Introduction
The transition matrix method (TMM) represents a target
by the linear map from incident-wave coefficients to scattered-wave
coefficients (Waterman 1969). Once constructed,
that map can be evaluated for different incident and receive directions
without resolving the boundary-value problem.
This coefficient map supports monostatic and bistatic scattering, rotations, and orientation averages when the retained basis and numerical solution are valid for those operations.
The useful basis depends on geometry. Spheres and spherical shells use spherical waves. Prolate spheroids use prolate spheroidal waves. Oblate spheroids can use spherical waves enforced on r(\theta). Finite cylinders favor cylindrical modes because the sidewall-endcap junction converges poorly in a smooth spherical basis (Varadan et al. 1982; Waterman 2009).
Medium indices, field conventions, and reporting quantities follow Notation and symbols. Boundary types are defined on the Boundary conditions page.
General single-target T-matrix formulation
Incident and scattered modal expansions
For time-harmonic pressure with implicit factor e^{-i\omega t}, the pressure in a homogeneous region satisfies the Helmholtz equation (Morse and Ingard 1968):
\nabla^2 p + k^2 p = 0.
In a modal T-matrix formulation, the incident and scattered fields are expanded as:
p^{inc} = \sum_{\nu} a_{\nu}\,\psi^{(1)}_{\nu}, \qquad p^{sca} = \sum_{\nu} f_{\nu}\,\psi^{(3)}_{\nu}.
Here \psi^{(1)}_{\nu} is regular and \psi^{(3)}_{\nu} is outgoing. The transition matrix is the linear map:
\mathbf{f} = \mathbf{T}\mathbf{a}.
For an axisymmetric target, azimuthal order m decouples and \mathbf T is block diagonal in m. If a_{m'n'}(\widehat{\mathbf k}_i) are the plane-wave coefficients for incident direction \widehat{\mathbf k}_i, the far-field amplitude in receive direction \widehat{\mathbf k}_s has the general form:
f(\widehat{\mathbf k}_s,\widehat{\mathbf k}_i) = \frac{1}{k_1} \sum_{mn}\sum_{m'n'} \mathcal Y_{mn}(\widehat{\mathbf k}_s) T_{mn,m'n'} a_{m'n'}(\widehat{\mathbf k}_i),
where \mathcal Y_{mn} includes the angular basis and outgoing-wave phase. Backscatter is the special case \widehat{\mathbf k}_s=-\widehat{\mathbf k}_i (Mishchenko et al. 2002).
Boundary conditions
Consider the four scalar acoustic boundary types:
fixed_rigid, pressure_release,
liquid_filled, and gas_filled.
The first two use only the exterior basis. For a rigid target, the normal velocity vanishes at the boundary, which in pressure language means the normal derivative of pressure vanishes. For a pressure-release target, the pressure itself vanishes at the surface.
For fluid- or gas-filled targets, an interior field must also be represented. The boundary conditions are then:
p^{ext} = p^{int}, \qquad \frac{1}{\rho_{ext}} \frac{\partial p^{ext}}{\partial n} = \frac{1}{\rho_{int}} \frac{\partial p^{int}}{\partial n}.
Density and sound speed therefore enter the modal boundary operator.
Spherical-coordinate branch
General axisymmetric surface formulation
Suppose the scatterer surface is axisymmetric and can be written in spherical coordinates as:
r = r(\theta).
The exterior regular and outgoing basis states are then built from spherical partial waves:
\begin{align*} \psi^{(1)}_{mn}(r,\theta,\phi) &= j_n(kr)\,P_n^m(\cos\theta)\,e^{im\phi}, \\ \psi^{(3)}_{mn}(r,\theta,\phi) &= h_n^{(1)}(kr)\,P_n^m(\cos\theta)\,e^{im\phi}. \end{align*}
Along the curved meridional profile r(\theta), the outward normal derivative is:
\frac{\partial}{\partial n} = \frac{1}{\sqrt{1 + \left[r_\theta / r\right]^2}} \left( \frac{\partial}{\partial r} - \frac{r_\theta}{r^2}\frac{\partial}{\partial \theta} \right).
where r_\theta=dr/d\theta.
A nonspherical boundary therefore couples radial and angular derivatives. The boundary residual is projected back onto the retained spherical basis to form each m-block.
Special geometries in the spherical branch
Sphere
For a sphere:
r(\theta) = a.
so r_\theta = 0 and the normal derivative reduces to the ordinary radial derivative. The spherical-coordinate T-matrix formulation therefore collapses to the classical spherical partial-wave problem.
Spherical shells
For a concentric shell with outer radius a and inner radius b, spherical symmetry keeps the T-matrix diagonal in (m,n). The diagonal coefficient is the exterior modal scattering coefficient obtained from the two interface systems. A fluid shell couples exterior, shell, and core acoustic fields. An elastic shell also carries longitudinal and transverse elastic potentials. The coefficient map is therefore:
f_{mn}=T_n a_{mn},
with T_n supplied by the corresponding shell boundary determinant. See VESM theory for fluid shells and ESSMS theory for elastic shells.
Oblate spheroid
For an oblate spheroid with axial semiaxis c and equatorial semiaxis a, where c \le a:
r(\theta) = \left( \frac{\cos^2\theta}{c^2} + \frac{\sin^2\theta}{a^2} \right)^{-1/2}.
Differentiating gives:
r_\theta = -\sin\theta\cos\theta \left( \frac{1}{a^2} - \frac{1}{c^2} \right) \left( \frac{\cos^2\theta}{c^2} + \frac{\sin^2\theta}{a^2} \right)^{-3/2}.
So the oblate branch remains a spherical-basis T-matrix formulation, but with the actual oblate meridional geometry entering through r(\theta) and r_\theta.
Finite cylinder
For a right circular finite cylinder with half-length a and radius b, the meridional surface can be written piecewise in spherical coordinates as:
r(\theta) = \min\left( \frac{a}{|\cos\theta|}, \frac{b}{|\sin\theta|} \right).
A ray from the origin first meets either an end cap or the sidewall. The profile is continuous but not differentiable at their junction, which slows convergence of a spherical-wave representation (Waterman 2009).
Monostatic reconstruction
Once the retained block coefficients are obtained, the backscatter amplitude is reconstructed by evaluating the outgoing expansion in the receive direction opposite to the incident plane wave. The backscattering cross section then follows from:
\sigma_{bs} = |f_{bs}|^2
and target strength is (MacLennan et al. 2002):
TS = 10 \log_{10}\left( \frac{\sigma_{bs}}{1\ \mathrm{m}^2}\right).
Cylindrical interpretation
For a finite cylinder, cylindrical partial waves describe the cross-section and an axial operator describes finite length. Near broadside in monostatic scattering, that operator reduces to the familiar finite-cylinder coherence factor.
The sidewall-endcap corners are the main obstacle. Angular products require particular care because a near-broadside coherence closure is not a general bistatic cylinder solution.
Prolate spheroid branch
Why spherical coordinates are not the best exact basis
A prolate spheroid is not a constant-r surface. So while spherical-wave expansions can still be written down, they do not align naturally with the geometry. This is exactly the regime where the classic spheroidal-coordinate transition-matrix literature becomes relevant (Varadan et al. 1982; Hackman and Todoroff 1984).
In prolate spheroidal coordinates, the boundary is the coordinate surface \xi=\xi_1.
Prolate spheroidal coordinates
Let q be the semifocal length and let (\xi,\eta,\phi) be prolate spheroidal coordinates. In Cartesian coordinates:
\begin{align*} x &= q\sqrt{(\xi^2 - 1)(1 - \eta^2)}\cos\phi, \\ y &= q\sqrt{(\xi^2 - 1)(1 - \eta^2)}\sin\phi, \\ z &= q \xi \eta. \end{align*}
The coordinate ranges are:
\xi \ge 1, \qquad -1 \le \eta \le 1, \qquad 0 \le \phi < 2\pi.
If the body surface is \xi = \xi_1, then the major semi-axis a and minor semi-axis b satisfy:
a = \xi_1 q, \qquad b = q\sqrt{\xi_1^2 - 1}.
so equivalently:
\xi_1 = \left[1 - \left(\frac{b}{a}\right)^2\right]^{-1/2}.
Spheroidal modal representation
In a homogeneous region, the separated pressure field is written as the product of a radial spheroidal function in \xi, an angular spheroidal function in \eta, and an azimuthal factor in \phi.
The resulting field expansion has the same logical structure as the generic T-matrix expression above, but the basis is geometry-matched.
For rigid and pressure-release prolates, the retained degrees remain effectively local in the exact spheroidal basis. For liquid- and gas-filled prolates, the interior and exterior reduced frequencies differ, so the angular bases no longer match exactly. This introduces overlap-driven coupling between retained degrees, exactly as in the exact prolate spheroidal modal-series solution (Hackman and Todoroff 1984; Spence and Granger 1951; Furusawa 1988).
Mathematical assumptions and scope
The formulations above assume linear time-harmonic acoustics, a single target, an axisymmetric geometry, and homogeneous properties within each region. Accuracy depends on modal truncation, boundary quadrature, and using a basis suited to the surface. Smooth spherical and spheroidal surfaces support general angular reconstruction once their blocks converge. A finite-cylinder near-broadside closure does not by itself establish a general bistatic cylinder operator.