Quantities

What is computed is selected by a quantity, which wraps the locations of observation:

F = field(ex, quantity)                # incident field
F = scatteredfield(sp, ex, quantity)   # scattered field
F = field(sp, ex, quantity)            # total field

The locations are Cartesian points, e.g., SVectors, in an array of any shape, see Defining Observation Points. The result has the same shape: per location, a vector quantity yields an SVector{3} of its complex Cartesian components, a scalar quantity a complex number.

Overview

Which scatterers are available for which excitation is summarized in the Feature Overview. The radar cross section is derived from the far field and has a function of its own.

Fields

The fields are evaluated at the given locations. Inside a scatterer, the total field vanishes where the scatterer is impenetrable, that is, inside a PEC, a sound-hard, or a sound-soft scatterer and inside a PEC core; inside a penetrable scatterer, such as a dielectric sphere, the field inside is returned.

The ScalarPotentialJump is the difference $\Phi_\mathrm{i} - \Phi_\mathrm{e}$ of the potential on the inner and on the outer side of the thin impedance layer.

Far Fields

The far field depends on the direction of the locations only, their distance from the origin being irrelevant. The factor $\mathrm{e}^{-\mathrm{j} k r} / r$ of the outgoing wave is omitted, that is, for the scattered electric field $\bm e^\mathrm{sc}$

\[\bm e^\mathrm{sc}_\infty(\hat{\bm r}) = \lim_{r \rightarrow \infty} r \, \mathrm{e}^{\mathrm{j} k r} \, \bm e^\mathrm{sc}(\bm r) \,,\]

and likewise for the acoustic pressure.

Traces

The traces are evaluated on the surface of the scatterer: only the angular position of each location is taken into account, its radial coordinate being replaced by that of the surface. Hence the locations may also be given by the points of a faceted surface mesh, which do not lie exactly on the surface.

• PressureTrace is the Dirichlet trace $\gamma_0 p$, the pressure on the surface.
• PressureNormalGradient is the Neumann trace $\gamma_1 p = \hat{\bm n} \cdot \nabla p$. By default, the normal $\hat{\bm n} = \hat{\bm r}$ of a sphere centered in the origin is employed. Normals can be passed instead, e.g., those of a surface mesh, PressureNormalGradient(locations, normals); for a spheroid, PressureNormalGradient(sp, locations) fills in its outward normals.
• PressureJump is the jump $[p] = p|_+ - p|_-$ across an open surface, that is, across a disc. Since the incident field is continuous across the disc, the jump of the total field equals that of the scattered one. For a closed surface an error is thrown.

Accuracy Settings

All functions accept the keyword argument parameter, e.g.,

F = scatteredfield(sp, ex, quantity; parameter=Parameter(-1, 1e-8))   # nmax, relativeAccuracy

where the relativeAccuracy determines when the evaluated series are truncated, see Parameter.