General Usage

Every computation combines three building blocks: an excitation, a scatterer, and a quantity to be evaluated at given locations. They are introduced by two simple examples, one for each physics; more details are provided afterwards.

Introductory Examples

Electromagnetic: Plane Wave and PEC Sphere

using SphericalScattering, StaticArrays

# define excitation: plane wave travelling in positive z-direction with x-polarization
ex = planeWave(frequency=10e6) # Hz

# define scatterer: PEC sphere
sp = PECSphere(radius = 1.0)

# define an observation point
point_cart = [SVector(2.0, 2.0, 3.2)]

# compute scattered fields
E  = scatteredfield(sp, ex, ElectricField(point_cart))
H  = scatteredfield(sp, ex, MagneticField(point_cart))
FF = scatteredfield(sp, ex, FarField(point_cart))

Acoustic: Plane Wave and Sound-Hard Sphere

# define the medium: air, with compressibility 1 / (ρc²) and mass density ρ
air = Medium(1 / (1.2 * 343.0^2), 1.2)

# define excitation: plane wave travelling in positive z-direction
ex = Acoustic.planeWave(frequency=1e3, embedding=air) # Hz

# define scatterer: sound-hard sphere
sp = HardSphere(radius = 0.1)

# define an observation point
point_cart = [SVector(0.2, 0.2, 0.3)]

# compute scattered pressure
p  = scatteredfield(sp, ex, Pressure(point_cart))
FF = scatteredfield(sp, ex, FarField(point_cart))

Defining Observation Points

In order to define points the StaticArrays package has to be used.

using StaticArrays

# defining a single point
point_cart = [SVector(2.0, 2.0, 3.2)]

# defining multiple points (along a line)
point_cart = [SVector(5.0, 5.0, z) for z in -2:0.2:2]

Defining an Excitation

For all available excitations a simple constructor with keyword arguments and default values is available:

The acoustic constructors are collected in the Acoustic submodule, so that their names do not clash with the electromagnetic ones.

Defining a Scatterer

For all available scatterers a constructor with keyword arguments is available, e.g.,

sp = PECSphere(radius=1.0)                                        # electromagnetic
sp = DielectricSphere(radius=1.0, filling=Medium(4ε0, μ0))

sp = SoftSphere(radius=1.0)                                       # acoustic
sp = Spheroid{SoundHard}(equatorialRadius=1.0, polarRadius=0.5)   # an oblate spheroid
sp = Disc(SoundSoft; radius=1.0)

The physics of the scatterer has to match that of the excitation. An overview of all constructors and of the type hierarchy behind them is given in Scatterers and Boundary Conditions.

Computing Fields

The incident, scattered, and total fields are computed by

F = field(ex, quantity)                # incident field: the excitation alone
F = scatteredfield(sp, ex, quantity)   # field scattered by the scatterer
F = field(sp, ex, quantity)            # total field: incident plus scattered

where the quantity wraps the locations, e.g.,

E  = field(sp, ex, ElectricField(point_cart))             # electromagnetic
H  = field(sp, ex, MagneticField(point_cart))
Φ  = field(sp, ex, ScalarPotential(point_cart))           # uniform static field
FF = scatteredfield(sp, ex, FarField(point_cart))

p  = field(sp, ex, Pressure(point_cart))                  # acoustic
γ₀ = field(sp, ex, PressureTrace(point_cart))             # on the surface
γ₁ = field(sp, ex, PressureNormalGradient(point_cart))    # on the surface
FF = scatteredfield(sp, ex, FarField(point_cart))

Which quantities are available for which excitation, and how the traces and far fields are defined, is described in Quantities. All functions accept the keyword argument parameter, which controls the truncation of the series, see Accuracy Settings.

Radar Cross Section

For an electromagnetic plane wave, the bistatic and the monostatic radar cross section are computed by

σ = rcs(sp, ex, points_cart)   # bistatic

σ = rcs(sp, ex)                # monostatic

Conversion Between Bases

Methods are provided to convert between Cartesian and spherical coordinates, whose convention is given under Spherical Coordinates:

point_cart = SphericalScattering.sph2cart.(point_sph)

point_sph  = SphericalScattering.cart2sph.(point_cart)

Converting fields:

F_cart = SphericalScattering.convertSpherical2Cartesian.(F_sph,  point_sph)

F_sph  = SphericalScattering.convertCartesian2Spherical.(F_cart, point_sph)

Plotting Fields

To visualize far-fields and near-fields the functions

patternPoints(;r=1.0, resolution=5)

phiCutPoints(ϕ; r=1.0, resolution=1)

thetaCutPoints(ϑ; r=1.0, resolution=1)

as well as the functions

plotff(F, points_sph; scale="log", normalize=true, type="abs")

plotffcut(F, points; scale="log", normalize=true, format="polar")

are provided (after loading the PlotlyJS package). For more details see the visualization of fields examples.