Modal series

Sphere and spheroid modal solvers separate variables. Finite-cylinder models add a length approximation. See Choosing a solver.

Sphere

For radius a and scattering angle theta, the implemented amplitude is

\[f_s(\theta)=-\frac{i}{k}\sum_{\ell=0}^{\ell_{\max}} (2\ell+1)P_\ell(\cos\theta)A_\ell.\]

Rigid and pressure-release coefficients are respectively

\[A_\ell^{\mathrm{rigid}}=-\frac{j_\ell'(ka)}{{h_\ell^{(1)}}'(ka)}, \qquad A_\ell^{\mathrm{soft}}=-\frac{j_\ell(ka)}{h_\ell^{(1)}(ka)}.\]

Fluid coefficients match pressure and density-weighted normal derivatives (Anderson, 1950). Solid-elastic coefficients also match displacement and stress (Hickling, 1962).

Impedance(zeta) is a locally reacting (Robin) boundary, ∂p/∂n=-ik\,p/\mathrm{zeta} at r=a:

\[A_\ell^{\mathrm{imp}}=-\frac{j_\ell'(ka)+ij_\ell(ka)/\mathrm{zeta}}{{h_\ell^{(1)}}'(ka)+i h_\ell^{(1)}(ka)/\mathrm{zeta}}.\]

zeta → ∞ and zeta → 0 recover A_ℓ^{rigid} and A_ℓ^{soft} above.

modal(Sphere(a), Rigid(), k; m_max = 40)
modal(Sphere(a), FluidFilled(g, h), k)
modal(Sphere(a), SolidElastic(g, hl, ht), k)

Increase m_max to check spherical-degree truncation across the required frequencies and contrasts. Resonances and extreme contrasts can cause ill-conditioning. See Convergence.

Non-plane-wave incidence

incident::IncidentField generalizes the sum above, replacing the plane wave's degree-independent phase with the field's own spherical-harmonic weight Qₗ/(2ℓ+1):

\[f(\theta)=\frac{1}{k}\sum_{\ell=0}^{\ell_{\max}}Q_\ell(-i)^{\ell+1}P_\ell(\cos\theta)A_\ell, \qquad Q_\ell^{\mathrm{plane}}=(2\ell+1)i^\ell.\]

SphericalWave(range) is a point source at -range * direction, with range > a, and propagation toward the origin along direction (Sapozhnikov and Bailey, 2013, Eq. 10):

\[Q_\ell^{\mathrm{spherical}}=ik(-1)^\ell(2\ell+1)h_\ell^{(1)}(k\cdot\mathrm{range}).\]

modal(Sphere(a), Rigid(), k; incident = SphericalWave(range))

The boundary coefficients are unchanged. The regular incident expansion converges inside the source range, while the scattered expansion remains valid beyond it. pressure adds the exact incident field for exterior total-pressure queries. Incident and total pressure are undefined at the source.

BesselBeam(angle) is a zeroth-order non-diffracting beam whose plane-wave components make angle (its half-conical angle β) with the observation axis:

\[Q_\ell^{\mathrm{Bessel}}=(2\ell+1)i^\ell P_\ell(\cos\beta).\]

angle=0 recovers a plane wave. Only zeroth-order, axisymmetric Bessel beams are supported.

Reusing direction, phase and amplitude

Built-in fields accept normalized Cartesian direction, complex amplitude and phase in radians. The scale is amplitude * cis(phase), giving pressure at the origin for plane and Bessel waves. Spherical waves use $p=A\exp(ikR)/R$, without a $1/(4\pi)$ factor.

beam = BesselBeam(pi / 6; direction = (1, 2, -1), amplitude = 0.4 + 0.7im, phase = 0.3)
k = 100.0
solution = modal(Sphere(0.01), FluidFilled(1.2, 0.9), k; incident = beam)
pressure(solution, (0.02, 0.003, 0.0))
incident_pressure(beam, k, (0.02, 0.003, 0.0))
incident_gradient(beam, k, (0.02, 0.003, 0.0))
incident_coefficient(beam, 2, k)  # Q_2 / 5 about the beam axis

Sphere-modal angle is measured from the field axis. Full BEM, closed-surface MFS and volume FEM also accept these fields. Set observation directions explicitly because their defaults follow solver incidence angles. PlaneWave() uses each solver's angle convention, with +x for standalone queries and sphere modal.

Custom fields provide incident_pressure and incident_gradient, or IncidentField(pressure, gradient) callbacks. Rebuild callbacks when frequency changes. Keep sources outside the target and any resolved exterior FEM domain.

Spheroid

Incident fields

Rigid, pressure-release and fluid-filled spheroids accept incident=field, including built-in fields and pressure/gradient callbacks. Projection retains both azimuthal sectors and complex phase. Fluid transmission uses the selected full or diagonal coupling.

beam = BesselBeam(pi / 6; direction = (1, 1, 0), amplitude = 1)
solution = modal(Spheroid(0.02, 0.01), Rigid(), 150.0;
    incident = beam, m_max = 8, n_max = 12,
    incident_n_eta = 48, incident_n_phi = 48,
    scatter_angle = 3pi / 4, scatter_azimuth = pi)
scattering_amplitude(solution)

Directions use the x-polar body frame. Set scatter_angle and scatter_azimuth explicitly when supplying an incident field. These solutions return far-field amplitudes only. Sources must lie outside the spheroid.

Refine incident_n_eta and incident_n_phi separately from modal orders. Their defaults are max(32,2n_max+12) and max(32,4m_max+4). The polar count must exceed n_max, and the azimuthal count twice the retained order. Full fluid coupling also requires refinement of n_quad.

Plane-wave series

Separation in prolate/oblate spheroidal coordinates yields angular and radial spheroidal wave functions. Rigid/soft boundaries decouple appropriate modes. A penetrable spheroid generally couples angular degrees because interior and exterior wavenumbers differ.

FluidFilled(...; coupling = :full) solves the off-diagonal coupling system. :diagonal neglects that coupling and changes the model approximation. Spheroidal wave functions are evaluated with SpheroidalWaves. Increase both m_max and n_max.

incidence_angle_sweep(body, boundary, k, angles; method=:modal) reuses the spheroidal basis for a full-coupling fluid spheroid at fixed frequency. Angles are radians from the symmetry axis. The result contains complex backscatter amplitudes and target strengths.

Higher orders can become unreliable when radial functions are poorly conditioned. Furusawa (1988) develops the prolate spheroidal fish-target models behind this treatment.

Finite and bent cylinders

The cylinder model combines a circular-cylinder coefficient sum with a finite axial coherence factor:

\[f_{\mathrm{bs}}\propto L\, \frac{\sin(kL\cos\beta)}{kL\cos\beta} \sum_m B_m(k a\sin\beta).\]

This lateral finite-length approximation omits end-cap scattering (Stanton, 1988). Use it near broadside and check short cylinders or oblique incidence against a closed-surface solver. Jech et al. (2015) limit its use as a reference to roughly 15 to 20 degrees from broadside.

For curvature radius rho_c, the bent-cylinder model (Stanton, 1989) multiplies the straight amplitude by L_effective / L, with

\[L_{\mathrm{effective}} = \int_{-L/2}^{L/2} \exp\left(i\frac{8k z_{\max}}{L^2}x^2\right)\,dx, \quad z_{\max}=\rho_c\left(1-\cos\frac{L}{2\rho_c}\right).\]

This correction applies near broadside and does not account for all effects of curvature.

modal(Cylinder(a, L), Rigid(), k) # straight
modal(Cylinder(a, L; radius_curvature = rho_c), Rigid(), k) # bent, near broadside

Shells and viscosity

Fluid shells match pressure and normal velocity (Jech et al., 2015, Venas and Jenserud, 2022). LayeredMaterial composes spherical fluid/elastic stacks around a fluid or vacuum core. Elastic shells generalize the Goodman and Stern (1962) formulation, whose original form assumes identical interior and exterior fluids.

modal(Sphere(a), Shelled(FluidLayer(g, h), FluidInterior(g_i, h_i), b_a), k)
modal(Sphere(a), Shelled(LayeredMaterial(FluidLayer(g1, h1),
    FluidLayer(g2, h2), interface_ratio), FluidInterior(g_i, h_i), b_a), k)
modal(Sphere(a), Shelled(ElasticLayer(g, cl, ct), FluidInterior(g_i, h_i), b_a), k)
modal(Sphere(a), Shelled(LayeredMaterial(FluidLayer(g1, h1),
    ElasticLayer(g2, cl, ct), interface_ratio), FluidInterior(g_i, h_i), b_a), k)

ElasticLayer defaults to interior_coupling=:generalized, using the actual cavity fluid (Stanton, 1990). :identical_fluid retains the original Goodman-Stern restriction. Mixed spherical stacks require :generalized, with pressure queries only in fluid regions. Nonspherical mixed stacks are unsupported.

The viscous-elastic model has a viscous outer layer, elastic wall and fluid core. Only the low-frequency monopole is supported.