus, so it likewise uses longitudinal and transverse potentials with both radial branches retained:
\Phi_3(r,\theta) = \sum_{m=0}^{\infty} \left[ F_m j_m(k_{L,3} r) + G_m y_m(k_{L,3} r) \right] P_m(\cos\theta)
\Psi_3(r,\theta) = \sum_{m=0}^{\infty} \left[ H_m j_m(k_{T,3} r) + I_m y_m(k_{T,3} r) \right] P_m(\cos\theta).
Gas-core field
The gas core contains the origin, so only the regular branch is admissible:
p_4(r,\theta) = \sum_{m=0}^{\infty} J_m j_m(k_4 r) P_m(\cos\theta).
This layered modal bookkeeping is why the spherical geometry remains tractable: all angular coupling disappears, and each mode is solved independently.
Interface conditions
Seawater-viscous interface at r = R_2
At the outer interface, the exterior fluid can support only acoustic pressure and normal fluid velocity, whereas the viscous layer can support normal and tangential stresses. The interface therefore enforces:
- continuity of normal traction,
- continuity of normal velocity,
- vanishing tangential traction on the seawater side.
In schematic form:
p_1 = -\sigma_{rr}^{(2)}, \qquad v_{r,1} = v_{r,2}, \qquad \sigma_{r\theta}^{(2)} = 0.
Viscous-shell interface at r = R_3
At the viscous-shell interface, both media can support normal and tangential motion and both can support traction. The interface therefore enforces:
u_{r,2} = u_{r,3}, \qquad u_{\theta,2} = u_{\theta,3}, \qquad \sigma_{rr}^{(2)} = \sigma_{rr}^{(3)}, \qquad \sigma_{r\theta}^{(2)} = \sigma_{r\theta}^{(3)}.
This is the main coupling point between viscous damping and the shell’s elastic resonance structure.
Shell-gas interface at r = R_4
At the inner interface, the gas cannot support shear traction. The shell-gas conditions are therefore:
\sigma_{rr}^{(3)} = -p_4, \qquad v_{r,3} = v_{r,4}, \qquad \sigma_{r\theta}^{(3)} = 0.
The gas core is thus coupled to the shell only through pressure and radial motion.
Mode-wise linear systems
Substituting the modal expansions into the interface conditions yields one independent linear system for each angular order m:
\mathbf{M}_m \mathbf{x}_m = \mathbf{F}_m.
The structure of \mathbf{x}_m depends on whether shear terms are active: for the monopole term m = 0, the tangential and shear branch drops out and the system reduces to a smaller 6\times 6 problem, whereas for m \ge 1 the coupled viscous and elastic shear branches are active and the system becomes a 10\times 10 problem.
This mode-wise reduction is one of the main advantages of the
spherical layering. The algebra per mode is more involved than for
SPHMS or ESSMS, but there is still no
cross-coupling between distinct angular orders.
Far-field backscatter
The exterior scattered field uses the outgoing spherical Hankel branch, so the far-field amplitude depends only on the exterior coefficients A_m^{(\mathrm{sca})}. Using the standard large-r asymptotics:
f_{\mathrm{bs}} = -\frac{i}{k_1} \sum_{m=0}^{M} (2m+1)(-1)^m A_m^{(\mathrm{sca})},
where the (-1)^m factor comes from evaluating the Legendre polynomials in the backscattering direction.
The backscattering cross-section and target strength are then (MacLennan, Fernandes, and Dalen 2002; Urick 1983; Simmonds and MacLennan 2005):
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}\right|^2, \qquad \mathrm{TS} = 10\log_{10}\left(\sigma_{\mathrm{bs}}\right).
Physical interpretation
The layered structure has a clear physical interpretation.
Gas resonance
The gas core supplies the main low-frequency compressibility contrast and is responsible for the strongest resonance tendency. In that sense, the gas plays the same qualitative role as a bubble or pressure-release cavity, but it is no longer isolated from the surrounding fluid directly.
Mathematical assumptions and scope
The model rests on the following assumptions:
- perfectly spherical concentric geometry,
- homogeneous material properties within each region,
- linear acoustics and linear elasticity,
- time-harmonic steady-state forcing,
- no nonspherical posture, taper, or internal asymmetry.
That combination makes VESM a layered spherical
resonance model, not a general fish-body solver. Its strength is that it
retains the physics of gas compression, shell elasticity, and viscous
damping within one separable spherical framework.