acoustotreams.special.ssw_psi

acoustotreams.special.ssw_psi(l, m, x, y, z, theta, phi, k)

Far-field amplitude of singular scalar spherical wave \(\psi\)

It is defined as

\[\psi_{lm}(x, y, z, \theta, \varphi) = \frac{\mathrm{(-i)}^{l+1}}{k} Y_{lm}(\theta, \varphi) \mathrm{e}^{-\mathrm{i} k \left(x \sin \theta \cos \varphi + y \sin \theta \sin \varphi + z \cos \theta\right)}\]

with acoustotreams.special.sph_harm().

This function describes a part of the solution to the vector Helmholtz wave equation in spherical coordinates that does not depend on \(r\) in the limit of \(kr \to +\infty\).

Parameters:

• l (int or array_like) – Degree \(l \geq 0\).

• m (int or array_like) – Order \(|m| \leq l\).

• x (float or complex or array_like) – X-coordinate of the source.

• y (float or complex or array_like) – Y-coordinate of the source. Has units of x.

• z (float or complex or array_like) – Z-coordinate of the source. Has units of x.

• theta (float or complex or array_like) – Polar angle (rad).

• phi (float or array_like) – Azimuthal angle (rad).

• k (float or complex or array_like) – Angular wavenumber in the medium. Has units of 1/x.

Returns:

complex