Notation and Symbols

Introduction

This is the landing page for the Physics Primer. It defines the symbols, units, medium numbering, and scattering-angle notation used by the remaining theory pages. A model-specific page may introduce additional symbols, but it should state explicitly whenever it departs from the notation below. Equations and physical interpretation remain on the topical pages rather than being repeated here.

The suggested reading order is:

Vectors, positions, and directions

Symbol	Meaning	Typical units
\mathbf{x}	field-point position	m
r=\lVert\mathbf{x}\rVert	distance from the target origin	m
\widehat{\mathbf{x}}=\mathbf{x}/r	observation-direction unit vector	dimensionless
\widehat{\mathbf{k}}_i	incident propagation-direction unit vector	dimensionless
\widehat{\mathbf{s}}	scattered or receive-direction unit vector	dimensionless
\mathbf{n}	unit normal, with orientation stated on the relevant interface	dimensionless
\Omega_i=(\theta_i,\phi_i)	incident direction	rad
\Omega_s=(\theta_s,\phi_s)	scattered or receive direction	rad
\psi=\cos^{-1}(\widehat{\mathbf{s}}\!\cdot\!\widehat{\mathbf{k}}_i)	scattering angle: 0 forward and \pi backward	rad

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.

Subscripts i and j denote arbitrary media. A subscript 1 always denotes the exterior medium in the Physics Primer and model-family theory pages.

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}
\rho'	acoustic density perturbation	kg m^{-3}
\sigma_{ij}	stress-tensor components	Pa
\mathbf{t}=\boldsymbol{\sigma}\mathbf{n}	traction acting on a surface with normal \mathbf{n}	Pa
\partial_n=\mathbf{n}\cdot\nabla	normal derivative	m^{-1} acting on the differentiated field

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}
\alpha_j	amplitude-attenuation coefficient	Np m^{-1}
\widetilde{k}_j	complex acoustic wavenumber	m^{-1}
\lambda_a=2\pi/k	acoustic wavelength in a lossless fluid	m
a	characteristic radius	m
L	characteristic length	m
ka	acoustic size based on a chosen radius	dimensionless

The wavelength is written \lambda_a here to distinguish it from Lamé’s first elastic parameter \lambda. A page using complex sound speed rather than complex wavenumber must state its sign convention explicitly.

Acoustic material properties

Symbol	Meaning	Typical units
\rho_j	equilibrium mass density in medium j	kg m^{-3}
c_j	compressional sound speed in medium j	m s^{-1}
\kappa_j	adiabatic compressibility of a fluid	Pa^{-1}
K_j	bulk modulus	Pa
Z_j	plane-wave acoustic impedance	Pa s m^{-1}
\eta_j	dynamic or shear viscosity	Pa s
\zeta_j	bulk viscosity	Pa s

Compressibility \kappa and bulk modulus K are distinct symbols.

Elastic material properties

Symbol	Meaning	Typical units
E_j	Young’s modulus	Pa
\nu_j	Poisson ratio	dimensionless
K_j	bulk modulus	Pa
G_j or \mu_j	shear modulus (\mu_j is Lamé’s second parameter)	Pa
\lambda_j	Lamé’s first parameter	Pa
c_{L,j}	longitudinal wave speed	m s^{-1}
c_{T,j}	transverse or shear-wave speed	m s^{-1}

Material contrasts

Symbol	Meaning	Typical units
g_{ij}	density contrast, \rho_i / \rho_j	dimensionless
h_{ij}	sound-speed contrast, c_i / c_j	dimensionless
\gamma_{\rho,ij}	density perturbation, (\rho_i-\rho_j)/\rho_i	dimensionless
\gamma_{\kappa,ij}	compressibility perturbation, (\kappa_i-\kappa_j)/\kappa_j	dimensionless

For example, g_{21} and h_{21} compare the first target region with the exterior medium. The symbols g_{32} and h_{32} compare the next region with the first.

Scattering quantities

Symbol	Meaning	Typical units
p_0	incident plane-wave pressure amplitude	Pa
f(\Omega_s\mid\Omega_i)	far-field scattering amplitude	m
\mathrm{d}\sigma/\mathrm{d}\Omega	differential scattering cross-section	m^2 sr^{-1}
f_{\mathrm{bs}}	backscattering amplitude	m
\sigma_{\mathrm{bs}}	backscattering cross-section	m^2
\sigma_s	total scattering cross-section	m^2
\sigma_a	absorption cross-section	m^2
\sigma_e	extinction cross-section	m^2
\mathrm{TS}	monostatic target strength	dB re 1 m^2
\mathrm{TS}(\Omega_s\mid\Omega_i)	directional or bistatic target-strength notation	dB
F or \mathcal{F}	dimensionless form function	dimensionless

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
\psi	angle between incident propagation and receive directions	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

Incident direction, target orientation, and receive direction are independent quantities. A monostatic geometry fixes the receive direction opposite the incident propagation direction. A bistatic geometry does not.

Basis and special-function notation

Symbol	Meaning
J_m, Y_m, H_m^{(1)}	cylindrical Bessel, Neumann, and outgoing Hankel functions
j_n, y_n, h_n^{(1)}	spherical Bessel, Neumann, and outgoing Hankel functions
P_n	Legendre polynomial
P_n^m	associated Legendre function
S_{mn}	angular spheroidal wave function
R_{mn}^{(q)}	radial spheroidal wave function of kind q
prime '	derivative with respect to the function’s complete argument unless stated otherwise

Normalizations for spheroidal functions and dimensionless form functions are given on the topical page where they are used.

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.