Acoustic Scattering Primer
Source:vignettes/acoustic-scattering-primer/acoustic-scattering-primer.Rmd
acoustic-scattering-primer.RmdIntroduction
Acoustic scattering is the disturbance produced when an incident sound field meets spatial changes in material properties or a boundary that constrains pressure and motion. The scattered field depends on frequency, target geometry and orientation, material contrast, boundary conditions, and both the incident and receive directions. Backscatter is one important directional case, not the definition of scattering as a whole (Pierce 1989; Colton and Kress 2013).
This page develops the common three-dimensional theory for a bounded target. It begins with the linear acoustic equations, constructs the exterior scattering problem, and then derives directional, bistatic, and monostatic far-field quantities.
See Notation and Symbols.
Linear acoustic field
Assumptions and perturbation variables
Let an inviscid fluid be initially at rest with equilibrium density \rho_0 and pressure P_0. A small acoustic disturbance is written:
P(\mathbf{x},t)=P_0+p'(\mathbf{x},t), \qquad \rho(\mathbf{x},t)=\rho_0+\rho'(\mathbf{x},t),
with particle velocity \mathbf{v}(\mathbf{x},t). Neglecting products of the perturbations gives the linearized continuity and momentum equations:
\frac{\partial \rho'}{\partial t} +\rho_0\nabla\!\cdot\!\mathbf{v}=0, \qquad \rho_0\frac{\partial\mathbf{v}}{\partial t}=-\nabla p'.
For a locally adiabatic fluid, p'=c^2\rho', where c is the small-signal sound speed. Taking the time derivative of continuity, substituting momentum, and eliminating \rho' yields:
\nabla^2p'-\frac{1}{c^2}\frac{\partial^2p'}{\partial t^2}=0.
This derivation assumes small disturbances, a stationary homogeneous background within each region, and a linear constitutive relation. Mean flow, strong nonlinearity, and continuously varying media require additional terms (Morse and Ingard 1968; Pierce 1989).
Time-harmonic convention
For a single angular frequency \omega=2\pi f, write the physical acoustic pressure as:
p'(\mathbf{x},t)= \Re\left\{p(\mathbf{x})e^{-i\omega t}\right\}.
Substitution into the wave equation gives the Helmholtz equation:
\nabla^2p+k^2p=0, \qquad k=\frac{\omega}{c}.
The corresponding particle-velocity phasor follows directly from the momentum equation:
\mathbf{v}=\frac{1}{i\omega\rho_0}\nabla p.
The sign convention matters because it fixes the phase of the outgoing wave and the exact form of the far-field asymptotics. Under e^{-i\omega t}, a plane wave traveling in direction \widehat{\mathbf{k}}_i is:
p^{\mathrm{inc}}(\mathbf{x}) =p_0e^{ik\widehat{\mathbf{k}}_i\cdot\mathbf{x}},
and an outgoing spherical wave is proportional to e^{ikr}/r.
The scattering boundary-value problem
Incident, scattered, and total fields
For a single target embedded in seawater, the total exterior pressure is split into incident and scattered parts:
p_1^{\mathrm{tot}}(\mathbf{x}) = p_1^{\mathrm{inc}}(\mathbf{x}) + p_1^{\mathrm{sca}}(\mathbf{x}),
where medium 1 is the surrounding seawater or other
ambient exterior fluid. The incident field is the solution that would
exist without the target. The scattered field is the target-induced
correction. Linearity permits this decomposition, but the boundary
conditions act on the total field (Morse and Ingard 1968; Colton and Kress 2013).
For penetrable targets, there may also be one or more interior fields:
p_2(\mathbf{x}), \quad p_3(\mathbf{x}), \quad \ldots
For an acoustic fluid region, p_j is a scalar pressure field. An elastic region instead requires displacement or scalar/vector potentials that support both longitudinal and transverse waves. The interface conditions couple these interior fields to p_1^{\mathrm{tot}}. Their physical forms are derived on the boundary-conditions page.
Exterior Helmholtz equation
In a homogeneous, source-free exterior fluid, each of p_1^{\mathrm{inc}}, p_1^{\mathrm{sca}}, and p_1^{\mathrm{tot}} satisfies:
\nabla^2 p_1 + k_1^2 p_1 = 0,
with exterior wavenumber:
k_1 = \frac{\omega}{c_1},
where c_1 is the exterior sound speed. A homogeneous interior fluid similarly satisfies:
\nabla^2 p_j + k_j^2 p_j = 0, \qquad k_j = \frac{\omega}{c_j},
with medium-specific sound speed c_j.
The boundary-value problem is completed by conditions on every interface and by an outgoing-wave condition at infinity. The geometry and boundary conditions, not the exterior Helmholtz equation alone, determine the scattered field.
Radiation condition
The scattered field must carry energy away from the target. For a bounded three-dimensional scatterer, the Sommerfeld radiation condition is:
\lim_{r\to\infty} r\left( \frac{\partial p_1^{\mathrm{sca}}}{\partial r} - ik_1 p_1^{\mathrm{sca}} \right) = 0
uniformly with respect to direction \widehat{\mathbf{x}}=\mathbf{x}/r. It excludes incoming waves from infinity and selects first-kind outgoing Hankel functions and the outgoing Green function for the adopted time convention (Sommerfeld 1949; Colton and Kress 2013).
Directional far-field scattering
Far-field amplitude
Let p_0 be the pressure amplitude of the incident plane wave. At distances large relative to both wavelength and target size, a bounded target has the asymptotic field:
\frac{p_1^{\mathrm{sca}}(\mathbf{x})}{p_0} = \frac{e^{ik_1 r}}{r} f(\Omega_s\mid\Omega_i) +O(r^{-2}), \qquad r\to\infty,
where \Omega_i=(\theta_i,\phi_i) specifies incident propagation, \Omega_s=(\theta_s,\phi_s) specifies observation, and f(\Omega_s\mid\Omega_i) is the complex far-field scattering amplitude. Since p_1^{\mathrm{sca}}/p_0 is dimensionless, f has units of length. Retaining both angular arguments is essential for a bistatic geometry (Bowman et al. 1987; Waterman 2009).
Define the scattering angle by:
\psi=\cos^{-1}\!\left( \widehat{\mathbf{s}}\cdot\widehat{\mathbf{k}}_i \right).
Forward scattering has \psi=0 and backscattering has \psi=\pi. For a nonspherical target, \psi alone is generally insufficient: two direction pairs with the same \psi can give different amplitudes because target orientation and azimuth also matter.
Differential and total scattering cross-sections
The time-averaged acoustic intensity is:
\mathbf{I}=\frac{1}{2}\Re\!\left(p\mathbf{v}^{*}\right).
For a plane wave in a lossless fluid, its magnitude is I_{\mathrm{inc}}=|p_0|^2/(2\rho_1c_1). Applying the far-field form to the outgoing wave gives I_{\mathrm{sca}}\sim |p_0|^2|f|^2/(2\rho_1c_1r^2). Therefore:
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} (\Omega_s\mid\Omega_i) =\lim_{r\to\infty} r^2\frac{I_{\mathrm{sca}}}{I_{\mathrm{inc}}} =\left|f(\Omega_s\mid\Omega_i)\right|^2.
The differential cross-section measures the scattered intensity into a chosen direction per unit incident intensity. Integrating over all receive directions gives the total scattering cross-section:
\sigma_s(\Omega_i) =\int_{4\pi} \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} (\Omega_s\mid\Omega_i)\,\mathrm{d}\Omega_s.
These are different observables: a target can have a deep null in one receive direction while still scattering substantial energy elsewhere. If the target absorbs energy, the extinction cross-section is \sigma_e=\sigma_s+\sigma_a, where \sigma_a is the absorption cross-section (Pierce 1989; Colton and Kress 2013).
Bistatic scattering
A bistatic experiment allows \Omega_i and \Omega_s to vary independently. Its fundamental quantities are the complex amplitude f(\Omega_s\mid\Omega_i) and the differential cross-section above. A directional logarithmic level may be written:
\mathrm{TS}(\Omega_s\mid\Omega_i) =10\log_{10}\!\left[ \frac{\mathrm{d}\sigma/\mathrm{d}\Omega} {1\ \mathrm{m}^2\,\mathrm{sr}^{-1}} \right].
The angular arguments must remain visible: this quantity is a directional scattering level, not a single intrinsic number for the target. Literature definitions of a “bistatic cross-section” sometimes include an additional 4\pi factor. The convention must therefore be stated whenever results from different sources are compared (Bowman et al. 1987; MacLennan et al. 2002).
Backscatter, cross-section, and target strength
In a monostatic geometry, the source and receiver are collocated in the far field. The receive direction from the target is opposite the incident propagation direction, so:
f_{\mathrm{bs}}(\Omega_i) =f(\Omega_s\mid\Omega_i) \big|_{\widehat{\mathbf{s}}=-\widehat{\mathbf{k}}_i}.
The fisheries-acoustics backscattering cross-section is the corresponding differential cross-section evaluated in that direction (MacLennan et al. 2002):
\sigma_{\mathrm{bs}}(\Omega_i) =\left|f_{\mathrm{bs}}(\Omega_i)\right|^2.
Target strength is its logarithmic level relative to 1\ \mathrm{m}^2:
\mathrm{TS}(\Omega_i) =10\log_{10}\!\left[ \frac{\sigma_{\mathrm{bs}}(\Omega_i)}{1\ \mathrm{m}^2} \right] =20\log_{10}\!\left[ \frac{|f_{\mathrm{bs}}(\Omega_i)|}{1\ \mathrm{m}} \right].
The reference area is required because a logarithm cannot be taken of a dimensional quantity. The older spherical scattering cross-section \sigma_{\mathrm{sp}}=4\pi\sigma_{\mathrm{bs}} is a different convention and must not be substituted into the equation above without conversion (MacLennan et al. 2002; Simmonds and MacLennan 2005).
Complex amplitudes are added before taking a squared magnitude when fields are coherent. Cross-sections may be averaged when phases are random or an ensemble average is intended. Target strengths in dB are not added or averaged as if they were linear energy quantities.
Scattering lengths and form functions
Many canonical-shape derivations report a dimensionless form function rather than the dimensional amplitude f. A common spherical-target convention is:
F(\Omega_s\mid\Omega_i)=\frac{2}{a} f(\Omega_s\mid\Omega_i), \qquad \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} =\frac{a^2}{4}|F|^2,
where a is a stated reference radius. Other authors choose f/a, include phase factors, or use a two-dimensional normalization for an infinite cylinder. A form function is therefore meaningful only together with its definition and reference length (Bowman et al. 1987).
Material contrast and interfaces
Material changes provide the source of scattering. For an inviscid acoustic fluid, the basic quantities are density \rho_j, sound speed c_j, compressibility \kappa_j=(\rho_jc_j^2)^{-1}, and plane-wave impedance Z_j=\rho_jc_j. Elastic regions additionally require moduli or longitudinal and transverse wave speeds. Viscous or absorbing regions introduce complex, frequency-dependent material response (Medwin and Clay 1998; Achenbach 1973).
Medium 1 denotes the exterior, with deeper regions
numbered inward. The directional density and sound-speed ratios are:
g_{ij} = \frac{\rho_i}{\rho_j}, \qquad h_{ij} = \frac{c_i}{c_j}.
For a one-region target, g_{21} and h_{21} compare the target with the exterior. These ratios affect interface transmission and reflection, whereas k_1a compares target size with wavelength. Equal acoustic impedance does not make two fluids identical: differing \rho and c can still change refraction and internal phase accumulation. The complete set of material and perturbation contrasts is tabulated on the notation page.
How scattering solutions differ
Every theory in the model library begins with the same incident/scattered-field decomposition but makes different choices in five places:
- Geometry. Canonical surfaces can separate in spherical, cylindrical, or spheroidal coordinates. Arbitrary bodies generally cannot.
- Boundary and material model. Rigid, pressure-release, fluid, elastic, viscous, and layered targets support different fields and interface conditions.
- Representation. A solution may use separated partial waves, integral equations, an incident-to-scattered coefficient operator, volume integrals, or high-frequency rays (Morse and Feshbach 1953; Waterman 2009).
- Approximation. Some formulations solve the stated idealized boundary-value problem exactly before numerical truncation. Others linearize weak contrast, retain selected modes, or invoke low- or high-frequency asymptotics.
- Observable. A calculation may retain the full complex directional amplitude, only monostatic backscatter, or an ensemble-averaged linear cross-section.
An “exact” series is exact only for its stated geometry, constitutive law, and boundary conditions. Its computed value still depends on truncation, special functions, quadrature, and finite-precision linear algebra. Those issues are treated in Numerical Methods.