---
title: "Scattering boundary conditions"
output:
  rmarkdown::html_vignette:
    toc: true
    number_sections: true
    fig_caption: yes
    fig_width: 3.5
    fig_height: 3.5
    dpi: 200
    dev.args: list(pointsize=11)
bibliography: REFERENCES.bib
link-citations: true
reference-section-title: References
vignette: >
  %\VignetteIndexEntry{Scattering boundary conditions}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
---

```{r article-links, include=FALSE}
article_root <- paste0(
  "https://brandynlucca.github.io/acousticTS/",
  "articles/"
)
notation_url <- paste0(
  article_root,
  "notation-and-symbols/notation-and-symbols.html"
)
primer_url <- paste0(
  article_root,
  "acoustic-scattering-primer/acoustic-scattering-primer.html"
)
```

# Introduction

Boundary conditions turn the governing wave equations into a particular
scattering problem. They state whether an interface can sustain pressure,
whether it can move, and which field quantities pass continuously between
adjacent media. A fixed-rigid surface, a pressure-release surface, and a
penetrable fluid target can therefore scatter very differently even when they
have the same geometry.

The conditions must act on the **total** field at an exterior boundary. The
incident and scattered fields are separated to solve the exterior problem, but
neither field generally satisfies the boundary condition alone [@Colton_2013].

::: {.note data-title="Notation"}
See [Notation and Symbols][n] and the
[Acoustic Scattering Primer][p].

[n]:
  `r notation_url`
[p]:
  `r primer_url`
:::

# Shared interface notation

Medium `1` is the exterior fluid. Target regions are numbered inward. For an
unlayered target, $\Gamma$ denotes the interface between media `1` and `2`.
For a shell with outer radius $a$ and inner radius $b$, the outer and inner
interfaces are $\Gamma_a$ and $\Gamma_b$.

The exterior pressure is decomposed as:

$$
  p_1^{\mathrm{tot}}
  =p_1^{\mathrm{inc}}+p_1^{\mathrm{scat}}.
$$

At a fluid-fluid interface, $\mathbf{n}$ is one unit normal used on both sides
of the interface. Its orientation does not change the continuity conditions as
long as it is used consistently. The normal derivative is:

$$
  \partial_n p
  =\frac{\partial p}{\partial n}
  =\mathbf{n}\mathbin{\cdot}\nabla p.
$$

With the $e^{-i\omega t}$ convention, the linearized momentum equation and its
phasor form are [@Morse_1968]:

$$
  \rho_j\frac{\partial\mathbf{v}_j}{\partial t}
  =-\nabla p_j,
  \qquad
  -i\omega\rho_j\mathbf{v}_j=-\nabla p_j.
$$

The fluid particle velocity is therefore:

$$
  \mathbf{v}_j
  =\frac{1}{i\omega\rho_j}\nabla p_j,
  \qquad
  v_{n,j}
  =\frac{1}{i\omega\rho_j}\partial_n p_j.
$$

Each homogeneous fluid pressure satisfies the Helmholtz equation:

$$
  \nabla^2p_j+k_j^2p_j=0,
  \qquad
  k_j=\frac{\omega}{c_j}.
$$

The directional density and sound-speed ratios used for adjacent media are:

$$
  g_{ij}=\frac{\rho_i}{\rho_j},
  \qquad
  h_{ij}=\frac{c_i}{c_j}.
$$

Thus, $g_{21}$ and $h_{21}$ compare the first target region with the exterior.
For a shelled target, $g_{32}$ and $h_{32}$ compare the core with the shell.

# Simple boundaries

## Fixed rigid

A fixed-rigid boundary cannot move in its normal direction. The normal fluid
velocity at the surface must therefore vanish:

$$
  v_{n,1}=\mathbf{v}_1\mathbin{\cdot}\mathbf{n}=0
  \qquad\text{on }\Gamma.
$$

The same limit can be expressed through the specific surface impedance:

$$
  Z_s=\frac{p_1^{\mathrm{tot}}}{v_{n,1}},
  \qquad
  |Z_s|\longrightarrow\infty.
$$

Substitution of the phasor momentum relation gives the Neumann boundary
condition:

$$
  \partial_n p_1^{\mathrm{tot}}=0
  \qquad\text{on }\Gamma.
$$

In terms of the incident and scattered fields, the prescribed data for the
scattered field are:

$$
  \partial_n p_1^{\mathrm{scat}}
  =-\partial_n p_1^{\mathrm{inc}}
  \qquad\text{on }\Gamma.
$$

The pressure itself need not vanish. Instead, the reflected field cancels the
incident normal velocity at the surface. In partial-wave formulations, the
Neumann condition fixes each scattering coefficient by excluding any total
field that would produce radial surface motion. At high frequency, a smooth
rigid surface supports a locally specular reflection without the pressure
phase reversal of a pressure-release surface.

## Pressure release

A pressure-release surface cannot sustain an acoustic pressure fluctuation.
It is the zero-impedance limit:

$$
  Z_s=\frac{p_1^{\mathrm{tot}}}{v_{n,1}},
  \qquad
  |Z_s|\longrightarrow0.
$$

For finite normal velocity, the total pressure obeys the Dirichlet condition:

$$
  p_1^{\mathrm{tot}}=0
  \qquad\text{on }\Gamma.
$$

The corresponding condition on the scattered field is:

$$
  p_1^{\mathrm{scat}}=-p_1^{\mathrm{inc}}
  \qquad\text{on }\Gamma.
$$

Normal motion is not constrained to vanish. The idealization is appropriate
when the material on the other side has negligible acoustic impedance relative
to the exterior fluid, as for an idealized gas boundary away from effects that
require an explicit interior solution. A locally reflected pressure wave
undergoes a phase reversal. In a modal solution, zero surface pressure replaces
the zero-normal-velocity condition used for a rigid target.

## Fluid-filled

A penetrable fluid target supports acoustic pressure on both sides of its
boundary. The exterior total field and interior field satisfy [@Anderson_1950]:

$$
  \nabla^2p_1^{\mathrm{tot}}+k_1^2p_1^{\mathrm{tot}}=0,
  \qquad
  \nabla^2p_2+k_2^2p_2=0.
$$

An inviscid fluid has Cauchy stress
$\boldsymbol{\sigma}_j=-p_j\mathbf{I}$. Continuity of normal traction therefore
requires pressure continuity:

$$
  p_1^{\mathrm{tot}}=p_2
  \qquad\text{on }\Gamma.
$$

The interface also admits neither separation nor interpenetration, so normal
particle velocity is continuous:

$$
  \mathbf{v}_1\mathbin{\cdot}\mathbf{n}
  =\mathbf{v}_2\mathbin{\cdot}\mathbf{n}
  \qquad\text{on }\Gamma.
$$

Using the momentum equation gives the derivative form of the second
transmission condition:

$$
  \frac{1}{\rho_1}\partial_n p_1^{\mathrm{tot}}
  =\frac{1}{\rho_2}\partial_n p_2
  \qquad\text{on }\Gamma.
$$

Pressure and normal motion are both generally nonzero. Their continuity couples
the exterior and interior solutions, while density and compressibility
contrasts determine the strength and phase of the reflected and transmitted
fields [@Medwin_1998]. Tangential velocity need not be continuous across an
ideal inviscid fluid-fluid interface because neither fluid transmits shear
traction.

# Acoustic shelled boundaries

<span id="inelastic-shelled-boundaries"></span>

An acoustic shell is a finite fluid layer, not an immobile shell and not an
elastic solid. Medium `2` occupies the region between $\Gamma_a$ and
$\Gamma_b$. It supports compressional pressure waves but no shear stress. The
shell field satisfies:

$$
  \nabla^2p_2+k_2^2p_2=0
  \qquad\text{between }\Gamma_a\text{ and }\Gamma_b.
$$

At the outer interface, pressure and normal velocity are continuous:

$$
  p_1^{\mathrm{tot}}=p_2,
  \qquad
  \frac{1}{\rho_1}\partial_n p_1^{\mathrm{tot}}
  =\frac{1}{\rho_2}\partial_n p_2
  \qquad\text{on }\Gamma_a.
$$

For a spherical shell, both independent radial solutions are admissible because
the layer does not include the origin. An axisymmetric shell field can be
expanded as:

$$
  p_2(r,\theta)
  =\sum_{m=0}^{\infty}
  \left[
    B_m\,j_m(k_2r)+C_m\,y_m(k_2r)
  \right]P_m(\cos\theta),
  \qquad b<r<a.
$$

Here, $j_m$ and $y_m$ are spherical Bessel functions of the first and second
kinds. The singularity of $y_m(k_2r)$ at the origin is irrelevant because
$r=0$ is outside the shell layer. The condition at $\Gamma_b$ determines the
relative weights of these two branches.

## Pressure release interior

For a pressure-release core, the shell pressure vanishes at the inner
interface:

$$
  p_2=0
  \qquad\text{on }\Gamma_b.
$$

In a spherical modal solution, a radial combination that satisfies this inner
condition identically is:

$$
  R_m^{\mathrm{pr}}(r)
  =j_m(k_2r)y_m(k_2b)-y_m(k_2r)j_m(k_2b).
$$

Its value at the inner radius is:

$$
  R_m^{\mathrm{pr}}(b)=0.
$$

The outer pressure and velocity conditions then couple this admissible shell
field to $p_1^{\mathrm{tot}}$. The shell can carry normal acoustic motion even
though the pressure is zero at its inner boundary. Its finite thickness and
wavenumber add propagation phase between the two interfaces. The resulting
response can contain cavity-like resonances that are absent from a single
pressure-release surface.

## Fluid-filled interior

If medium `3` is a fluid or gas core, it supports its own Helmholtz field:

$$
  \nabla^2p_3+k_3^2p_3=0
  \qquad\text{inside }\Gamma_b.
$$

Pressure and normal velocity are continuous at the inner interface:

$$
  p_2=p_3,
  \qquad
  \frac{1}{\rho_2}\partial_n p_2
  =\frac{1}{\rho_3}\partial_n p_3
  \qquad\text{on }\Gamma_b.
$$

For a spherical core containing the origin, regularity excludes the spherical
Bessel function of the second kind. The core expansion is therefore:

$$
  p_3(r,\theta)
  =\sum_{m=0}^{\infty}D_m\,j_m(k_3r)P_m(\cos\theta),
  \qquad 0\leq r<b.
$$

Together, the two conditions at $\Gamma_a$ and the two at $\Gamma_b$ determine
the exterior scattering field, the two shell branches, and the regular core
field for each mode. The shell is not assigned infinite impedance, and its
normal velocity is not set to zero. Resonance and interference arise from
compressional propagation through the shell and core rather than from elastic
shear or bending waves.

# Elastic shelled boundaries

## Elastic shell boundary

An elastic shell supports longitudinal and transverse motion. Its displacement
$\mathbf{u}(\mathbf{x},t)$ follows from balance of linear momentum and Hooke's
law. For a homogeneous isotropic shell, the time-domain Navier equation is
[@Achenbach_1973]:

$$
  (\lambda+2\mu)\nabla(\nabla\mathbin{\cdot}\mathbf{u})
  -\mu\nabla\mathbin{\times}(\nabla\mathbin{\times}\mathbf{u})
  -\rho_s\frac{\partial^2\mathbf{u}}{\partial t^2}
  =0.
$$

For a displacement phasor with $e^{-i\omega t}$ dependence, this becomes:

$$
  (\lambda+2\mu)\nabla(\nabla\mathbin{\cdot}\mathbf{u})
  -\mu\nabla\mathbin{\times}(\nabla\mathbin{\times}\mathbf{u})
  +\rho_s\omega^2\mathbf{u}
  =0.
$$

The Helmholtz decomposition separates dilatational and equivoluminal motion:

$$
  \mathbf{u}=\nabla\Phi+\nabla\mathbin{\times}\mathbf{\Psi},
  \qquad
  \nabla\mathbin{\cdot}\mathbf{\Psi}=0.
$$

The scalar and vector potentials satisfy separate Helmholtz equations:

$$
  \nabla^2\Phi+k_L^2\Phi=0,
  \qquad
  \nabla^2\mathbf{\Psi}+k_T^2\mathbf{\Psi}=0.
$$

Their wavenumbers and wave speeds are:

$$
  k_L=\frac{\omega}{c_L},
  \qquad
  k_T=\frac{\omega}{c_T},
  \qquad
  c_L=\sqrt{\frac{\lambda+2\mu}{\rho_s}},
  \qquad
  c_T=\sqrt{\frac{\mu}{\rho_s}}.
$$

The infinitesimal strain and Cauchy stress tensors are:

$$
  \boldsymbol{\varepsilon}
  =\frac{1}{2}\left[
    \nabla\mathbf{u}+(\nabla\mathbf{u})^{\mathsf T}
  \right],
  \qquad
  \boldsymbol{\sigma}
  =\lambda(\nabla\mathbin{\cdot}\mathbf{u})\mathbf{I}
  +2\mu\boldsymbol{\varepsilon}.
$$

At either fluid-solid interface, let $\mathbf{n}_s$ point from the solid shell
into the adjacent fluid. The fluid stress is
$\boldsymbol{\sigma}_f=-p_f\mathbf{I}$. Continuity of traction is then the
vector condition:

$$
  \boldsymbol{\sigma}\mathbf{n}_s
  =-p_f\mathbf{n}_s.
$$

Separating that relation into normal and tangential parts gives:

$$
  \mathbf{n}_s\mathbin{\cdot}
  \boldsymbol{\sigma}\mathbf{n}_s=-p_f,
  \qquad
  (\mathbf{I}-\mathbf{n}_s\mathbf{n}_s^{\mathsf T})
  \boldsymbol{\sigma}\mathbf{n}_s=\mathbf{0}.
$$

The second condition states that an inviscid fluid applies no tangential
traction. It does not require the tangential solid displacement to vanish.
Continuity of normal velocity supplies the remaining kinematic condition:

$$
  \mathbf{v}_f\mathbin{\cdot}\mathbf{n}_s
  =-i\omega\mathbf{u}\mathbin{\cdot}\mathbf{n}_s.
$$

Written in terms of fluid pressure, the same condition is:

$$
  \frac{1}{i\omega\rho_f}\partial_{n_s}p_f
  =-i\omega u_n,
  \qquad
  u_n=\mathbf{u}\mathbin{\cdot}\mathbf{n}_s.
$$

At the outer shell surface, $p_f=p_1^{\mathrm{tot}}$. At the inner surface,
$p_f=p_3$. The normal $\mathbf{n}_s$ points radially outward at $r=a$ and
radially inward at $r=b$. Stating the normal this way removes any ambiguity in
the pressure-traction sign.

For a solid elastic target with no core, these conditions are imposed only at
the exterior surface. Regularity at the origin excludes singular longitudinal
and transverse radial solutions. The resulting fluid-solid coupling is the
starting point for classical elastic-sphere and elastic-cylinder scattering
theory [@Faran_1951].

For an axisymmetric shell, normal traction, normal velocity, and one tangential
traction condition are imposed at each radius. These six scalar conditions
couple the exterior acoustic field, both elastic potentials in the shell, and
the interior acoustic field [@Goodman_1962]. They admit resonances governed by
shell thickness, density, longitudinal and transverse wave speeds, and fluid
loading. This is the essential distinction from an acoustic shell, which has
no transverse potential and cannot support shear or bending motion
[@Stanton_1990].

# Viscous interfaces

A compressible Newtonian viscous fluid supports deviatoric stress as well as
pressure. If $\eta$ is the shear viscosity, $\zeta$ is the bulk viscosity, and
$\mathbf{D}$ is the rate-of-deformation tensor, its Cauchy stress is
[@Pierce_1989]:

$$
  \boldsymbol{\sigma}_v
  =-p_v\mathbf{I}
  +2\eta\mathbf{D}
  +\left(\zeta-\frac{2}{3}\eta\right)
  (\nabla\mathbin{\cdot}\mathbf{v}_v)\mathbf{I}.
$$

The rate-of-deformation tensor is:

$$
  \mathbf{D}
  =\frac{1}{2}\left[
    \nabla\mathbf{v}_v+(\nabla\mathbf{v}_v)^{\mathsf T}
  \right].
$$

At an interface with an inviscid acoustic fluid, let $\mathbf{n}$ point from
the viscous medium into the inviscid fluid. Normal traction and normal velocity
are continuous:

$$
  \mathbf{n}\mathbin{\cdot}\boldsymbol{\sigma}_v\mathbf{n}
  =-p_f,
  \qquad
  \mathbf{v}_v\mathbin{\cdot}\mathbf{n}
  =\mathbf{v}_f\mathbin{\cdot}\mathbf{n}.
$$

Because the inviscid fluid cannot apply shear traction, the tangential traction
on the viscous side vanishes:

$$
  (\mathbf{I}-\mathbf{n}\mathbf{n}^{\mathsf T})
  \boldsymbol{\sigma}_v\mathbf{n}=\mathbf{0}.
$$

At a bonded viscous-fluid and elastic-solid interface, both velocity and
traction are continuous. With $\mathbf{n}$ used consistently on both sides,
the phasor conditions are:

$$
  \mathbf{v}_v=-i\omega\mathbf{u}_s,
  \qquad
  \boldsymbol{\sigma}_v\mathbf{n}
  =\boldsymbol{\sigma}_s\mathbf{n}.
$$

These vector conditions couple normal and tangential motion. Viscous
compressional and shear branches are generally attenuating, so the interface
can broaden and damp resonances that would be sharper in a lossless elastic
shell [@Feuillade_1998].

# Circumferential waves and resonance families

Boundary conditions do not introduce circumferential waves as separate terms.
Instead, they determine which wave families the complete field solution can
support and how those waves couple to the exterior fluid. Geometry, frequency,
material properties, and attenuation then control their phase, propagation
distance, and resonance structure [@Uberall_1973].

| Phenomenon | Boundary setting | Physical interpretation |
|---|---|---|
| Franz or creeping waves | Smooth convex rigid or pressure-release boundaries | Exterior diffracted waves that follow the surface into the geometrical shadow while continually radiating into the surrounding fluid. |
| Rayleigh-type waves | Fluid-loaded elastic solids and sufficiently thick elastic shells | Surface-localized elastic motion formed from coupled longitudinal and transverse fields. |
| Lamb-type waves | Thin or moderately thin elastic shells | Dispersive guided shell waves whose motion depends on the conditions at both shell surfaces. |
| Whispering-gallery waves | Penetrable fluid or elastic targets | High-order internal or surface-guided waves retained near the boundary by repeated refraction or reflection. |
| Glory | Spherical or nearly axisymmetric targets | A directional enhancement produced when circumferential contributions converge coherently near a symmetry direction. It is an interference or focusing effect, not a separate wave family. |

These labels describe identifiable contributions within an exact modal solution.
They are not additional boundary conditions. For example, the longitudinal and
transverse potentials of an elastic target already contain its surface and
guided-wave contributions once the traction and velocity conditions have been
enforced. Separating those contributions generally requires modal phase,
resonance-pole, time-domain, or asymptotic analysis. Surface-elastic-wave
interpretations of elastic-sphere and elastic-cylinder backscatter provide a
classical example [@Marston_1988].

# Weak, fluid-like

Weak fluid-like scattering is an approximation regime, not a replacement for
the fluid-fluid transmission conditions. For spatially varying density and
compressibility, the source-free pressure equation can be written as:

$$
  \nabla\mathbin{\cdot}
  \left(\frac{1}{\rho}\nabla p\right)
  +\omega^2\kappa p=0,
  \qquad
  \kappa=\frac{1}{\rho c^2}.
$$

When medium `2` differs only slightly from medium `1`, its total internal field
may be replaced at first order by the incident field:

$$
  p_2(\mathbf{x})\approx p_1^{\mathrm{inc}}(\mathbf{x}).
$$

The density and sound-speed ratios must remain close to unity:

$$
  g_{21}=\frac{\rho_2}{\rho_1}\approx1,
  \qquad
  h_{21}=\frac{c_2}{c_1}\approx1.
$$

The corresponding density and compressibility perturbations are:

$$
  \gamma_\rho
  =\frac{\rho_2-\rho_1}{\rho_2}
  =1-\frac{1}{g_{21}},
  \qquad
  \gamma_\kappa
  =\frac{\kappa_2-\kappa_1}{\kappa_1}
  =\frac{1}{g_{21}h_{21}^2}-1.
$$

The weak-contrast assumption requires:

$$
  |\gamma_\rho|\ll1,
  \qquad
  |\gamma_\kappa|\ll1.
$$

The scattered field is then the first-order response to the residual density
and compressibility contrasts [@Chu_1999]. Strong impedance jumps, internal
multiple scattering, and resonant behavior violate this approximation even
though the exact fluid-fluid boundary conditions remain valid.