Acoustic Scattering Primer
Source:vignettes/acoustic-scattering-primer/acoustic-scattering-primer.Rmd
acoustic-scattering-primer.RmdIntroduction
The notation and reporting conventions on this page follow standard time-harmonic scattering texts and elastic-wave treatments (Morse and Ingard 1968; Morse and Feshbach 1953; Urick 1983; Pierce 1989).
This article is the shared physics starting point for the model-family theory pages. The goal is to make the recurring acoustic field definitions, medium-indexing rules, and reporting quantities explicit in one place, so that the family pages can stay focused on the mathematics specific to each geometry or approximation regime.
The package theory pages all assume linear, time-harmonic acoustics. They then change one or more of the following ingredients:
- the geometry,
- the boundary conditions,
- the material layering, or
- the approximation used to simplify the full boundary-value problem.
This primer defines the common exterior/interior pressure-field problem before those model-specific choices are introduced.
Time-harmonic convention
Throughout the package theory pages, the physical acoustic pressure is represented by the real part of a complex time-harmonic field:
\Re\left\{p(\mathbf{x})e^{-i\omega t}\right\},
where \mathbf{x} is position, t is time, and \omega = 2\pi f is angular frequency. Working with the complex phasor p(\mathbf{x}) is what converts the linear acoustic wave equation into the Helmholtz equation (Morse and Ingard 1968).
The sign convention matters because it fixes the phase of the outgoing wave and the exact form of the far-field asymptotics. All theory pages below follow this same e^{-i\omega t} convention.
Exterior and interior pressure fields
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 always the surrounding seawater or
other ambient exterior fluid. The incident field is the field that would
exist if the target were absent. The scattered field is the additional
field generated by the target (Anderson 1950; Faran 1951).
For penetrable targets, there is also one or more interior fields:
p_2(\mathbf{x}), \quad p_3(\mathbf{x}), \quad \ldots
The detailed interpretation of those interior fields depends on the
model: for a fluid sphere or fluid cylinder, medium 2 is
the interior fluid; for a shelled problem, medium 2 is the
shell and medium 3 is the shell interior; and for a
viscous-elastic layered problem, numbering continues inward as
needed.
Exterior Helmholtz equation
In the homogeneous surrounding fluid, the time-harmonic pressure satisfies:
\nabla^2 p_1 + k_1^2 p_1 = 0,
with seawater wavenumber:
k_1 = \frac{\omega}{c_1},
where c_1 is the exterior sound speed.
For homogeneous interior fluid regions, the same equation holds in each interior medium:
\nabla^2 p_j + k_j^2 p_j = 0, \qquad k_j = \frac{\omega}{c_j},
with medium-specific sound speed c_j.
For elastic and viscous regions, the governing equation is no longer a single scalar Helmholtz equation. Instead, the displacement or potential fields split into compressional and shear branches, each with its own wavenumber. The model-specific theory pages derive those branches explicitly.
Radiation condition
The scattered field must represent energy leaving the target rather than arriving from infinity. In exterior scattering theory, that is enforced by the Sommerfeld radiation condition:
\lim_{r\to\infty} r\left( \frac{\partial p_1^{\mathrm{sca}}}{\partial r} - ik_1 p_1^{\mathrm{sca}} \right) = 0.
This condition is what selects outgoing Hankel functions in modal solutions and outgoing Green’s functions in boundary-integral formulations (Sommerfeld 1949; Morse and Ingard 1968).
Far-field scattering and target strength
Far-field amplitude
At distances large compared with both the wavelength and the target size, the scattered field approaches an outgoing spherical wave:
p_1^{\mathrm{sca}}(\mathbf{x}) \sim \frac{e^{ik_1 r}}{r} f(\theta_s,\phi_s \mid \theta_i,\phi_i), \qquad r \to \infty,
where (\theta_i,\phi_i) describe the incident direction, (\theta_s,\phi_s) describe the receive or scattering direction, and f is the far-field scattering amplitude.
The exact basis used to compute f
depends on the model. In spherical, cylindrical, spheroidal, and
T-matrix formulations, that amplitude is reconstructed from modal
coefficients (Bowman,
Senior, and Uslenghi 1987). In approximate models such as
DWBA, TRCM, or KRM, it is
assembled from asymptotic or perturbative expressions.
Backscatter, cross-section, and target strength
For monostatic target-strength work, the receive direction is opposite the incident direction. The backscattering amplitude is therefore:
f_{\mathrm{bs}} = f(\theta_s,\phi_s \mid \theta_i,\phi_i) \Big|_{\text{monostatic receive direction}}.
The linear backscattering cross-section is:
\sigma_{\mathrm{bs}} = \left|f_{\mathrm{bs}}\right|^2,
and the target strength is (Urick 1983; Simmonds and MacLennan 2005; MacLennan, Fernandes, and Dalen 2002):
\mathrm{TS} = 10 \log_{10}\left(\sigma_{\mathrm{bs}}\right).
Some model families instead work with a scattering length or form function. In those cases the algebra may be written as 20\log_{10}|f_{\mathrm{bs}}|, but the underlying quantity remains the squared-magnitude backscatter referenced through the same linear-to-log conversion.
The practical rule is simple: average or combine scattering in the linear domain, using \sigma_{\mathrm{bs}} or another linear amplitude measure, and report target strength in dB only after the linear quantity of interest has been formed.
Medium numbering and contrast notation
Exterior medium is always 1
To keep notation consistent across theory pages, medium
1 is always the surrounding seawater or ambient exterior
fluid.
The numbering then increases inward: medium 2 is the
first target region encountered from the outside, medium 3
is the next deeper region, medium 4 is the next deeper
region after that, and so on.
That rule is used even when different model families describe very
different targets. For example, an unshelled fluid sphere uses media
1 and 2, an elastic shell with internal fluid
uses media 1, 2, and 3, and a
viscous-elastic layered sphere may use media 1,
2, 3, and 4.
Contrast ratios
The density and sound-speed contrasts between media i
and j are written as:
g_{ij} = \frac{\rho_i}{\rho_j}, \qquad h_{ij} = \frac{c_i}{c_j},
where \rho_i and c_i are the density and sound speed of medium
i.
This notation is deliberately directional. For example, a body relative to seawater is described by g_{21} and h_{21}, a shell relative to seawater is also described by g_{21} and h_{21} when the shell is the first target region, an internal fluid relative to the shell is described by g_{32} and h_{32}, and an internal fluid relative directly to seawater is described by g_{31} and h_{31}.
When a model uses only one target region, the preferred notation is therefore g_{21} and h_{21} rather than bare g and h.
Coordinate-specific scale factors versus material contrasts
Some pages also use symbols such as h_\xi, h_\eta, or h_\phi for metric scale factors in curvilinear coordinates. Those are not sound-speed contrasts. Whenever both appear in the same page, the contrast notation is kept as h_{ij} to avoid ambiguity.
What changes from model to model
Once the common exterior problem is fixed, model families differ in a small number of principled ways:
- Basis choice. Spheres use spherical functions, cylinders use cylindrical functions, and spheroids use spheroidal functions or geometry-aware T-matrix operators (Bowman, Senior, and Uslenghi 1987).
- Boundary conditions. Rigid, pressure-release, fluid-filled, gas-filled, viscous, and elastic interfaces each enforce different conditions at the target surface.
- Geometry representation. Canonical models solve exact boundary-value problems on idealized shapes, while approximate models use segmented profiles, perturbative line integrals, or ray-style sums.
-
Approximation regime.
DWBA,SDWBA,HPA,KRM,TRCM, and related models are not exact boundary-value solutions. They keep only the dominant physics of a stated regime.
The theory pages are therefore best read as answers to a common question: starting from the shared exterior scattering problem, what additional physical assumptions make this model solvable or useful?