Module vectorwaves 

Vector waves one at a time: the vector spherical harmonics and the spherical, cylindrical and plane vector waves in their local component frames, with pullbacks.

Upstream: treams.special.vsh_X, vsh_Y, vsh_Z, vsw_*, vcw_*, vpw_* and sph_harm.

vector_wave evaluates one wave of a Family at six complex arguments, vector_wave_pullback gives the gradients of all six, and vector_wave_array broadcasts both over arrays of labels and arguments. Spherical waves are in spherical components (r, theta, phi), cylindrical waves in (rho, phi, z) and plane waves in Cartesian components, as in treams.

The inline tests module checks the waves against the Cartesian fields of crate::fields, their pullbacks against finite differences, the small-argument branches and the normalization of the harmonics.

Structs

VectorWaveResidual

What vector_wave_array saves for its pullback: the family, the labels, the convention and the arguments, each one value for all outputs or one per output.

WaveLabel

The integer labels of one vector wave.

Enums

Family

The kind of vector wave vector_wave evaluates.

Constants

PARALLEL 🔒

Vector-wave arrays evaluate in parallel from this many elements.

Functions

angular 🔒

The normalized scalar spherical harmonic Y_lm(theta, phi) and, if tangential and l > 0, the angular functions pi = m Y_lm / sin(theta) and tau = dY_lm / dtheta, both divided by sqrt(l (l + 1)): [Y, pi, tau], or [Y, 0, 0] otherwise.

checked 🔒

cylindrical 🔒

Cylindrical M and N waves at (kz, krr, phi, z, k), or their helicity combinations; krr is k_rho rho, as in upstream vcw_*.

evaluate 🔒

fixed_plane_pullback 🔒

Pullback to (k, r) of a plane wave p exp(i k·r), given the polarization jet p in the wave vector.

fixed_plane_pushforward 🔒

Directional product of a shared polarization and the point-dependent phase.

harmonics 🔒

The scalar harmonic Y and, if tangential, the angular functions pi and tau of label; None when |m| > l.

plane 🔒

Plane wave p exp(i k·r) at (kx, ky, kz, x, y, z), with the chain rule through the polarization p(k).

pullback_impl 🔒

vector_wave_pullback with derivatives in the first N arguments, the ones the family reads; the others get zero.

sph_harm

Scalar spherical harmonic Y_l^m(theta, phi) at the polar angle theta and the azimuth phi.

sph_harm_from_vsh_z

The scalar spherical harmonic Y = -i Z_r of a Family::HarmonicZ wave Z (treams.special.vsh_Z), whose only nonzero component is the radial one, Z_r = i Y.

sph_harm_from_vsh_z_pullback

Pullback of sph_harm_from_vsh_z: the Family::HarmonicZ wave cotangent [i cotangent, 0, 0] for the scalar-harmonic cotangent.

spherical 🔒

Spherical M and N waves at (kr, theta, phi), or their helicity combinations.

tangential 🔒

The tangential X and Y vector harmonics from [Y, pi, tau].

vector_wave

Evaluate a vector wave in its local component frame.

vector_wave_array

vector_wave over arrays of labels and arguments, each one value for all outputs or one per output; the residual keeps only the inputs.

vector_wave_pullback

The gradients of all six complex arguments of vector_wave for a cotangent of the wave. A real physical input takes the real part of its gradient.

vector_wave_pushforward

Directional derivative in the wave’s six complex arguments. One jet slot propagates the supplied direction through the same kernels as the value.