Introduction
The elastic-cylinder modal series solution (ECMS)
combines the phase-shift solution for an infinite elastic circular
cylinder with a near-broadside finite-length factor (Faran 1951; Stanton 1988). The cross-sectional
solve contains longitudinal and transverse waves. The finite-length
closure is approximate.
The resulting model is exact in the local circular cross-section and approximate only in the way finite length is reduced to a coherence factor.
Medium indices and reporting quantities follow Notation and symbols. The fluid-solid interface conditions are given on the Boundary conditions page.
Exterior fluid and elastic interior
Exterior acoustic field
In the surrounding seawater, the pressure satisfies:
\nabla^2 p_1 + k_1^2 p_1 = 0, \qquad k_1 = \frac{\omega}{c_1},
where c_1 is the exterior sound speed.
Elastic interior
Inside the solid cylinder, the displacement field \mathbf{u}_2 satisfies the Navier equation:
(\lambda_2 + 2\mu_2)\nabla(\nabla\cdot\mathbf{u}_2) - \mu_2 \nabla\times(\nabla\times\mathbf{u}_2) + \rho_2 \omega^2 \mathbf{u}_2 = 0,
where \lambda_2 and \mu_2 are Lamé parameters and \rho_2 is the solid density.
The Helmholtz decomposition is:
\mathbf{u}_2 = \nabla \Phi_2 + \nabla\times \mathbf{\Psi}_2,
the solid supports a longitudinal branch and a transverse branch with wavenumbers:
k_{L,2} = \omega\sqrt{\frac{\rho_2}{\lambda_2 + 2\mu_2}}, \qquad k_{T,2} = \omega\sqrt{\frac{\rho_2}{\mu_2}}.
Unlike a fluid cylinder, the solid supports shear as well as compression.
Infinite-cylinder modal representation
Exterior cylindrical waves
For an incident plane wave near broadside, the exterior field is expanded in cylindrical harmonics:
\begin{align*} p_{1,\mathrm{inc}}(r,\phi) &= \sum_{m=0}^{\infty} \epsilon_m i^m J_m(k_1 r)\cos(m\phi), \\ p_{1,\mathrm{sca}}(r,\phi) &= \sum_{m=0}^{\infty} \epsilon_m i^m B_m H_m^{(1)}(k_1 r)\cos(m\phi). \end{align*}
Here J_m is the regular cylindrical Bessel function, H_m^{(1)} is the outgoing Hankel function, \epsilon_m is the Neumann factor, and B_m is the scattered coefficient of order m.
Elastic interior potentials
The longitudinal and transverse interior potentials may be written schematically as:
\begin{align*} \Phi_2(r,\phi) &= \sum_{m=0}^{\infty} \epsilon_m i^m C_m J_m(k_{L,2} r)\cos(m\phi), \\ \Psi_2(r,\phi) &= \sum_{m=0}^{\infty} \epsilon_m i^m D_m J_m(k_{T,2} r)\sin(m\phi). \end{align*}
with the angular parity chosen so that the resulting displacement components match the cylindrical symmetry of order m.
Regularity at the axis excludes singular radial functions from the solid interior.
Boundary conditions at the cylinder wall
At the cylinder surface r = a, the elastic-solid and exterior-fluid fields must satisfy three conditions:
- continuity of normal velocity,
- balance of normal traction with acoustic pressure,
- vanishing tangential traction because the exterior fluid is inviscid.
With the e^{-i\omega t} convention, normal-velocity continuity is:
\frac{1}{i\omega\rho_1}\frac{\partial p_1}{\partial r} = -i\omega u_{r,2}.
Normal-traction balance is:
p_1 = -\sigma_{rr}^{(2)}.
The inviscid exterior also requires:
\sigma_{r\phi}^{(2)} = 0.
Substitution gives an independent three-coefficient system for each azimuthal order m: one scattered acoustic amplitude and two elastic-potential amplitudes.
Phase-shift representation
Elastic-cylinder theory is conveniently expressed through an order-dependent phase shift \eta_m. The backscattering contribution of order m is:
(-1)^m \epsilon_m \sin\eta_m \left(\cos\eta_m - i\sin\eta_m\right).
Separating the angular and Neumann factors, define the modal coefficient:
B_m = \sin\eta_m e^{-i\eta_m}.
The phase shift measures the change in an outgoing cylindrical partial wave caused by the elastic boundary. Resonances produce rapid variation of \eta_m with frequency or modal order.
Finite-length closure
The modal derivation above is exact for an infinite circular
cylinder. ECMS inherits the same near-broadside
finite-length closure used by FCMS, namely:
\frac{\sin(k_1 L \cos\theta)} {k_1 L \cos\theta},
where L is cylinder length and \theta is the incidence angle relative to the cylinder axis.
The finite-cylinder backscattering amplitude is therefore:
f_{\mathrm{bs}}^{(\mathrm{straight})} = \frac{L}{\pi} \frac{\sin(k_1 L \cos\theta)} {k_1 L \cos\theta} \sum_{m=0}^{\infty} (-1)^m \epsilon_m \sin\eta_m e^{-i\eta_m}.
This is the straight solid-cylinder branch of ECMS.
Uniformly bent extension
For a uniformly bent elastic cylinder, the local phase shifts remain those of the straight cross-section. Curvature changes the phase relationship among axial positions.
This gives the bent-cylinder coherent-length correction:
f_{\mathrm{bs}}^{(\mathrm{bent})} = \frac{L_{\mathrm{ebc}}}{L} f_{\mathrm{bs}}^{(\mathrm{straight})}.
Here L_{\mathrm{ebc}} is the complex equivalent coherent length of the bent axis. The curvature correction therefore multiplies the elastic straight-cylinder kernel rather than replacing it.
Target strength
Once the complex backscattering amplitude is known, the linear backscattering cross-section and target strength are (MacLennan et al. 2002):
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}\right|^2, \qquad \mathrm{TS} = 10\log_{10}\left(\frac{\sigma_{\mathrm{bs}}}{1\ \mathrm{m}^2}\right).
The target-strength definition is unchanged relative to the fluid cylinder families. The extra physics is in the elastic phase shifts and, where used, the bent-axis coherent-length correction.
Assumptions and intended regime
The family rests on the following assumptions:
- circular homogeneous solid elastic cylinder,
- longitudinal and transverse waves supported in the interior,
- linear acoustics in the exterior fluid and linear elasticity in the solid,
- near-broadside finite-length closure,
- curvature, when present, enters through axial coherence rather than a new local cross-sectional solve.
The local circular-cylinder physics is exact within these assumptions. The finite-length and curvature closures are intended for incidence near broadside, not end-on incidence.