Notation and symbols
Source:vignettes/notation-and-symbols/notation-and-symbols.Rmd
notation-and-symbols.RmdIntroduction
The notation on this page follows standard scattering texts for Helmholtz, elastic-wave, and basis-expansion formulations (Morse and Ingard 1968; Flammer 1957; Waterman 2009).
This page is the compact notation guide for the package theory articles. The model-family pages still define symbols locally when needed, but the symbols listed here are intended to carry the same meaning everywhere unless a page explicitly says otherwise.
Medium indexing
The package theory pages use one exterior-to-interior convention:
| Symbol | Meaning |
|---|---|
1 |
surrounding seawater or ambient exterior fluid |
2 |
first target region encountered from the exterior |
3 |
second target region encountered from the exterior |
4 |
third target region encountered from the exterior |
Examples: an unshelled sphere uses media 1 and
2, an elastic shell with internal fluid uses media
1, 2, and 3, and a
viscous-elastic layered sphere can use media 1,
2, 3, and 4.
Shared field variables
| Symbol | Meaning | Typical units |
|---|---|---|
| p_1^{\mathrm{inc}} | incident pressure in seawater | Pa |
| p_1^{\mathrm{sca}} | scattered pressure in seawater | Pa |
| p_1^{\mathrm{tot}} | total exterior pressure, p_1^{\mathrm{inc}} + p_1^{\mathrm{sca}} | Pa |
| p_j | pressure in medium j | Pa |
| \mathbf{u} | elastic displacement vector | m |
| \mathbf{v} | fluid particle velocity | m s^{-1} |
| \mathbf{n} | outward unit normal | dimensionless |
| \sigma_{ij} | stress-tensor components | Pa |
Frequencies, speeds, and wavenumbers
| Symbol | Meaning | Typical units |
|---|---|---|
| f | acoustic frequency | Hz |
| \omega | angular frequency, 2\pi f | rad s^{-1} |
| c_j | sound speed in medium j | m s^{-1} |
| k_j | acoustic wavenumber in medium j, \omega/c_j | m^{-1} |
| k_{L,j} | longitudinal elastic wavenumber in medium j | m^{-1} |
| k_{T,j} | transverse or shear wavenumber in medium j | m^{-1} |
| a | characteristic radius | m |
| L | characteristic length | m |
| ka | acoustic size based on a chosen radius | dimensionless |
Material properties and contrasts
| Symbol | Meaning | Typical units |
|---|---|---|
| \rho_j | density in medium j | kg m^{-3} |
| \kappa_j | compressibility in medium j | Pa^{-1} |
| \lambda_j | Lamé’s first parameter in elastic medium j | Pa |
| \mu_j | shear modulus in elastic medium j | Pa |
| g_{ij} | density contrast, \rho_i / \rho_j | dimensionless |
| h_{ij} | sound-speed contrast, c_i / c_j | dimensionless |
Important examples: body relative to seawater is expressed with g_{21} and h_{21}, a shell relative to seawater is likewise expressed with g_{21} and h_{21} when the shell is the first interior region, an internal fluid relative to the shell is expressed with g_{32} and h_{32}, and an internal fluid relative directly to seawater is expressed with g_{31} and h_{31}.
Scattering quantities
| Symbol | Meaning | Typical units |
|---|---|---|
| f(\theta_s,\phi_s \mid \theta_i,\phi_i) | far-field scattering amplitude | m |
| f_{\mathrm{bs}} | backscattering amplitude | m |
| \sigma_{\mathrm{bs}} | backscattering cross-section, |f_{\mathrm{bs}}|^2 | m^2 |
| \mathrm{TS} | target strength, 10 \log_{10}(\sigma_{\mathrm{bs}}) | dB |
Angles and directions
| Symbol | Meaning | Typical units |
|---|---|---|
| \theta_i, \phi_i | incident polar and azimuthal angles | rad |
| \theta_s, \phi_s | scattered or receive polar and azimuthal angles | rad |
| \theta | body or target orientation angle when a page uses a 2D axisymmetric reduction | rad |
| \beta | local body-tilt angle along a segmented or curved centerline | rad |
Coordinate-specific notation
Some theory pages use geometry-matched coordinates that introduce additional symbols:
| Geometry | Symbols | Meaning |
|---|---|---|
| sphere | (r,\theta,\phi) | ordinary spherical coordinates |
| cylinder | (r,\phi,z) | ordinary cylindrical coordinates |
| prolate spheroid | (\xi,\eta,\phi) | prolate spheroidal coordinates |
| oblate spheroid | (\xi,\eta,\phi) or r(\theta) | oblate geometry in a spheroidal or spherical representation |
When a page uses scale factors such as h_\xi, h_\eta, or h_\phi, those are coordinate metric coefficients, not sound-speed contrasts. The contrast notation always keeps the medium subscripts explicitly: h_{21}, h_{32}, and so on.