Introduction
The elastic-cylinder modal series solution (ECMS) is the
solid-cylinder counterpart to the fluid FCMS family. It
combines the exact phase-shift treatment of an infinite elastic circular
cylinder with the same near-broadside finite-length coherence factor
used for finite cylinders in the classical Stanton formulations (Faran 1951; Stanton 1988a, 1988b).
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.
This page follows the shared package notation: medium 1
is the surrounding seawater and medium 2 is the solid
elastic cylinder.
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.
Using the usual Helmholtz decomposition:
\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}}.
The presence of both branches is the main physical distinction from fluid cylinder theory: the interior can support 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.
Schematically:
\frac{1}{i\omega\rho_1}\frac{\partial p_1}{\partial r} = i\omega u_{r,2}
p_1 = -\sigma_{rr}^{(2)}
\sigma_{r\phi}^{(2)} = 0.
Substituting the cylindrical-wave expansions into these conditions yields the elastic cylinder coefficient problem for each order m. The algebra is more involved than for a fluid cylinder because the solid interior contributes both longitudinal and transverse potentials, but the essential structure remains the same: each azimuthal order produces an independent coefficient system.
Phase-shift representation
Rather than carrying the raw coefficient system directly, elastic cylinder theory is often expressed in terms of an order-dependent phase shift \eta_m. The corresponding backscattering contribution of order m is:
(-1)^m \epsilon_m \sin\eta_m \left(\cos\eta_m - i\sin\eta_m\right).
Equivalently, the modal coefficient may be written as:
B_m = (-1)^m \epsilon_m \sin\eta_m e^{-i\eta_m}.
This representation is useful because it makes the elastic-cylinder physics look like the shift of an outgoing cylindrical partial wave relative to the free-space solution. Resonant features appear as rapid variation in the phase shifts across modal order and frequency.
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 same logic used in bent fluid cylinder theory applies at the level of axial coherence. The local cross-sectional elastic phase shifts are still governed by the same elastic cylinder boundary conditions; curvature mainly changes the phase relationship between different axial positions.
That leads to the same coherent-length correction used in bent-cylinder modal theory:
f_{\mathrm{bs}}^{(\mathrm{bent})} = \frac{L_{\mathrm{ebc}}}{L} f_{\mathrm{bs}}^{(\mathrm{straight})}.
where L_{\mathrm{ebc}} is the 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, 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).
The target-strength definition is unchanged relative to the fluid cylinder families; the extra physics is entirely 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.
These assumptions are what make ECMS the natural
solid-cylinder analogue of FCMS: exact local
circular-cylinder physics, approximate finite-length closure, and
strongest physical relevance near broadside rather than at end-on
incidence.