Calculates the far-field scattering amplitude and related quantities for a finite cylinder using the modal series solution, supporting various boundary conditions (rigid, pressure-release, liquid-filled, and gas-filled).
Usage
This model is accessed via:
target_strength(
...,
model="fcms",
boundary,
sound_speed_sw,
density_sw,
m_limit
)Arguments
boundaryBoundary condition at a cylindrical surface. One of
"fixed_rigid","pressure_release","liquid_filled", or"gas_filled". See the boundary conditions documentation for more details on these different boundary conditions.sound_speed_swSeawater sound speed (\(m~s^{-1}\)).
density_swSeawater density (\(kg~m^{-3}\)).
m_limitOptional model truncation limit used to cap the number of modes in the numerical calculation.
Theory
The modal series solution for a finite cylinder expresses the backscattering amplitude as:
$$ f_{bs} = -\frac{L}{\pi} \frac{\sin(k L \cos \theta)}{k L \cos \theta} \sum_{m=0}^{\infty} i^{m+1} B_m $$
where \(L\) is the cylinder length, \(k\) is the wavenumber in the surrounding medium, and \(\theta\) is the angle between the cylinder axis and the incident wave direction. The coefficients \(B_m\) depend on the boundary condition at the cylinder surface.
Boundary Conditions and Modal Coefficients
Rigid (fixed) cylinder: The normal velocity at the surface is zero. The modal coefficient is: $$ B_m = (-1)^m \epsilon_m \frac{J_m'(K a)}{H_m^{(1)'}(K a)} $$ where \(J_m\) and \(H_m^{(1)}\) are the cylindrical Bessel and Hankel functions of order \(m\), the prime denotes differentiation with respect to the argument, \(K = k \sin \theta\), and \(a\) is the cylinder radius. The Neumann factor is \(\epsilon_0 = 1\), \(\epsilon_m = 2\) for \(m \geq 1\).
Pressure-release cylinder: The acoustic pressure at the surface is zero. The modal coefficient is: $$ B_m = (-1)^m \epsilon_m \frac{J_m(K a)}{H_m^{(1)}(K a)} $$
Fluid-filled (or gas-filled) cylinder: Both pressure and normal velocity are nonzero at the surface. The modal coefficient is: $$ B_m = -\epsilon_m / (1 + i C_m) $$ where $$ C_m = \frac{ \left[ J_m'(K' a) Y_m(K a) \right] / \left[ J_m(K' a) J_m'(K a) \right] - g h \left[ Y_m'(K a) / J_m'(K a) \right] }{ \left[ J_m'(K' a) J_m(K a) \right] / \left[ J_m(K' a) J_m'(K a) \right] - g h } $$ Here, \(Y_m\) is the cylindrical Bessel function of the second kind, \(K' = K / h\), \(g\) is the density contrast (target to medium), and \(h\) is the sound speed contrast (target to medium).
Modal Truncation
The modal sum is truncated at a maximum order determined by \(m_{\max} = \max(\lceil k a \rceil) + 10\), which is sufficient for convergence in most practical cases.
References
Stanton, T.K. (1988). Sound scattering by cylinders of finite length. I. Fuid cylinders. The Journal of the Acoustical Society of America, 83: 55-63.
Stanton, T.K. (1989). Sound scattering by cylinders of finite length. III. Deformed cylinders. The Journal of the Acoustical Society of America, 85: 232-237.