Calculates the far-field scattering amplitude and related quantities for a solid elastic (calibration) sphere using a modal series solution.
Usage
This model is accessed via:
target_strength(
...,
model="calibration",
sound_speed_sw,
density_sw,
adaptive = TRUE
)Arguments
sound_speed_swSeawater sound speed (\(m~s^{-1}\)).
density_swSeawater density (\(kg~m^{-3}\)).
adaptiveLogical. If
TRUE, extend the partial-wave sum beyond the initial \(\mathrm{round}(ka)+10\) modal cap until the tail term falls below the internal convergence threshold. IfFALSE, use the original fixed modal cutoff only.
Theory
The calibration sphere model computes acoustic scattering from a solid elastic sphere by expanding the incident and scattered fields in terms of spherical Bessel and Hankel functions and Legendre polynomials. Both compressional and shear waves within the sphere are included, and the appropriate boundary conditions are enforced at the sphere-water interface.
The dimensionless frequency parameter is defined as \(q = ka\), where \(k\) is the wavenumber in water and \(a\) is the sphere radius. The longitudinal and transverse wave numbers inside the sphere are
$$ q_1 = \frac{qc}{c_1} \\ q_2 = \frac{qc}{c_2}, $$
where \(c_1\) and \(c_2\) are the longitudinal and transverse sound speeds in the sphere, respectively.
The far-field backscattering form function is given by:
$$ f_\infty(q) = -\frac{2}{q} \sum\limits_{\ell=0}^{\infty} (-1)^\ell (2\ell+1) \sin \eta_\ell \exp(i \eta_\ell) $$
The phase angle for each mode is then given by:
$$ \tan \eta_\ell = -\frac{B_2 j_\ell'(q) - B_1 j_\ell(q)}{B_2 y_\ell'(q) - B_1 y_\ell(q)} $$
where the phase angle \(\eta_\ell\) is determined by the boundary conditions and the material properties of the sphere and surrounding fluid. The phase angle \(\eta_\ell\) is calculated using a series of coefficients. The auxilliary quantities used for determining the boundary conditions and subsequently \(\eta_\ell\) are reported in MacLennan (1981).
References
Hickling, R. (1962). Analysis of echoes from a solid elastic sphere in water. The Journal of the Acoustical Society of America, 34: 1582-1592.
MacLennan D. N. (1981). The theory of solid spheres as sonar calibration targets. Scottish Fisheries Research No. 22, Department of Agriculture and Fisheries for Scotland.